Linear systems of ordinary differential equations: with periodic and quasi-periodic coefficients

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M A T H E MAT I C S I N SCIENCE AND ENGINEERING A S E R I E S OF MONOGRAPHS AND T E X T B O O K S

Edited by Richard Bellman University of Southern California 1.

2. 3. 4.

5. 6.

7. 8. 9.

10. 11.

12. 13.

14. 15. 16.

17. 18. 19. 20. 21.

22.

TRACY Y. THOMAS. Concepts from Tensor Analysis and Differential Geometry. Second Edition. 1965 TRACY Y.THOMAS. Plastic Flow and Fracture in Solids. 1961 RUTHERFORD ARIS. The Optimal Design of Chemical Reactors: A Study in Dynamic Programming. 1961 JOSEPH LASALLEand SOLOMON LEFSCHETZ.Stability by Liapunov’s Direct Method with Applications. 1961 GEORGE LEITMANN (ed.) . Optimization Techniques: With Applications to Aerospace Systems. 1962 RICHARDBELLMANand KENNETHL. COOKE.Differential-Difference Equations. 1963 FRANK A. HAIGHT.Mathematical Theories of Traffic Flow. 1963 F. V. ATKINSON.Discrete and Continuous Boundary Problems. 1964 A. JEFFREY and T. TANIUTI. Non-Linear Wave Propagation: With Applications to Physics and Magnetohydrodynamics. 1964 J U L I U S T. T o u . Optimum Design of Disital Control Systems. 1963 HARLEY FLANDERS. Differential Forms: With Applications to the Physical Sciences. 1963 SANFORD M. ROBERTS. Dynamic Programming in Chemical Engineering and Process Control. 1964 SOLOMON LEFSCHETZ. Stability of Nonlinear Control Systems. 1965 DIMITRIS N. CHORAFAS. Systems and Simulation. 1965 A. A. PERVOZVANSKII. Random Processes in Nonlinear Control Systems. 1965 MARSHALL C. PEASE,111. Methods of Matrix Algebra. 1965 V. E. BENES.Mathematical Theory of Connecting Networks and Telephone Traffic. 1965 WILLIAM F. AMES. Nonlinear Partial Differential Equations in Engineering. 1965 J. A C Z ~ LLectures . on Functbnal Equations and Their Applications. 1966 R. E. MURPHY.Adaptive Processes in Economic Systems. 1965 S. E. DREYFUS.Dynamic Programming and the Calculus of Variations. 1965 A. A. FEL’DBAUM. Optimal Control Systems. 1965

MATHEMATICS I N S C I E N C E A N D E N G I N E E R I N G 23. 24.

25. 26.

27. 28. 29. 30.

A. HALANAY. Differential Equations: Stability, Oscillations, Time Lags. 1966 M. NAMIKOZUZTORELI. Time-Lag Control Systems. 1966 DAVIDSWORDER. Optimal Adaptive Control Systems. 1966 MILTONASH. Optimal Shutdown Control of Nuclear Reactors. 1966 D I M I T R IN. S CHORAFAS. Control Systrm Functions and Programming Approaches. ( I n Two Volumes.) 1966 N. P. ERUGIN. Linear Systems of Ordinary Differential Equations. 1966 SOLOMON MARCUS.Algebraic Linguistics; Analytical Models. 1966 A. M. LIAPUNOV. Stability of Motion. 1966

I n preparcltiorz A. K A U F M A N N Graphs, . Dynamic Programming, and Finite Games MINORU URABE. Nonlinear Autonomous Oscillations A. K A U F M A Nand N R. CRUON.Dynamic Programming: Sequential Scientific Management GEORGELEITMANN (ed.) . Optimization: A Variational Approach Y. SAWAGARI, Y. SUNAHARA, and T . NAKAMIZO. Statistical Decision Theory in Adaptive Control System9 MASUNAO AOKI.Optimization of Stochastic Processes F. CALOGERO. Variable Phase Approach to Potential Scattering 1. H. AHLBERG, E. N. NILSON,and J. L. WALSH.The Theory of Splines and Their Application J. K U S H N E Stochastic ~. Stability and Control HAROLD

COPYRIGHT 0 1966,

B Y ACADEMIC PRESSINC. ALL RIGHTS RESERVED. NO PART OF T H I S BOOK MAY BE REPRODUCED I N A N Y FORM, BY PHOTOSTAT, MICROFILM, OR A N Y OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC.

1 1 1 Fifth Avenue, New York, New York 10003

*

*

United Kinydoin Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W . l

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 66-23935 Originally published as: “Lineynyye Sistemy Obyknovennykh Differentsial’nykh Uravneniy s Periodicheskimi i Kvaziperiodicheskimi Koefitsiyentami” Press of Acad. Sci. BSSR. Minsk, 1963

PRINTED I N T H E UNITED STATES O F AMERICA

Author’s Comments

In 1956 a monograph by the author, The Lappo-Danilevskiy method in the theory of linear differential equations, was published (University of Leningrad Press). It was written* exactly twenty-five years to the day after the death of the outstanding Russian mathematician I. A. Lappo-Danilevskiy in commemoration of that event. It was assumed that this work would be published in the periodical Uspekhi matematicheskikh nauk or in the series Problemy sovremennoy matematiki, which is published under the direction of the preceding periodical. Therefore, the exposition was extremely brief and in many places almost sketchy. The present monograph is a revision of that work. Here, the material is expounded in much greater detail and the content of the book has been considerably supplemented. Therefore, even its name has been changed. The author is aware that even now greater detail might be in order in certain portions of the exposition, for example, in Sections 1, 37, 43, 44, and others. However, the reader can find greater detail on the subject of Section 37 in the work by Lyapunov. Here, we needed this only for the validity and completeness of the solution of problems on the bounded solutions of the equation x + p ( f ) x = O with periodic function p ( f ) of varying sign. We have not dealt with other methods of solving this problem but refer the reader to the survey article by V. M. Starzhinskiy.

.?he monograph was written in 1954. An address on the subject in question was presented A@ 20, 1954, at the Scientific Section of the Physico-mathematical sector of the Academy of Sciences of the Kazakh SSR [66]. vii

Contents

Author's Comments Introduction I. Functions of a Single Matrix 2. Auxiliary Theorems 3. Functions of Several Matrices and of a Countable Set of Matrices 4. Classes of Systems of Linear Differential Equations That . Can Be Integrated in Closed Form Are That Equations Differential Linear of 5. Other Systems Integrable in Closed Form 6; The Construction of Solutions of Certain Linear Systems of Differential Equations in the Form of a Series of Several Matrices (of a Series of Compositions) 7. Solution of the Poincar~-Lappo-Danilevskiy Problem 8. Formulation of Certain Problems of Linear Systems of Differential Equations with Real Periodic Coefficients 9. Solution of the Problems Posed in Section 8 on the Basis of Real Functions 10. Expansion of an Exponential Matrix in a Series of Powers of a Parameter ll. Determination of the Coefficients in the Series Expansion of an Exponential Matrix 12. Approximate Integration of Equation (10.1) 13. The Case in Which Po(t), P1(t), ... , Pm(t) in Equation (10.1) Are Constants 14. The Case in Which Po is Constant and expPo tis a Periodic Matrix in Equation (10.1) 15. An Example Illustrating Section 14 16. 17. 18. 19.

Canonical Systems The System (16.3) With Po = P1 = ... Artem'yev's Problem The Theory of Reducible Systems

v xi 1

22 33 36 41

44 49

56 60

68 75

82 85 89 90

101

= Pm-1 = 0

ix

105

106 109

Introduction

In Section 1the fundamentals of the theory of functions of a single matrix a r e expounded along the lines of Lappo-Danilevskiy. A method based on Lappo-Danilevskiy’s formulas is presented for constructing Lagrange’s minimum polynomial from a matrix A, where this polynomial is a function of the matrix A. We show that the analytic continuation of f ( A ) , which is obtained by using Lagrange’s formula, produces all possible values o f f ( A ) including irregular ones if f ( A ) is a multiple-valued function. We give the general representation of the function Y = In X , we show when this function can possess real values, and we indicate the form of the principal and regular value. In Section 2 we study the problem of expanding a matric function (where the e.

Xk

are matrices) in a series of powers of

Here, we treat both regular and irregular values of f

In Section 3 we treat functions of several matrices, following Lappo-Danilevs kiy. In Sections 4 and 5 we give some classes of systems of differential equations that are integrable in closed form. These classes of equations canbe of significance in the construction of approximate solutions of systems of linear differential equations. In Section 6 we present some general theorems on the series expansion of an integral matrix of a linear system of differential equations with respect to a parameter E that appears in the coefficients of the system. Basically, we follow the procedure of Lyapunov. In Section 7 we give certain results concerning the solution of the Poincard-Lappo-Danilevskiy problem that a r e obtained from the analytic theory of linear systems of differential equations. Specifically, we shall present methods of constructing an exponential matrix W that characterizes the multivaluedness of the integral matrix X (z) = ( z -a)W.N (z -a ) in a neighborhood of a xiii

XiV

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

singular point z = a, where N ( z - a ) is a single-valued matrix in a neighborhood of the point z = a. We present the general representations of W for the case in which z = a is a regular singular point (Lappo-Danilevskiy) and also for the case of an irregular singular point [5, 291. We give LappoDanilevskiy’s expansions of W in a series with respect to small parameters that appear in the coefficients of the system of differential equations. A s mentioned elsewhere, all this is of significance in the theory of linear systems with periodic coefficients. In Section 8 we pose certain problems in the theory of linear systems of differential equations with real periodic coefficients. The problem is that of constructing an integral matrix X (f) in the form (Floquet’s prob1em)X ( t ) = exp At . N ( f ) , where A is a constant matrix and N ( f ) is a periodic matrix. Here, we show when it is possible to have A and N ( t ) real. We also show what the period of the function N ( t ) is and under what conditions A is a regular value of In X (2x), where X (2n) is an integral transformation, that is, a matrix by which we multiply the integral matrix X (t) that is normalized at the point t = O whenwe increase t by the period 2 x of the coefficient matrix. In Section 9 we give the general solution of the problem posed in Section 8. Specifically, we exhibit the procedure for constructing W with the aid of formulas that are found in [5, 291 for the general representation of In X and a r e listed in Section 1. We give an example in which we obtain a real matrix for Wand a matrix N ( t ) with period twice as great as the period of the coefficient matrix of the given system of differential equations. For a system of the form

(where A is a parameter), we present a procedure for finding the matrices Wand N @)in the form of series in powers of the parameter 1. A bound is given for the radius of convergence of these expansions in terms of the maximum absolute values of the elements of the coefficient matrix and for the region of convergence of the series constructed by Lappo-Danilevskiy which represent an exponential transformation (see Section 7). It turns out that, in seeking the invariants of the matrix A (see Section 8), we can use the expansion obtained by Lappo-Danilevskiy for a certain singular exponential matrix W = H. ,We shall present certain inequalities of a general form concerning a 2 X 2 matrix A for which the characteristic numbers possess specified properties. In Sections 10 and 11 we present various methods of constructing matrices A and N (t) (Sections 8) for a system of the

INTRODUCTION

general form

dX df

-=

k=O

in the form of series in terms of the parameter s. In Section 12 we give approximate methods of constructing an integral matrix, These methods a r e based on the fact that, in the representation of the integral matrix

X (t)= exp At - N (t), where

k=O

k =O

we take segments of the series for A and N ( t ) . It turns out that we can sometimes, as a preliminary, represent the given system of differential equations in a form such that, for c = O , we obtain a system that is integrable in closed form and in which A and N (f) a r e easily found. Here, the systems studied in Sections 4 and 5 a r e of significance. In Section 13 we study separately the case in which the first m cl matrices Po (f),P I (t),..., P, (t) in the system studied in Section 10 a r e constants. We also present systems in which the coefficient matrix of the system of differential equations is not a holomorphic function of the small parameter E. In Section 14 we examine the case in which Po is a constant matrix and the matrix exp .Pot is periodic. In Section 15 we give an example of a system of two equations (illustrating Section 14) in which the matrix is Po = 1101 All. We consider various methods of representing the integral matrix X (f) in the form

X (t)= exp At . N (t) including the method of taking 2 x A in the form of an irregular value of In X ( 2 x ) , where X ( 2 x ) is an integral transformation of the matrix X (f). In Section 16 we study canonical systems of linear differential equations dX = x (Po + dt

c 0

k=l

Pk (t)

Ek

mi

LINEAR SYSTEMS

OF ORDINARY DI F F ER E N T I A L EQUATIONS

with periodic matrices Pk (t)= Pk (1 + 2x) and constant matrix Po.. We give the conditions (of N. A. Artem’yev) under which the integral matrix X ( t ) of such a system will be bounded. In Section 17 we examine the system studied in Section 16, this time under the condition that Po = PI = ... = P,-, = 0. In Section 18 we consider Artem’yev’s problem on the conditions for boundedness of the integral matrix of the canonical system

where P(i

+ 2 r , rl, ...*rv,4 = PV, rl, ..., rv,

ZJ

and pl,..., pv, E a r e parameters. Here, the problem concerns the region of values of the parameters rl, ..., rvre, inwhich the integral matrix X ( t ) is bounded. In Section 19 we find that the entire set of real matrices Z ( t ) that, together with Z - I ( t ) , are bounded and that map the system dX dt

5

X P (t), P (t

+2r) = P (f)

into the system

dY = Y B dt

with real canonical matrix B according to the formula X = YZ ( t ) . In Sections 20 and 21 we study a system of the form (20.1)

where the matrices Pk (t) are quasi-periodic: pk(t)

=

’

cP& (k)& k t

.

Here, the ) ::C a r e constant matrices and the pk a r e real numbers. Such systems were first considered by 1. Z. Shtokalo [lo,381. In these two sections we also study Shtokalo’s method of finding the conditions under which the integral matrix X (t)-+O as t -* 03 (Theorem 24.1). In Sections 22-24 Shtokalo’s method is used to obtain certain approximate integral matrices (20.1). Bounds for *thee r r o r in the ‘

xvii

INTRODUCTION

approximations of these solutions are given. We construct approximate solutions and conditions for the asymptotic stability of solutions of certain nonlinear systems of differential equations. W e pose certain problems for nonlinear systems of differential equations in Section 24. A criterion found by A. E. Gel’man [40] is given for the reducibility of a linear system with quasi-periodic coefficients. In Section 25 we obtain other approximate forms of solutions (and under different hypotheses) based on methods of Shtokalo and N. N. Bogolyubov [ 79,801. In Section 26 we look at a problem of B. P. Demidovich for finding the conditions for boundedness, for small values of O , of the integral matrix of the system X P (t), dt where the matrix P(t)is periodic with period -= dx

P ( t ) d t = M as

o

and where

o - + 0.

O

0

In Section 27 we consider in detail a particular problem of Artem’yev and we present a second problem (considered by various authors [49]), another formulation of which leads to Artem’yev’s problem. In Section 28 we show the connection between the Poincar6 Lappo-Danilevskiy problem and the Floquet problem on the representation of an integral matrix of the system dX -= dt

in the form

X P ( t ) , P (t

+ 2 z) = P ( t )

-

X = exp At N (t), a constant matrix and N ( t ) is periodic. We show how

where A is it is possible to solve the Floquet problem in certain cases by using the methods of solving the PoincarB-Lappo-Danilevskiy problem by using certain general considerations pointed out by Lyapunov. Here, certain formulas obtained by Lappo-DanilevsMy a r e simplified. We show that in these questions, the exponential transformation of a special integral matrix obtained by him can be of value. In Section 29 we give some general tests for boundedness or unboundedness and periodicicity of solutions of linear systems of two differential equations with periodic coefficients.

xviii

LINEAR SYSTEMS

OF O R D I N A R Y D I F F E R E N T I A L EQUATIONS

In Section 30 we look at the problem of boundedness and periodicity of solutions of systems of the differential equations studied in Sections 3 and 4. In Section 31 we investigate questions of boundedness and periodicity of solutions of a system of two differential equations and consider an example. With the aid of a singular exponential transformation of Lappo-Danilevskiy, we show the connection between the parameters of the system under which there exists a periodic solution with specified period (equal to o r a multiple of the period of the coefficient matrix). An approximative form of this periodic solution is constructed. In Section 32 we find more conditions of periodicity of solutions of the system considered in Section 3. In Section 33 we study the equation

i 4-p ( f ) x = 0, p ( f

+ 1) = P(t).

(33.1)

Following Lyapunov, we shall study the question of the boundedness of the solutions of this equation. For the case in which there is a one-parameter family of solutions possessing the property that x + O a s t + co, we find the entire set of initial values x(O), x ' ( 0 ) of such solutions. We find the characteristic numbers of solutions of this equation. In Section 34 we establishthe conditions under which all solutions of Eq. (33.1) are bounded and the conditions under which there a r e periodic solutions. We find the set of all initial values of the periodic solutions. In Section 35 we find the regionofvalues of the parameters E , p, and h that appear in the equation

.. x

+P ( f ,

E,

p, h ) x = 0,

corresponding to which there are periodic solutions with periods commensurable with the period of the function P ( f , E, p, .)A Methods of constructing these solutions are given. We show that the periodic solutions of the system of n equations

-dX - X P ( t , E ) , P ( t + 2 z,

E)

=P(t, E)

dt

can be represented in the form of series of positive powers of converge in the same region in which the series

k=O

E

that

INTRODUCTION

XiX

converges. From this, we obtain the region of convergence of the series representing periodic solutions of the system of n equations

where p is defined a s a function of E in such a way that periodic solutions exist. In Section 36 we show that the system of two linear differential equations

with periodic matrix P ( t ) = P ( t -!-0 ) does not have solutions with period incommensurable with the period of w . This shows that all periodic solutions of the equation were found in the preceding sections. We show that this assertion does not hold for a system of n equations when n > 2. We show how one may find for such a system conditions under which it does o r does not have periodic solutions with period incommensurable with the period of the matrix P(t). We consider the question of periodic solutions of linear systems with nonperiodic coefficient matrix. In Section 37, following Lyapunov, we present methods of solving questions on the existence of bounded and periodic solutions of the equation ;c'+ p ( t ) x = 0

with a periodic function p ( t ) of variable sign. In Section 38 we describe a transformation introduced by V. M. Starzhinskiy that maps a system of two linear homogeneous differential equations with periodic coefficient matrix into the equation + p (t)x = 0 where p (1) is a nonnegative periodic function. In Sections 39 and 40 we construct a transformation of a system of two linear differential equations with periodic coefficient matrix that maps this system into a canonical system with periodic coefficient matrix. In Section 41 a remark is made on the transformation of a system of R linear equations into a canonical system. In Section 42 we find necessary and sufficient conditions for the roots of a polynomial with real coefficients to lie on the unit circle. We present a method for showing the existence of roots of this polynomial inside the unit circle. In Section 43 we study the behavior of roots of a polynomial as functions of a -parameter E . appearing in the coefficients of the

x

xx

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

polynomial. We find conditions under which the roots of the poiynomial will lie on the unit circle for small values of E if they lie on the unit circle for E = 0. We give a method for finding the entire region of variation of t on thereal axis that will allow these roots to remain on the unit circle. In Section 44 we consider the system of linear differential equations dX dt

-=

m

Pk ( I )t k ,

x p (t. c), P (f, E ) =

(44.1)

k=O

+

where the matrices P (t 2 n) = P (t). We present methods for finding the conditions under which the integral matrix of the system (44.1) possesses the property that X (t) --0 a s t -*aor that X (t)will undergo bounded oscillations. We shall study the canonical andnoncanonical systems (44.1). These questions are solved on the basis of the integral matrix; that is, their solution will not involve an exponential transformation as was the case in the preceding sections. Therefore, all the series inpowers of 2. with the aid of which all these questions are solved converge in the region in whichthe series in the system (44.1) converges. In Section 45 we use these same methods to answer the question as to when the integral matrix X ( t ) 0 as t -00. Here, we also use the paper of I. S. Arzhanykh [70] on finding conditions that the parameters of the system of differential equations

-

where the matrix P ( t i-2%. P1, pv, 4 = P ( t , Pll P V l 4. must satisfy for the integral matrix X (t)to have bounded oscillations. Here, we assume this system to be canonical. The method of solving this question is the same a s in Sections 44 and 45. In Section 46 we give a second method of solving Artem’yev’s problem. In Section 47 we draw certain conclusions from [32]regarding the theory of implicit functions. Specifically, we study the implicit function y = y (x), defined by the equation ...I

p m (4 Y)

+ Prn+1(x, Y) + ...

a * * *

= 0,

(47.40)

where PR(x, y) a r e kth-degree homogeneous polynomials with real coefficients and the series (47.40) converges in a neighborhood of the point x = 0, y = 0. We find necessary and sufficient conditions

INTRODUCTION

xxi

-

for the existence of real functions y satisfying the equation that approach 0 a s x 0. We shall find all such solutions. We determine the entire region of convergence of series representing such functions. We show that none of functions y = y(x) defined by equation (47.40) have singular points x .= in the region of convergence of the series (47.40) with the property that the function y(x) does not * have a limit a s .r-+x. In Sections 48 and 49, implicit functions x = x(zt,y=y(z) defined by Eqs. (48.4) and (48.5) are studied in detail. Implicit functions x (z:,y (2)defined by the equations

where the functions 0 (x, y, z ) and F ( x , y, z ) a r e not holomorphic in a neighborhood of the point x = y = z = 0, a r e studied, for example, in Sections 2 and 5- of [32]. We do not go into this question here, although these cases a r e related to the content of the present book in the study of differential equations the right-hand members of which satisfy certain relevant hypotheses. In the present book we have not touched on the questions of exact expansions (for example, as in [76-781 ) or questions of the asymptotic behavior of solutions of linear differential equations (as, for example, in [45]).

1. Functions of a Single Matrix

In the present book, we shall use matrix calculus. Therefore, we shall begin by explaining certain facts related to the theory of functions of a matrix.* We assume that matrix algebra and the reduction of matrices to canonical form is already known to the reader. We shall consider only square matrices. A function f ( A ) of a matrix A is said to be analytic if it can be represented in the form of a Taylor series with numerical (scalar) coefficients in a neighborhood of a matrix of the form al, where a is a number and I is the unit matrix (of appropriate dimensions), that is, if it can be represented in the form

k=O

where the ak a r e numbers (possibly complex). The function f ( A ) is called an entire function if the series (1.1) converges for all finite values of the matrix A . If A is a matrix, the expression exp A = eA means, by definition, the sum of the matrix series m

k=O

We note that the series (1.2) converges for every matrix A. whose elements a r e complex numbers. In other words, the function defined by the series (1.2) is an entire function. An analytic function f ( A ) possesses the property that f (SAS-') = Sf ( A ) S-' ,

(1.3)

where S is an arbitrary matrix with nonzero determinant D ( S ) . This follows from [SAS-'IR =SARSS-' ( k = 0, I , 2 ,...1 . *See also [ 1-31.

2

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

We write the characteristic equation of the matrix A of order n (- 1 )" D ( A - A I) = A" $- a, An-'

+ ...+ a,, = 0.

(1.4)

Here, the a, (for k = 1, 2, ..., n ) a r e the basic symmetric functions of the characteriptic numbers A,, ..., i,,of the matrix A. (These functions a r e known up to sign from algebra.) At the same time, the ak a r e polynomials in the elements of the matrix A. We know that the characteristic equation of the matrixSAS-l coincides with (1.4) for an arbitrary matrixS c D (S)#O. We shall call the basic symmetric functions of il,..., A,, the invariants of the matrix A. If the ak and a in the series (1.1) a r e symmetric functions of Al, .... A,,, then f(A)obviously possesses the property (1.3). However, if the ak and adepend on A,, ..., ,,?. but a r e not symmetric functions, then, in general, f ( A ) in formula (1.1) is defined only when we state how the characteristic numbers Al, ..., An are numbered. This numbering may be such that the property (1.3) will be conserved even for nonsymmetric functions ak (A1, ..., ),,). For example, this will be the case if the numbers I,,, ..., A,, are numbered in accordance with some numerical properties that they possess. However, i f they a r e numbered in order of their position in the canonical form of the matrix A=SJS-', their numbering will depend on the choice of the matrix S and the matrix f ( A ) will no longer possess the property (1.3). Lagrange's formula, which makes it possible to represent an analytic function of a matrix / ( A ) in the form of a polynomial in the matrix A , is familiar to us. In the case in which the characteristic numbers kl,..., A,, of the matrix A a r e distinct, this formula takes the form

Under all other assumptions regarding the characteristic numbers Air..., A,, , we can also construct apolynomial of degree not exceeding n- I for f ( A ) ; for example, we obtain such a polynomial by taking the limit in (1.5). However, it is always possible* to construct this polynomial in such a way that its degree will be less by 1 than the sum of the highest orders of the elementary divisors belonging to the different characteristic numbers of the matrix A. *See also Section 6 i n (3hapter IV and Sections 1 and 2 i n Ckapter V of the book by

F. R Gantmakher [3].

FUNCTIONS OF A S I N G L E MATRIX

3

Thus, if the matrix A. has, for example, only one characteristic number and the highest order of the elementary divisor is equal to 2, then it is possible to construct for f(A)a first-degree polynomial in A. We obtain such minimum polynomials P ( A ) as follows. Suppose that we have a matrix of order n

where J is a quasi-diagonal canonical matrix:

and J , (A) is a pth-order Jordan matrix the elements* (Jp('))kt of which a r e determined by the equations (Jp

(i))kk

= k,

(JP

('))k-t-lvk

=

and

(JP(i)Ikl=O,

if

(k--l)(k-f-l)+-O.

If we have a matrix

where the cpR are numerical symmetric functions and P ( A ) is the minimum polynomial for f (A), then, on the basis of (1.3), n-1

But for the analytic function f (J)we have the Lappo-Danilevskiy formula 1, p. 431

\

f ( J ) = [Gp, (f(h))*. . . 9

GPm(f('rn))19

(1.9)

where the pth-order matrix Gp(f (k)) is defined by *By ( E l k l , we mean that element of the matrix E in the krh row and fth column,

4

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L EQUATIONS

o

f' o I! G,

00.)) =

~

f (i.)

f"(h)

21

I!

o ...

0

f (1)...

0

n

. . . . . . . . . . .. (1) ... f (1) (P - 111 I!

f(P-1)

In particular,

P ( J ) = IGp, V'(h)),

Sa.9

Gpm(P(hm))].

(1.10)

Since, on the basis of (1.8), f ( J ) = P ( J ) , we have

( k = 1, 2 ~ * *m). -~

G p ~ ( f ( ~ k= ) ) Gp,(P(hk)) Remark 1.1.

If A,

= A,

(1.11)

and p1 >/ p,, then the equality GP, ( f ( h 1 ) ) = GP, ( P ( h ) )

implies GP, cf (*,I) = Gp, ( P O.,)),

which is obvious from the structure of these matrices. Suppose now that we have an nth-order matrix

with distinct kl, ..., h,,. From Eqs. (1.11) we obtain a system of linear equations for finding the coefficientscp,(&,...,km)ofLagrange's polynomial that appear in the formula (1.7):

( k = 0 , 1, 2,..., pv-

1;

v =

1, 2,..., m).

The determinant A (h,, ...,A,,) of the coefficients for unknown (P,,-~ ,..., rpr, 'po is constructed as follows. The first row is of the form 1, kl, 1;. ..., ).:-I. Below it a r e p1 - 1 rows which a r e obtained successively by differentiating the first row (p, - 1) times. The remaining characteristic numbers ..., hm also form, respectively, p 2 , . . . , pm rows. Thus,

5

FUNCTIONS OF A SINGLE M A T R I X

A (A1,

... , A,)

=

1: ...

1

Al

0

1 2A1.. .

I

. . . . .. 0 0

.

. .

(n - 1) A:2-

.

. .

.. . . . . .

( n - I ) ...(n-p,+

)lAP;:,

It can be shown that

where a is a constant independent of A,, ..., im. 0, and all the coefficients* Consequently, A (Al, ..., J),. )\,)of Lagrange’s polynomial can be found in terms of

A1, ...,

A~

and

/(k)

(A,,)

(4

= 1,

..., m ; k = 0,

’k

(Al, ...,

1 , ..., pv - 1).

On the basis of Remark 1.1, the Lagrange polynomial that we have constructed is valid also for the case in which the matrix A in the expression [ ( A ) is of any order whatever greater than n, but it will have only the characteristic numbers Ll, ... , A, that correspond to the sets of elementary divisors whose orders do not exceed p,. ..., I);, respectively. Thus, we have constructed Lagrange’s polynomial of lowest degree for the matrix function /(A). This construction of Lagrange’s formula is also possible for a function of a matrix A that is of the form m

where the ab(d) are functions of the invariants** of the matrix A, that is, the aR (A) are symmetric functions of the characteristic numbers of the matrix A. This follows from the fact that such a function /(A) possesses the property (1.3). A consequence of Remark 1.1. It follows from Remark 1.1 that Lagrange’s polynomial of degree n 1 for an arbitrary matrix A of order n can be constructed a s follows. Let Al, ..., A,,, denote

-

*If PI = p ~ , then q~().I, ..., ),1 is a symmetric function with respect to A19 and ).*# which is obvious from Eqs. (l.lll). Therefore, if p1 = ... =. ,p thenqk ().I. .... ),.) will also be a symmetric function with respect to hl, ..., A., (If pI=pr IS odd, then, by exchanging ).I and ).* we change the sign both in A ().I, ...,),.) and in the numerator of ‘$‘k ().I.

.-t

Am)),,

*Were, by invariants” i s meant an arbitrary symmetric function

6

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

the distinct characteristic numbers of the matrix A corresponding to (1) (m) (m) the set of elementary divisors of orders pi'', ..., p k , ,..., pl ,..., Pk, , respectively. Consider the canonical matrix J withdistinct characteristic numbers I.,,..., ,X to each of which corresponds a single

+

elementary divisor of order p!" ... f pi:); ...; pim' t...+pi",' , respectively. In accordance with Remark 1.1, the Lagrange polynomial of degree n - 1 constructed for the matrix J according to formula (1.7), in which the denominators of the coefficients q a ( k l . ..., k,) are given by formula (l.l12),will also be suitable for the matrix A. We call the reader's attention to the following fact. In the case in which all the characteristic numbers A,, ..., A,, of the matrix A are distinct, if we write Lagrange's polynomial for f ( A ) in the form

we see that the coefficients (pm (A~, ..., A,,) are symmetric* functions of A,, ..., kn., In many cases, this enables us to express the coefficients q& directly in terms of the invariants of the matrix A , that is, directly in terms of the elements of the matrix A. When we write f ( A ) in this form, we avoid the necassity of evaluating the roots of the nth polynomial, which is important in many cases. We shall later use this phenomenon, in particular, when seeking a solution of a system of linear differential equations with constant coefficients. We denote the elements of the matrix A by a,,, where k is the number of the row and 1 the number of the column containing the element ak,. Sometimes, we shall denote the matrix A by lk.zkl11. In this notation, all the elements of the matrix 1/r11a r e equal to I; IAl is the matrix whose elements a r e equal to the absolute values of the elements of the matrix A. The inequality I A I < B, where B is a matrix with positive elements, means that the absolute values of the elements of the matrix A do not exceed the corresponding elements of the matrix B.

we know [I] that if a complex power series j ( z ) a radius of convergence p, then the series [ ( A ) =

-

=2](1kzk

has

k=0

Ak

, where

k=O

*We see this from the footnote on page 5, when the &t (AI, functions of A,, . . ., A,for multiple characteristic numbers also.

. . .,

A n ) are symmetric

7

FUNCTIONS OF A SINGLE MATRIX

1: I ,

A is a matrix, converges absolutely for IAI < -p

where n

is the order of the matrix A . In general, this series converges for matrices A whose characteristic numbers A,, ..., A, lie in the circle of convergence off (2). If a functionof a matrix A is given by the series (1.1) in a neighborhood of the matrix a1 (or in a neighborhood of the zero matrix if a- Oj, then the values of f ( A ) are obtained for other values of the matrix A by analytic continuation of the : na series of the elements of the matrix A. Lagrange's formula also enables us to carry out this analytic continuation with the aid of the analytic continuation of only n functions of a single variable f (A1), ..., f(AJ. For a second-order matrixA ,Lagrange's formula takes the form

Iff (A) = exp :At, then

exp At =

e x p A,t A1

-e x p Ad - A,

AS

A, exp Alt + A1 exp Aat -Aa A1

If A1 and A, approach the same value A, which is nonsingular for f (A), while remaining on a single Riemann plane of the function f (A),

then (1.14) becomes f(A)=f(A).I--f'(A)-I+f'(A)A.

(1.15)

Let us now write Lagrange's formula in the form (1.13) for a thirdorder matrix A :

where

8

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

In accordance with what was said above, the coefficients of all powers of the matrix A both in formula (1.14) and in formula (1.16) a r e symmetric functions of the characteristic numbers of the matrix A. Let us suppose that the function f ( z ) is multiple-valued. Then, a s we shall immediately show, the limiting value of f (A) a s A A, is determined not only by the value '4 but also by the path along which A-A, in the space of elements of the matrix A. Suppose that ,:A ..., k: a r e the characteristic numbers of the matrix A,,. If L:, ...,:A are all distinct and if the f(AOk) a r e finite for k = 1, ..., n, then the limiting value of /(A,,) will be finite, as can be seen from Lagrange's formula. If some of the numbers ,:A ..., :A a r e equal but situated on a single Riemann plane of the function f (z) and if the derivatives of the appropriate order of f ( z ) a r e finite, then f (A,) will also be finite and will be given by the corresponding form of Lagrange's formula. For example, in the case of second-order matrices, we obtain (1.15). Let us suppose that some of the characteristicnumbers I:, ...,:1 a r e equal and situated on different sheets of the Riemann surfaces* of the function f (z). Then, some (or even all) of the elements f k I (A) of the limit matrix

-

f (A,) = It f k I ( A 0 ) It may be infinite. However, if f(A,) is a finite matrix, the limiting value, as Lappo-Danilevskiyuhas shouwn,will depend not only on the matrix A, and the values f ( h ) .....f ( U , but also on the choice of the matrix S , which reduces the matrix & to the canonical form (1.6). This limiting value can be obtained by taking the limit in Lagrange's formula. Suppose that a matrix A is of the form A = S [AI, ..., I,] S-'.

Then, in accordance with formulas (1.3), (1.5) and (1.9), we have f ( A ) = S[f('d,

***I

f ( U 1 S-'

=P(A,

f ( 4 , -..*f ( h ) ) y

(1.17)

where P ( A , f (I1),..., f ( k n ) ) is Lagrange's polynomial. Let us suppose that the matrix A approaches A, in such a way that, close to the matrix &, we have

*mat is, there existhi = APsuch that, for example, f (A; ) # \(A;

).

9

FUNCTIONS OF A S I N G L E MATRIX

or /(A)= P ( 4 f ( V t

...) f (An))

+P(A

~

(1.18)

an)r

l - . r* v

where!&), ..., f(A,) are the valuesoff(z)on the original sheets of the Riemann surface close to the points A,! ..., 1;. When some of the characteristic numbers A,! ...,:A coincide, the polynomial P(A, f(U, ..., f (A,)) approaches a limiting form of Lagrange’s polynomial a6 XI-+&; however, P ( A , all ..., an) may approach a matrix whosevalue depends on the choice of the matrix S . Following Lappo-Danilevskiy, we shall call these values of f (A) irregular. In general, suppose that A =S

IJp,

oil),

..*¶

Jp,,,

(Am)l

S-’*

Then, we have

H%re, if f ( A s ) = f ( h , ) for 1, =I A,, then f ( A ) is a regular value. If a, where a is a constant, is the value of f ( z ) on an is the arbitrary Riemann sheet, then f ( k ) ( ~ )= f(”(z), where z-coordinate on the Riemann sheet. Then, the common value

f ( z ) = f (2)

+

z

.

f (4=S [f (Jp, (U),* .. f (Jp,

+S

[a1 Ip,,

..., am Ip,]

(Am))I

+

S-’

(1.181)

S-’,

where I, is the unit pth-order matrix. If ak # at for kk = A,, then f ( A ) is, by definition, an irvegular value that depends on the choice of s. Let A be a second-order matrix. Then, on the basis of (1.14) and (1.18), we have

f (A) = p (A, f (Al), f (&I) where

and

+ P (A, a’.

4

9

(1.19)

10

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

If we set (1.20)

we obtain

Then, A,=e-b,

A2=a+26,andthematrixP(A, al, %)isoftheform

11

2% +a1

a2-Ql

11

We have

-.

)il-+a,A,4a and . A - A , =a1

as b 0. Here, suppose that a, -* nr, and a2 -.m, 4 m, and the matrix S is fixed. Then, in accordance with (1.15), we obtain the limiting value of f(A,J in the form

ts

2m,+m, 3

3 S-1

2 %-m1

3

m, i-2m1

3

F U N C T I O N S OF A SINGLE M A T R I X

I1

or

The second term is a matrix R . that depends on the choice of the matrix S which reduces the matrix.& = a1 = SaIS' to canonical form. The characteristic numbers of the matrix R a r e m, and Therefore, by changing the value of the matrix S , the matrix H can also be written R - S [m,, m,] S-l. Consequently, formula (1.22) can be written in the form

&.

f (4) = f ( 4 I +s

[m,,

m2l

S-l.

(1.23)

where S is an arbitrary matrix with nonzero determinant. In the case in which A, and La approach a value a while remaining on a single sheet of the Riemann surface of the function f (z), we have m, = m, = m and R = mI ;that is ,f (A,,) no longer depends on the matrix S. Formula (1.23) can also be obtained from (1.17) with

f(h)= f ( 4

+ m,, f (La) = f ( 4+ ma.

Here, a different path A +a1 is taken since, in this case,

and S is fixed, whereas, in deriving (1.22), we have

as 6 + 0. Consequently, the matrix that transforms the matrix A to canonical form is changed (along with 6). However, it should be noted that, in the space of elements of the matrix A , there is also a path A + A,, such that the matrix of f ( A ) has certain values approaching infinity. If A, and i2 approach a single value A in such a way that they remain on different sheets of the Riemann surface of the function

12

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

f ( z ) (€or example, if the inequality f(1,) # f(1J remains valid even when = k2 = A) and if the limit matrix A is of the form

for some nonzero p , that is, if the characteristic number A corresponds to a nonprime elementary divisor, then some of the elements of the matrix f ( A ) always approach infinity.* It follows from this that the limiting form of Lagrange's polynomial (1.15) assumes all finite values of the matrix f(&) or f(S&S-l). These last two assertions follow from formula (1.21). On the basis of (1.19) and (1.21), it is easy to see that, for themultiple-valued function f ( z ) , the formula (1.23) assumes all possible finite values of the matrix f (A I). This is true because the general finite value of the matrix f (11) can, according to (1.19), be written in the form f ( 1 1 ) = f ( A ) I + P @ I , ml, m3. where P ( l 1, m,, mJ. is obtained from the matrix

with characteristic numbers a, and ag as a,- m, and If the matrix A is of order n, then, obviously, f ( A ) + f ( A ) I + S [ m , ,..., m,,]S-'

a1 -. m,.

as A-.AI.

(1.23

1)

With regard to the exponential function, we note also that

(1.24

*Even in the general case of an nth-order matrix, this will always be true when the characteristic numbers A1 and A, coincide with respect to the coordinates, forming a nonprime elementary divisor though remaining on differentsheets of the Riemann surface of the functionf ( t ) [ 4 ] . We shall also call these values of f ( A ) irregular.

FUNCTIONS OF A SINGLE MATRIX

13

If the matrices A and B commute, that is, if

AB = BA,

(1.25)

then, eA+’ = eAeB. If the determinant of the matrix A is nonzero, then InA=B

(1.26)

is defined as a solution of the equation

eB = A. The principle value of In A vanishes at A = I Under condition (1.25), we have

(1.27)

.

InAB=InA+lnB

(1.28)

(on the basis of (1.30), In A and In B commute). Therefore, InY-l=-InY

+ S [ i n , , . . . , m,]S-’2zi

and, in particular, InY-* = - InYfor suitablevalues of the logarithm on the right and on the left. Here, the mk are integers. In a neighborhood of A = I, the expansion (1.29)

is valid, giving the principle value ofiln A. If the characteristic numbers A,, ... , A, of the matrix A a r e distinct, then, from Lagrange’s formula,

We obtain all values of In A bymeans of analytic continuation and a

limiting process on the basis of this formula. It is shown inarticle [ 51 that it is possible to construct Lagrange’s polynomial for In A in which the coefficients a r e expressed directly in terms of the invariants of the matrix A. Thus, when we seek to find In A , we do not need to find the roots of the characteristic polynomial of the matrix A. Let us find this polynomial for In A , following the reasoning used in Section 1 of article [5].

14

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

...,

We denote by xl, x, the characteristic numbers of the matrix X. Then, in accordance with Lagrange's formula, we have

Y = In X

= anel

Xn-' +

X"-a

+ ... + ao,

(1.31)

where the a, a r e rational functions of x, and In xi. Let us express explicitly the 01, in terms of the invariants of the matrix X . On the basis of formulas (1.6) and (1.9), we see that the In x, are the characteristic numbers of Y. We introduce the notations n

n

k= 1

1=1

where the z, and the zh&are, respectively, the characteristic numbers and the diagonal elements* of the matrix 2;

where D ( X ) is the determinant of the matrix X. We see from (1.31) that the matrices X and Y can be reduced to triangular form with the aid of the same matrix S.. Therefore, we have a(Y)= Inx,... x, = ~ , , - ~ a , - ~ -k ann-aafl-S ... aOn

+ +

..,.....................

+

an-An-8

}

(1.32)

I

n

+ + - 0 -

"oats-,.

I

Let us find an explicit expression for the left members in terms of the invariants of the matrix X . With this in mind, we take In xi in the form (1.33) ~

Were, the equation written down follows from (1.4).

.

15

FUNCTIONS OF A SINGLE MATRIX

If we substitute this value into the left-hand members of Eqs. (1.32), we obtain

7

a (Y)= In x1

...x,

= In D (X)

.................. . . . .

5

J2

[(Xi

- 1)1+1

0 i=I

+ l(1-

1) ( X i - 1)t-l 21

+ - (I!1X i -

+ ...

+(Xi-

I)t

+

1)]X

x [l + t ( X , - l)]-'dt .........

.......... n

C x?-'ln I==I

x: =

ii i=1

%?--I

1

+

(Xi

1 +'t

Cons ide r

- 1) dt= f(x, - 1) (Xi

- 1p-' + ... + ( X i (Xi

- 1)

- 1) dt. /

16

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

where the S , a r e the basic symmetric functions of ( x , - I), . . . , (x,

n n

.sn=:

n

(Xi

n

- 1 ) . s,.-.I ==P n ( X j - I ) , ...,S 1 * E ( X , - I)&=Y i=1 j # i

I= 1

-1),

I.

i= I

Furthermore, we have

i= I

2

x

n-I

n

(Xi

-1 ) k

I= I

n-I

t' S " , i

v=o

==

C

t p A;,

SOi-: I

p=o

Here, S I , ~is obtained from Sl by omitting terms containing (xiand n

n

i= I

i= I

It is easy to see that

C S , , , =: i=I

By using this result, we obtain

I

n-1

s, (n--1).

1)

FUNCTIONS OF A S I N G L E MATRIX

x n

Here, 6, =

(xi - l)k. and by 80, we mean 8, = 1

1=I

17

+ k since

n

I=I

i= 1

The quantitites 6, a r e rational functions of S,,, (for m = 1. ..., n). The quantities S , in thecharacteristic equation of the matrix ( X - I) a r e found to be rational functions of the elements of the matrix ( X - I). Thus, the left-hand members of Eqs. (1.32) are found in the form [(on the basis of (1.34), (1.35), and (1.36)] f + I n-1

o(X'Y) =

J

EZ

0

f(1- 1) ... k tpA; (f-k+l)!

2

dt ( 1 s I, ..., n-l),

(1.37)

tPSP

p=o

where the A! are rational functions of the elements of the matrix ( X - I) and hence of the matrix X . If we now determine the quantitites a, (for i = 0. ..., n - 1)from the linear system (1.32), we find them directly in terms of the

invariants of the matrix X. Let us denote the determinant of the coefficients of the system (1.32) for unknown ao, a1,..., an+ in terms of A(XL= A. If xI,..., x, are all distinct, then A(X) # 0. Let u s denote by A, the determinant obtained from A ( X ) by replacing the ith column with the left-hand members and let us denote by A, the determinant 3, in which the quantities a ( X f Y)are replaced with the expressions given by (1.37). Then, we obtain

Finally, we have*

--x AIXi. n-I

Y=hX=

Ll

(1.39)

,=o

We note that the denominator of the fraction constituting the integrand in (1.37) is n

n (l+t(x,-

I= I

n

l))=Z PSP. P=o.

*This formula, in accordance with the consequence of Remark 1.1 becomes a finite limit formula for multiples of w,. . . . , x,.

18

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

Therefore, this denominator has rootst, = ( 1 --&'(fork= 1, ..., n). Since 0 < t < 1 , the integral (1.37) is real and finite if the characteristic number x, (fork = 1. ..., n ) of the matrix X is nonnegative. This leads to the following remark. Remark 1.2. InX given by formula (1.39) is real if none of the characteristic numbers xl,..., x,, are negative. If any of these numbers are negative, then the path of integration [OJ] in (1.37) [or (1.33)] should be taken in the complex plane. But then, as can be shown, 1nX would have to be complex also. We shall not prove this here since we shall soon prove it by using other considerations. For a second-order matrix X, formula (1.39) takes the form

1nX =

+

(S,

+ 2) InD -2M 4s, - s:

X+

+ 2 )M -S:

(S1

- 2S1+ 254- 2 4s, -s;

(1.40) ?

where S 2 = = D - o + 1, a = x 1 + x 2 ,

S,=a-2,

D=x,x,

and S2P+ Slf

0

+1

x,we have 1nX = a 2 X 2+ a, X + ao,

For a third-order matrix

(1.41)

where a y , al, and a. are determined from the equations

and the free terms Po, P , , and P2are determined by the equations

19

FUNCTIONS O F A S I N G L E MATRIX

Mo = 6; - 36162

+ 26; + 363 -462 + 61,

+ 26162 -6163 - 26: + 2 4 -668, 63 (6; + 261 -- 262 + 3), -3, 62 = - 2U1+3, 63 = + -

M i = 6fb2

M 2

03 ~1 US - 1. 02 6, = 01 Here, al, ag, and 13 a r e polynomials in the elements of the matrix X that a r e defined a s the coefficients of the characteristic equation of the matrix X D - A I) = k3 - o1h2 + u ~ A- a3 = 0,

(x

+

+

+

and, at the same time,a, = x, + x, x,, u2 = x,x, x,x, x ~J? = ~ x1xfi3, , where x,, x2, and xs a r e the characteristic numbersof the matrix X, that is, the roots of the equation - D (X - A I) = 0. Let us examine in greater detail the form of In A when the real matrix A has negative characteristic numbers. Suppose that A = SBS-l, where B is a quasi-diagonal real matrix; B = [B,,..., B,] and B , (for Y = 1, ..., k ) arerealsquarematrices either with a single elementary divisor corresponding to the real characteristic number A, o r with two elementary divisors corresponding to the two complex conjugate characteristic numbers A, and Ak+l. Here, the matrices S may also be assumed real. On the basis of (1.3), we have 1nA = S [In&, ..., 1nBJ S-'.

(1.42)

We take* as real the values of In B , for those B, that have characteristic numbers A, not equal to negative numbers. Suppose now that B , corresponds to a negative characteristic number h, of the matrix A. Then, the matrix B, has only one elementary divisor and B , = S,J (A,) S - I ,

In B , = S,ln J (A,) ST'= S , In [ - J (A,) .(- I)] S;' = =S,

[In (- J ),A(

+ x it] S;*.

(1.43)

H e r e , S , is a real matrix with nonzero determinant and ln(-J(A,)) is a real matrix since the characteristic number of the matrix - J (A,) is equal to - A, 7 0. If we substitute the value of (1.43) into (1.42), we obtain

In A = Al

+

x iSL (0, 1) S-l,

(1.44)

where A, is a real matrix having characteristic numbers equal to In h, if i,is not equal to a negative number and ln(-L,) if A, < O . *According to formula (1.39).

20

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

L ( 0 , 1) is a real diagonal matrix of order n in which we have 1 at positions corresponding to the roots iv ~0 and 0 at all other positions. We note also that the matrix .4, obviously commutes with the matrix ::iSL (0, 1) S-l = i s A2. Let us write (1.44) in the form InA=A1+xiA,.

(1.45)

Here, the matrix A, has characteristic numbers equal only to zero and unity. We call this value of In A the principul vuhce. Remark 1.3. We might have set In B, = S, [In (- J (i.,)) - x ill STI

for some (or even all) of the negative characteristic numbers kv. Then, at the corresponding positions in the matrix L (0, I), we would ~ 0 we , could also have had -1 instead of 1. Of course, when have set

or In 5, = S., Iln (- J (J.,,)) -t (212 --- 1) ?: i]S;l,

(1.452)

where n is an integer (positive or negative). The matrix A, can also be taken as follows: A? = S L 10, 2m, (2n + 1). (2n, - 1)j S-l,

(1.46)

where L is a diagonal matrix the elements in which a r e equal to 0 or 2m if they correspond to nonnegative characteristic numbers and to (2n f ’ l ) o r ( 2 4 -- 1) if they correspond to negative numbers; the numbers m , n , and n, a r e integers. Obviously, formulas (1.451), (1.4Ei2), and (1.46)yield a value of In A other than the principal value. W e note also that In A is a multiple-valued function and that, in accordance with (1.18) o r (1.231), all thevaluesof lna I are obtained in the form Ina I = I In a

+ 25: is [ml,..., m,]S-l,

(1.47)

where S-l is an arbitrary matrix such that D ( S ) i 0 and ml,..., m, a r e arbitrary integers. Lappo-Danilevskiy termed such values of In a1 “irregular” when the numbers ml,..., m, a r e not all equal, a s was noted earlier. For a second-order matrix A = a I with a = 1, by setting h a = In 1 = 0. we obtain

F U N C T I O N S OF A S I N G L E M A T R I X

In I = 25r is [m,, m,]

S-I.

21

(1.48)

In particular, (1.49) These values of In I a r e real for an arbitrary real matrix S . Fora = - I, we have In a = x i and I n [ - - I ] = i x S [ 2 m l + 1, 2m,+ 11s-’.

(1.50)

If S is an arbitrary real matrix, then

has a real value. Remark 1.4. If the matrix A has an even number of negative characteristic numbers, then, for example,* on the basis of (1.51), we can set the imaginarypartoflnd equal to 0 in (1.43), (1.44), and (1.45). But, when we do this, we will obtain a nonprincipal (and irregular) value of In A. In conclusion, we note that the functionf(A)of a matrix A can be defined by using, for example, Lagrange’s formula (1.52) Here, the scalar functions (P~().~,..., Ln) are defined in terms of / ( I ) and its derivatives in a neighborhood of the characteristic numbers XI, ..., 1, of the matrix A. It is in just t$is way that Gantmakher [ 31 defines f (A). Thus, we have the values off (A)when f (4, together with the relevant derivatives (that is, those that appear in the construction of Lagrange’s formula), is determined in a neighborhood of the characteristicnumbersil,..., 4, or, as Gantmakher says, f (A) is defined on the spectrum of the matrix A. We note that in this case,

+

*Since In I (A ) = In (- J (A )) In (- 1.I*,,,) = In ( - I (A,)+ [In (- 1 .I,)...., In (- 1 *IS)]. Her:, In (-I .In) is Eiven by formula (1.51) and the matnx [In (- 1 *II),..., In (- I *Is)] is a quasi-diagonal mth-order matrix.

22

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

in a neighborhood of each characteristic number lk, one may take a different f k (2); that is, f l (z),..., fn (z) a r e not necessarily elements of the same analytic function f(z).

2. Auxiliary Theorems Let f (2) denote an analytic, possibly multiple-valued function that is holomorphic in a neighborhood of points a&(where k = 1, ..., n). Suppose that the series

converges for 1 ~ 1< E ~where , the XI, are nth-order matrices that do not depend on E and the characteristic numbers xi =xi(€) of the matrix X a r e such that x i ( 0 ) = a&. Let us define a function Y = f ( X )in a neighborhood of E = 0 on the spectrum of the matrix X in such a way that f ( x k ( 0 ) ) = f ( x l (0)) if x&(0) = xl (O).* Theorem 2.1. The matrix Y can be represented in the form of a convergent series

Proof: We have

where the scalar quantitites

(Pk

a r e defined** by the equations

The determinant of the coefficients of the unknowns (Pk is nonzero for distinct characteristic numbers .Y~,..., xn. *In other words, Yo = 1 (Xo) is a regular value (see p. 9). **See formulas (L32).

23

AUXILIARY T H E O R E M S

Because of the hypotheses made, both s ( X k ) and the left-hand members of these equations are single-valued* functions of i in a neighborhood of E=O. Since the Lagrange polynomial f (X)takes on 8 finite limiting form** when the characteristic numbers of the matrix X coincide [l,2, 31, it follows that the and f ( x )a r e holomorphic functions in a neighborhood of E = 0. We note that, when there are multiple characteristic numbers (E)

we can first define xk xi+i

= Y I+~(E) .

=Xk(8,

= Xi+i(s)

T)

= ... = x j + p ( E )

by setting, for example,

IT,...,

= X j + p ( E ) -t b p r .

Here, if the given root belongs to a circular system of 9 elements, then eachelement xy)(fork = 1,..., 9 of this circular system will also be a root of multiplicity p ; that is, we shall have*** xj2,=xj$)*=

...= xjyn (k = 1 (...,9).

Then, we need to consider the characteristic numbers x 1+1! ~-xi?, )

(2)

+

61:

,..., xjTP = x I+P ! ~ (E)+)

bpT

(k = 1,..., 9).

We shall have n distinct characteristic numbers. The left-hand members of Eqs. (2.4) and the a ( X k ) will be single-valued functions of €forall1.1 G, T~. For7 = 0,we obtain T k = qk(8, 0) a s singlevalued functions of E in a neighborhood of E 0. The existence of a limiting finite form of Lagrange's formula can, obviously, be used in a different manner. For Y = In X, the theorem**** is also obvious on the basis of (1.39). For second- and third-order matrices X, it is obvious on the basis of formulas (1.40) and (1.41).

-

Were, as we can see, every circular system [6] of characteristic numbers r k ( t ) (for R = p , p 1 , ..., p rn) is symmetric. *YThese limiting values of v k can be found, for example, by using the corollary to Remark 1.1. **+?herefore, (Pk (A, ,..., A,,) in (1.8) will also be a symmetric function of il ,... kQ, (which belong to a single circular system) and (Pk (xl (c). .... x,,, (E)) w i l l be a single-valued ) i l l also be finite for function of c in a neighborhood of c = 0. It is easy to see that i p k ( ~ w = 0 if any of the XI (0), ..., x,,, (0)coincide. Theorem (2.1) follows from this. ****For the case in which Y = In X, this theorem is studied i n [14] for X, = I . It is basically proven for arbitrary X , in [7]and[4] by different methods. However, Artem'yev initiated essentially this type of investigating procedure in his works [8,-9], as did &tokalo in [lo]. In [7] the question of the expansion of the functionln (zX k €")in a series in terms of the parameter L and of the characteristic numbers of that function is studied for the first time in complete detail. Unfortunately, this work was unknown to me prior to the end of 1956. Therefore, it is not mentioned in [4], which was completed in 1954 (15. footnotes on p. 211 in [ll] and on p. 5 in [4]. In [12] the results of other authors [13] are repeated.

+

+

.

24

LINEAR S Y S T E M S OF ORDINARY D I F F E R E N T I A L E Q U A T I O N S

For the principal value of Y = InX, these formulas enable us also to find the expansion (2.2). We can find a nonprincipal value by use of (1.18) o r by- making the substitution f ( x , ) = Inx,+ 2 m a i with suitable constants m in (2.4). Remark 2.1. Theorem 2.1 remains valid in the case in which different holomorphic functionsf, (z) are taken in the definition o f f (X) in a neighborhood of the limitingvalues of x, (0) = a, since the values of f ( z ) at the points belonging to a single circular system of roots of the characteristic equation X"+a,-,(E)X"-l+...

+a,(E)X+a,(i)=O

(2.5)

of the matrix (2.1) are calculated with the aid of the function f ( z ) = f,(z), which is given by its element in a neighborhood of the point Uj. This remark follows from the fact that the left-hand members of Eq. (2.4) and the o(Xn)are symmetric functions of Xj+i,...l Xi+p.

Remark 2.2. The series (2.2) converges at least in the circle r < el in which there is no more than one branch point of the roots of the characteristic equation (2.5) if the roots x, (E), ..., x, (E) do

151

not assume singular values of the function f (2) at these values of e. Consequently, the series (2.2) converges at least in the circle lel < r < E , in which the discriminant A ( € )of Eq. (2.5) does not vanish for e f 0. On the other hand, if x,(O), ..., xn (0) a r e distinct, the series (2.2) will also converge in the circle la1 \< r < E , in which there is no more than one zero of the discriminant A ( E ) . (However, in a neighborhood of E = 0, the functions f (xk ( 8 ) ) must be such that the value of f (X(8)) will be regular in a neighborhood of a branch point of el.) However, the radius of convergence of the series (2.2) may be greater than this, a s will be shown at the end of this section. Theorem 2.2. Suppose that the function Y = f ( X ) referred to in Theorem 2.1 is defined in a neighborhood of E = 0 on the spectrum ~ ~ ( 2 ) ...., x,, (2) in such 0 way that f ( x , (0))= f (x,(0)) ifxr(0)= xi (0). (In the case Y = InX, the principal values of lnx,(e),..., Inx,(a)) are taken, f o r example.) I f the characteristic numbers x , ( ~ ) ( f o kr =l,...,n)Of the matrix (2.l)do not assume singular values of the function f(z)zn the region I E 1 c el (in the case of Y = In X , we must havexh (E) =&0 in the region 1.1 < sI),then the invariants of the matrixY = f ( X ) can be W represented in the f o r m of series pkek in the circle IEI 4 r < in which there is no more than on&%anch point e,, of the roots of the characteristic equation (2.5). *If c = Ois a branch point, then so = 0. If the point e,, 4-3 is a branch point in the circle thenx,(O), ..., ~~(0)aredistinctandweneedto takef(xk(EOj= / ( X , ( E ~ )ifxk(Eo) ) < Xl(t0).

[cI < r ,

25

AUXILIARY T H E O R E M S

This theorem follows from Remark 2.2, but we shall prove it anew. Proof: The characteristic numbers x k ( s ) (for k = I , ..., n) of the matrix (2.1) a r e determined from Eq. (2.5), where the uk(e)(for k = I , ..., n) a r e holomorphic functions of E in the regionlel < E ~ . It follows from this that the characteristic numbers in the region 1.1 < el have only algebraic singular points and, for every e0 in the region I = I < el, a r e representable in the form of series of integral powers of (e - E ~ or ) ( B - E ~ ) ' / ~ , where k is a positive integer less than n. The characteristic numbers .of the matrix Y a r e equal to f(xl ( E ) ) ,...,f (x, ( E ) ) . The invariants ok (for k = I ,...,n) of the matrix Y are known symmetric Rth-degree polynomials in fl (xl (e)),...,f n (xn (E)):

The algebraic singular point so (closest to E = 0) of the functions isnotasingularpointofok(e)since the functions a k ( E ) are single-valued in a neighborhood of the point €0 (because they are symmetric functions of f(x1),..., f(xn), and the & ( E ) do not assume singular values of f (2) in the region 181< E ) . The assertion follows from this. For example, suppose that there are two branch points E~ and E, of the roots of Eq. (2.5). If, in a neighborhood of the point el, we take values of f ( x h ( s ) ) in such a way that the left-hand members of Eqs. (2.4) a r e single-valued (that is, f(X(e)) is a regular value in a neighborhood of the point EJ, then these values of f (xk (E)) in a neighborhood of c, may be such that f (X (€))willbe a nonregular value in a neighborhood of the point E ~ . But then, in a neighborhood of this point, the coefficients yk(e) may be nonsingle-valued. Cn the other hand, if there is no more than one branch point in the region IB I < r , then f (X( 2 ) ) will be a regular value when the function f (X(e)) is continued analytically. Example: xl(c), ..., x,,(E)

Suppose that In u1 (1) = In vz (1) = 0. Then, D(I.) is a single-valued function in a neighborhood of h = 1. But, if we extend D(A) into a neighborhood of 1. = - 1, we obtain D ( A ) = =I n @ +

jf1.a-

-

1)[In(h--1/):~----1)-2211i],

In(- l)=-xi

26

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L EQUATIONS

since Inv, = lne"

+ri,

Inv, = In e-i~ + - x i

a s A -c - 1. If A now moves around the point h = - 1 at a close distance, then D(h) becomes D(A) = In (A -/ E l ) [In (A

+f K 1 )

- 2rr i],

that is, D(k) is a nonsingle-valued function in a neighborhood of A = - 1 . On the other hand, i f we assume D(~)single-valued in a neighborhood of A = - 1 , then D(A) is nonsingle-valued in a neighborhood of A = 1 . If D (A) is expanded in a neighborhood of A = 0, then the radius of convergence is I ).I < 1.. But if we take D (iin ) a neighborhood of k = 2 in such a way that 1nV. is regular in a neighborhood of A = 1, then the expansion in powers of ( k - 2) will converge for I?.- 2 I < 3. If we take

then D(l) = In(eA+ l/e"- l).ln(ex - /

. g x xm)

and D(A) can be expanded in powers of 1: D (A) =

,,k&,and this &=I

series will converge for I A l < x since there is only one branch point >.= 0 in this region. However, we need to remember that the m

radius of convergence of the series

c pk$ in Theorem 2.2 may be k=O

larger than this. This will be shown at the end of this section. Let us now consider the question of expanding an irregular value of a function of a matrix in a series in terms of a parameter. Consider the 12th-order matrix

where the matrices Xk are independent of E . Let us denote by the elements of the matrix X. Suppose that the elementary divisors of the matrix (2.6) are primes and that the characteristic numbers

27

AUXILIARY THEOREMS

...,x, (e) a r e holomorphic functions in a neighborhood of E = 0. Suppose that a function f ( 2 ) (in general, multiple-valued) is holomorphic in a neighborhood of the characteristic numbers z ak = %&(O)-(fOr k = 1, ..., n) of the matrix Xo. Then, we have Theorem 2.3. The function

x1(E),

-

can be represented by a convergent series

x YkEk, OD

Y=

Yo = f (XO),

k=O

where f (X,) i s any, possibly irregular, value ( i f ak = a,). Proof: We have

x,=so [a,,..., a,] So' X = S (e) [xl(s),..., x, ( e ) ]S-' x~ (E)

+ xh (0)

= ak, S ( e ) -+ So

as

(E) 2

-.,

0

I

(2.9)

and the elements of the matrix S ( 8 ) are holomorphic functions in a neighborhood of E = 0. On the basis of (1.3) and (1.9), Y = S ( e ) If@l(9)...., f

(X"(C))lS-l(E).

(2.10)

We note that here we can set So = I. Specifically, we may write

Since

the problem amounts to examining the function f

x,

). Here,

= [al,..., a,,]: that is, here we have So= I. This enables us to assume that the elements xkl of the matrix (2.6) approach 0 as €+Owhen k # 1. The theorem follows from (2.9) and (2.10). If f ( X , ) in (2.8) is irregular, it depends on So = S (0).

28

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Remark 2.3. Instead of requiring thatthe elementary divisors of the matrix X be prime, we may require that the characteristic numbers x& (E) be holomorphic in a neighborhood of E = 0 and that the canonical structure of the matrix*

X

= S (€1[Ji (xi (E)),...,

remain the same for all

leJ

J , (xm

(€))I

S-'

(E)

< R. m

Example: Y = In X , X =

X k E&, X, = exp A . Suppose that the k d

canonical structure of the matrix

x

-

S (E)

[Jp,

( x (€)I,...,

Jpv

( x , (E1)IS-l (€1

remains invariant for all !el < R and that among the characteristic numbers a, = x, (0) (for k 2 ' 1 ,...,' v ) of the matrix A there are some such that ak -a, = 2m x i

( m- an integer).

A = So [Jp, (ai),:..,Jpv (a,,)] S,',

S (E)

--•

So

+

as E --t 0 and ak = bk 2m, 5i i (for k = 1, ...,v) ,where the m, are integers and 6k - 6, f 2m 5c i for nonzero integral m. Then, co

Y =E

(2.11)

YkEk,

k=O

Y,=InexpA+S,

[2mlxiI

2m,xiIp,]S;',

where InexpA = S o [ J p ,(61),..., Jp. (6,)1S;1 is the principal value of In exp A. R e m a r k 2.4. Wenotethatthepoints=E1at whichA(E,)=O can be a singular point for the series (2.2) such that, in a neighborhood of E = E ~ ,the series (2.2) obtained by analytic continuation of the series (2.2) constructed in a neighborhood of the point E = 0 will be an irregular value of Y = f (A' ( E ) ) . Here, the canonical structure of f (X (€1) may be the same as forX ( E ) (that is, it may have the same s e t of elementary divisors) o r i t may be different in accordance with what was said between formulas (1.23) and (1.24). In thesecond case, *?he existence of holomorphic functions S (€)and S-'(e)in a neighborhood of L= 0 was first shown by Y u S. Bogdanov in 1947, but his proof was not published (see [95]).

29

AUXILIARY THEOREMS

that is, if certain nonprime elementary divisors of the matrix X (E) correspond to certain elementary divisors of the matrixY = f ( X (€1) then the limiting values of certain elements of the matrix f ( X ( 2 ) ) will be infinite a s e - 7 ~ ~In. the first case, on the other hand, the norm of the matrix f (X(s)) may be either bounded o r unbounded a s E , E~ However, we need to keep in mind that the series (2.2) can converge in the circle I E I ;: 1 E, I (where A (el) = 0)even in the case in which the value of f (X( q ) )is irregular at the point E = el. This case is noted by Theorem 2.3 and Remark 2.3. Following the reasoning on pp. 86-87 of [14], let us consider separately the case of second-order matrices and Y = f ( X ) = In X. Here, we have

.

where x1 and x, are roots of the equation x2 - a ( X )x

+ D (X)= 0, 2 x = a ( X ) k f A *-

(E),

A (i)= a2 (X)- 40 (X).

Suppose (as we shall always assume) that X(E) is real and that (2.13)

D ( X ) = XlX, # 0

for all values of e in question. According to what was said above, the only singular points 8 = E ~ of the series (2.2) can be points at which x, = x2, that is, points at which A ( € 0 ) = a2 ( X b0))- 4 D (X(eo))

= 0.

(2.14)

In view of (2.12), the singular points E = E~ a r e the points at which x, = x i and the arguments of xl and xi differ only in sign. On the other hand, if Eq. (2.14) is not satisfied in the circle of convergence of the series (2.1), then the series (2.2) will converge in the same circle as does (2.1). Let us suppose that D (X)= x I ( E ) x, (e) = 1.

Then,

and the only singular values E

= E;

are the points at which

(2.15)

30

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

A@) = 0 ’ ( x ( ~ ) ) - 4 = 0 .

(2.16)

We see from (2.15) that,inthiscase,the arguments of n; and x, are always of opposite sign Therefore, in this case, the singular values of c0 a r e those points at which x, = x, and the arguments of xl and x, are nonzero. In particular, those points at which a(X

(E))

= - 2,

(2.17)

may be singular values of t since, at such points, x, = x, = - 1 and their arguments are x and - x . However, the roots 8 = of Eq. (2.17) a r e not necessarily singular points for the series (2.2) since, in a neighborhood of the point i= E l , an arbitrary irregular value of In X (c) can be expanded in a series of positive powers of E - E ~ if, for example, for a = i l (where ( A ( k ) ( ~is ) the kth derivative), we have A(E,) = 0 and A ( k ) ( ~ l= ) 0 (k = 1.2,

..., 2m - l),

A ( 2 m ) ( ~ 1#) 0,

~ ) xl(e) where m is a positive integer. In this case, x1(z1) = X ~ ( E but and x2(e) are holomorphic in a neighborhood of the point E =r E ~ Therefore, an arbitrary irregular value of 1nX (E) can be represented in a neighborhood of the point E = el in the form of a series of positive powers of t -fl. Suppose, for example, that we have Y = l n x ( ~ ) ,where

Here, the characteristic numbers xl (E) and x2 (E) are

We have x, = x, if E = -3, -1, or 1. In a neighborhood of the point I, the functions x,(~)andx,(~)are holomorphic. Here, if we take a regular value of In X(B) in a neighborhood of the point E = 1, we obtain the series

g=-

(2.18)

.

31

AUXILIARY T H E O R E M S

which converges in the region1 E I < 3 although this value of In X ( E ) is irregular in a neighborhood of the point E = - 1 because, as we go from the point P = 1 along the real €-axis to the point E = - 1, we obtain x, = x, = - 1 with the distinct arguments x and-= since x; approaches the point x, = 1 while remaining in the upper half-plane and x, in the lower. We can see this by performing the calculation.

-

Consider

Here, if we take the principal values of the logarithm, we obtain Inx, - Inx, = 21nxl, Inx,

=

- In x,.

Therefore, from (2.12), we have y =x (E)

2 1 n x1 x, - x ,

x, - x ,

in x,.

If we denote the elements of the matrix Y by Y k l ,we obtain

or

32

LINEAR S Y S T E M S OF ORDINARY D I F F E R E N T I A L E Q U A T I O N S

from which we see that all the elements of the matrix Yare holomorphic functions in the region I E I *< 3. On the other hand, if we take a regular value of Y=lnX(E) in a neighborhood of the point E = -- 3, then Y = In X ( E ) can be represented in the form of a series

c 00

Y=

pk ( 8

k=O

4-3 ) k ,

+

which converges in the region I s 3 I < 4 since the closest singular point is a = 1 because, in accordance with Theorem 2.3, the matrix Y will again be holomorphic in a neighborhood of the point L = - I. Let us now consider that case of a second-order matrix in which, in the series (2.1), we have X , = I and D ( X ( t ) ) = 1, In this case, the characteristic numbers x, and x2 of the matrixX can be found from the formula 2x = a (X(E)) _+

f.2

( X (a)) - 4

,

where a (X(0))=2; that is, we have x2(0) = x1(0) = 1. If we take the principal value of Y = In X (s), we have the convergent series (2.2). If the function o ( X ( z ) )increases from 0 with increasing 6, there will be no singular points of the series (2.2) on the positive half of the E-axis. If the function 5 ( X ( 5 ) ) does not vanish for real values of E , there will be no singular points of the series (2.2) on the real e-axis. This is true because the only possible singular points of E = eJ a r e those points at which a(X(s1)) = 2 o r .(X(il))=-2. If A ( t ) remains nonnegative as B varies along the real axis starting at t = 0 when A (0) = 0, xl (0) = x, (0) = 1, and the arguments of xl(0) and x,(O) a r e equal, then x,(a) and x,(E) remain on the real axis. Therefore, their arguments cannot differ. For x l ( ~to) be equal to xz(i) and for their arguments to become different, it is necessary that x1(e) = xZ(+ - 1 when the arguments of x1 ( 8 ) and X ~ ( E )are equal to c and -n.This is possible only when a (X(z)) = - 2. Thus, the singular point E = Elof the series (2.2) that is closest to c = 0 can only be apoint at which3 ( X (el)) = - 2. From this it follows that, if o (X (t)) does not vanish on the real €-axis, the series (2.2) cannot have a singular point on it. For example, this will be the case if the elements x12 and xzl of the matrix X(E)a r e of the same sign. To see this, note that, if o ( X (E)) = xl1 xzz= 0 , we have (since D(X ( E ) ) = 1 and hence xl1x2,= ."1$zl -k 1)

-

- x:,

= x21x1,

+1

so that xll is imaginary, which is impossible for real X(E). Let us exhibit a region of convergence of a series (2.2) on the basis of bounds for the elements of the matrix X(E)in the case in which X, in the series (2.1) is I. Since Y = I n X ( t ) (the principal value) can be represented in the form (2.19)

FUNCTIONS O F S E V E R A L M A TRICES

33

it is clear that the series (2.19) converges in the case in which the maximum absolute value of the characteristic number of the matrix (X(E)-I is less than l/n. If this inequality is satisfied, the series (2.2) converges. A bound for the maximum absolute value of the characteristic numbers of a matrix A with positive elements appears, for example, in [l, 31. Sometimes, a bound for the maximum absolute value x ( 8 ) of the characteristic numbers of a matrix X(E) can be obtained by using a series of the form 01

I

+C

A k Ek.

that majorizes the series (2.1). We shall use this later

k=l

(see Sect. 10). In Section 34 we shall also exhibit cases in which A (E)# 0 in the entire region in question (except possibly at the point E = 0). Consequently, the series (2.2) will converge in the same region* as does the series (2.1).

3. Functions of Several Matrices and of a Countable Set of Matrices Lappo-Danilevskiy first [l] began to examine functions of m matrices X1, ..., X, of order n and constructed a theory of such functions. Specifically, he studied functions of matrices&, ..., X,,,

q=l

11

...i,

where the a a r e complex numbers and j , ... j. range independently of each other over all possible values from 1to m. Lappo-Danilevskiy called the series (3.1) a “series of compositions.” Following LappoDanilevskiy, let us write the series (3.1) in the form m

where

it

...i,

It is easy to see that the series (3.1) is a set of a particular form of n8 series of m 2independent variable elements of the matrices

x,,..., x,.

*All cases in which ,p1 and ps remain complex for Eq. (33.1)

34

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

Let 1x1 denote the matrix whose elements are equal to the absolute value of the corresponding elements of the matrix X. We indicate that none of the elements of the matrix 1x1 exceed )! > O , where 11 pII is a- matrix whose elements are all equal to p, by writing

1x1 IIPII.

(3.4)

If the series (3.1) converges in the region

I XI I < IIPi

11s

(3.5)

then the function F(X,, .... X,) is said to be holomorphic in a neighborhood of zero matrices. If I all .../,I
and if the series r a ( v ) P, where E is a complex number, converges

2

in the region I E I< np, then the series (3.1) converges in the region 1 x 1 1

+

t ..- IXmI
(3.6)

and is said to be uniformly holomorphic in the region (3.6). If the series (3.1) converges for arbitrary finite matrices XI, .., X , , it is called an entire series. As Lappo-Danilevskiy showed, the usual theorem on the uniqueness of the expansion of a function in a power series does not hold for functions of several matrices. We do, however, have the following valid assertion. I f a function of several matrices F (X,, ...,X,) can be represented in the f o r m (3.1) f o r arbitrary arrangement of the matrices X,, ...,X,,, , then the coefficients a/, ...j,are uniquely determined. In other words, i f

-0

v=O

f o r arbitrary arrangement of the matrices X1 ,...,X,,, , then a], ...j , = PI, ... Let us denote the infinite sequence of matrices X,, X,, .., (0

I XpI converges, then the sequence of matrices

by X. If the series PSI

X is said to be regular. Let us denote by a g , a,,, numbers. If the series

...pr

(for v = 1, 2, ...) certain complex

F U N C T I O N S OF S E V E R A L M A T R I C E S

35

converges, let us consider the series

2 ... 2 x,, ...X,

apl

...pv = [Xa], , IXal0 = no

(3.7)

lap, ... pvI
If

and m

m

Let us now consider a seriesfor aregular sequence of matrices

-=

assuming that I apl ...*,, I a("). If this series converges for every set of matrices XI,X, ,... satisfying the condition W

p= I

we shall say that the series (3.8) converges in a neighborhood of zero matrices X. On the other hand, if the series (3.8) converges for every positive p when condition (3.9) is satisfied, then the series (3.8) is said to be an entire series. We note that if the series

has radius of convergence n p, then the series (3.8) converges in the region (3.9). If the equation

36

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS m

W

holds for a regular sequence of matrices X of arbitrary order, then ap,

... P" = BPI ... Pv 4. Classes of Systems of Linear Differential Equations That Can Be Integrated in Closed Form

Suppose that the elements Xkl(t) of a matrix X a r e functions of Let us write this matrix in the form

t.

Then, we define the derivative of the matrix X with respect to t a s that matrix whose elements are the derivatives of the corresponding elements of the matrix X :

dt If we have matrices

X and Y that are functions of t , then

If we have m matrices X l ( t ) , ..., X,,,(t), then

Suppose that the matrices dt

(k = I , ..., m)

C L A S S E S O F S Y S T E M S O F LINEAR DIFFERENTIAL EQUATIONS

37

dX Consequently, if the matrix X (t)commutes with its derivative -, then dt dex -ex _ dt

dX dt

-=dX

dt

ex.

This follows immediately from the definition of the function ex on the basis of (1.2). By definition, we also have

Consider the system of linear differential equations

-dX1 - xlP11(0 i-.. dt

*

+ xnPn1(t)v

.............. du, = XlPI, (t) + ... + xnPm (t).

(4.1)

dt

This system has n linearly independent solutions xkl, ...sXkn ( k = 1 , ...,n), which can be written in the form of a matrix

in which each element Xkl is the 1 thfunction of the k,th solution of the system (4.1). Consider also the coefficient matrix of the system (4.1) P (1) =

1

PI,

( 4 3

..., PIn (4

I/

....... .

pn1 ( f ) ,

Pnn (t)

(4.4)

If we substitute successively the solutions of (4.2) into Eq. (4.1), we obtain na equations, which may be written in matrix form as follows: dX = X P . -

(4.5)

dt The matrix X is called an integral matrix. Since the matrix X ( t ) is composed of linearly independent solutions of the system (4.1),

38

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

it follows that D ( X ( t ) ) 4 0 in the region of continuity of the matrix P (t). We shall say that an integral matrix X ( t )that is equal to the unit matrix I at t -0 (that is, X (0) = f)is normalized at the point t -0. If X (1) is a normalized integral matrix at the point t =0, then every other integral matrix X , ( t ) can be expressed in the form X , ( t ) = AX ( t ) , where A is a constant matrix. Let us note certain cases in which the integral matrix X can be found in closed form. Let us suppose that the matrix P ( t ) in the matrix equation (4.5) possesses the property that

P(t)

r

P(t)df = JP(t)dt*P(t).

(4.6)

that is, let us suppose that the matrix P ( t ) commutes* with its integral. Then the integral matrix X that is normed at the point t =O can be obtained in the form [l] t

X = exp

Pdt,

(4.7)

n

since, on the basis of (4.6), dX -exp df In particular, if

-

f Pdt.P=XP. n

(4.8) p = A 'f ( t ) , where A is a constant matrix and "p(t) is a numerical function of t , then, in accordance with (4.7), we obtain t

X

= exp A

v(t)df.

(4.9)

n

Suppose now that ( t ) =1 and that the characteristicnumbers kl, .... i.,. of the matrix A a r e distinct. Then, in accordance with Lagrange's formula, we obtain from (4.9)

*A thorough study of the structure of the matrices P (1) possessing the property (4.6) i s made in the article by Bogdanov and Chebotarev [15]. This sndy i s closely connected with the work of V. V. Morozov, in which a study i s made of matrices possessing the property that P (tl) P (tt) = P (ts)P (tl). Reference [15] includes a bibliographyon matrices possessing the property (4.6). A special case of (4.6) i s studied by V. Amato [16]. Linear systems that can be integrated in finite form were also studied in [17-19].

C L A S S E S O F S Y S T E M S O F LINEAR D I F F E R E N T I A L EQUATIONS

If we write X in the form

X=

A"-'

39

+ yn-sA"-' + ...

A+ yo, then the 'pr (for k =O, 1. ..., n-I) will ke symmetric functions* This enables us to obtain the integral matrix X without of A,, ..., in. finding the characteristic numbers of the matrix A.. We note that the second-order matrices P ( f ) possessing the property (4.6) a r e of the form ?,,-I

(4.11) where 'pl(f) and qr(1) are numerical functions of t and where bl and b2 a r e constants.** Thus, when P ( t ) in thematrixequation(4.5) is of the form (4.11), we have X in the form (4.12)

(4.13) k= I

where A, (for k = 1, ..., rn) a r e constant commutative matrices and the qk(t) a r e numerical functions, then condition (4.6) is obviously satisfied. Fedorov [ 201 obtained the following interesting generalization of Lappo-Danilevskiy's system, for which we have a solution analogous to (4.7). Suppose that the matrix P ( t ) in Eq. (4.5) possesses the property

L.[P(f)B"]=O @ = I ,

(4.14)

2, ...).

where = P ( f ) , MN - NM 1 [MN] dt and L is a constant vector, that is, a matrix all of the elements of which are zero except those in one row and the elements in that row are independent oft. Then, *This is obvious from (2.4). We note that Shtokalo [ 101 obtained a solution to (4.10) in the form X =

1 2

(pE-A)-* ePfdp, where c is a closed contour encircling all the

% 1 (C)

characteristic numbers of the matrix A. *Were, we do not include the case ofPu = Pzl -0and P11 P12. The elements of the matrix (4.11) satisfy the equations4 1 PZZ= b # u , PU =b&

-

+

.

40

LINEAR S Y S T E M S OF ORDINARY D I F F E R E N T I A L EQUATIONS

X =LexpB

(4.15)

is a solution of Eq. (4.5). To see this, note that, on the basis of (4.14),we have

LPB" = LBnP

(4.16)

and

-

LB ... B P B ...B = LPBm+" = LBm+" P .

-n

m

Therefore, since

it follows that

where X is given by Eq. (4.15). From (4.16),we have LPB" = LB"-' PB. Therefore, L [PB" 1 = LB"-' [PB].

(4.17)

Since LBk [PB] =O (k = 1,

..., m),

(4.18)

where m is the degree of the minimum polynomial* Bm + z,Bm-'

+

a2 Bm"

+ ... + am-l B + a,,,

=0,

(4.19)

satisfied by the matrix B (see [21]and also Sect. 1 of the present book), it follows that Eq. (4.18) holds for k > m. Thus, we have obtained the Theorem (Fedorov). I f the matrix P(t)in (4.5)is such that there exists a constant vector L satisfying E q s . (4.18), then Eq. (4.5) has a solution of the form (4.15). If the vector L has v arbitrary pararrzters, we obtain from (4.5)v linearly independent solutions. From this it follows that the order of the system (4.5)is depressed by v units. *The

ck

in (4.19) are scalar functions of the characteristic numbers of the matrix B .

OTHER S Y S T E M S O F L I N E A R D I F F E R E N T I A L E Q U A T I O N S

41

If the matrix P ( t )is of second order and if there exists a constant vector L satisfying the equation L [PB] =O> then the system (4.5) can be integrated.

5. Other Systems of Linear Differential Equations That Are Integrable in Closed Form We note another case in which the solution of the system (4.5) can be obtained in closed form [22]. Suppose that we have a system of linear differential equations

where ‘9, (f) and ‘pz (t)are continuous functions oft and where U, and U2 a r e constant matrices possessing the property that

u, (UZU,- UJJJ - (UzU1- UlU,)u, =o.

(5.2) We note that if the matrices U, and U2commute, then condition (5.2) is satisfied. But then condition (4.6) is also satisfied, so that this case reduces to that already examined. Now, let us suppose that the matrix U,, in addition to satisfying condition (5.2), possesses the property that

u&J,- UlU2 = P(U1). (5.3) where P(U,) is a polynomial in U,with numerical coefficients. If to every characteristic number of the matrixU,there corresponds only one elementary divisor, then condition (5.3) is satisfied. Under conditions (5.2) and (5.3), the integral matrix X ( t ) , normalized for t = 0 , is obtained in the form* *Fedorov presented a moregeneral case than the system (5. l), in which X (t)is obtained in closed form 1231. Morozov [24] has found necessary and sufficient conditions for U and U,under which we have (5.4). Salakhovaand Chebotarev [25] have found necessary and sufficient conditions for the system

where Ai and E j are constant matrices and

to have a solution of the form

This condition reduces to satisfaction of the equation [ A ( t ) , [ B ( f ) , A ( t ) ] ]=o for arbitrary ‘pi ( t ) and t#i (i).

42

LINEAR SYSTEMS

OF ORDINARY D I F F E R E N T I A L EQUATIONS

where

Let U, and U2 be two second-order matrices of the forms

Then, conditions (5.2) and (5.3) a r e satisfied since to the characteristic number of the matrix U,there corresponds one elementary divisor. In this case, on the basis of (1.24), we have

and

Consequently, Eq. (5.4) can be written in the form

/10

c J e(b*-’I) L~ U)

X(t)=e

O

‘pl

( t ) dt

o O//

p,f t ) p

a (1)

so that, on the basis of formulas (1.24), we finally have

From this we obtain the sum of the diagonal elements and the determinant of the matrix X ( t ) in the form

43

OTHER S Y S T E M S O F LINEAR D I F F E R E N T I A L EQUATIONS

D(X (f))= @%t)

LZ( t ) .

e(br+bt)

(5.9)

We shall have occasion to use these formulas in what follows. We have considered the case of the system (5.1) in which the elements of the second-order matrices U1and U, in the upper righthand corner a r e equal to 0. We have already mentioned the special case of condition (5.2) in which the matrices ul and U2 commute. In this case, the integral matrix of the system (5.1) can be written in the form t

U, v1 ( t ) dl

+ Uc

t 91 (0 dl 0

0

X(t)=e The general case, in which condition (5.2) is satisfied but the second-order matrices U,and U2 do not commute and do not simultaneously have a zero element on the diagonal, can be written in the form (5.10)

where the elements a , b , c , and d are related by c,b,

+ blc2 =0,

49b2 -c, (al -d,)$ =O.

Consequently, the general form of such matrices can be written (5.11)

The characteristic numbers of the matrix U, are of the form ii,=b+mn,

h,=b-mn

(5.12)

and those of the matrix O', a r e of the form

El = a + cm. & = a We may write

From formula (1.3) we have

+cm.

(5.13)

44

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

= A-I

1+ a

cm

2cme2"Lt

0

(1)

a+cm

This formula enables us to write the solution of (5.4) in the form

(5.15) A.

X A-l

From this formula, we obtain a

(X( t ) ) = e

L, (1) +(b+mn)

L:, ( I )

and (X(W

=

e2

(0

+

+p+cmJ

Em) L, ( t )

+

L,

W+Ib--mn) rz(fJ

2bL, (f)

(5.16) (5.17)

6. The Construction of Solutions of Certain Linear Systems of Differential Equations in the Form of a Series of Several Matrices (of a Series of Compositions) Linear systems of differential equations possess the distinctive property that their general solution can be expressed in the form of certain series that converge uniformly in every closed interval in which the coefficients of the system a r e continuous. Suppose that the matrix P ( t ) in Eq. (4.5) is continuous in €he interval 0 Q t Q p. Then, we obtain the matrixX ( t )that is normalized at the point t = O in the form m

x = p k ( t ) ,&(t)=I, k=O

C O N S T R U C T I O N OF S O L U T I O N S OF C E R T A I N L I N E A R S Y S T E M S

45

where

and the series (6.1) converges uniformly in the interval 0.5 f 4 . p since it is majorized by the series Y = e Mt ,

(6.3)

where M is a constant matrix the elements of which a r e positive numbers equal to the maxima of the absolute values of the corresponding elements of the matrix P ( t ) (that is, IP(t) I 4 M ) . Thus, we have a bound for the speed of convergence of the series (6.1). To prove this assertion, we need to satisfy the equation

dX =XPA dt

(6.4)

(where h is a parameter) formally with the series m

To determine the coefficients formula

x&(l),we

obtain the recursion

t

Xk ( t )=

xk-1

(f)

(Idf)

0

From this, we obtain the inequalities

Consequently, the series (6.5) i s majorized by the series

that is,

1 2 X k (t)

1’1 &

k-0

2& k=O

Mkhk lk

= eM

’‘ .

46

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

It follows from this that the series (6.5)converges uniformly in the interval 0 ,< t < p for every value of A including the value h = 1. This proves the assertion. * Let us suppose that m

&=I

where the A, a r e constant matrices and the y k (f) a r e numerical functions that a r e continuous in the interval 0 4 f 4 p. Then, in the series (6.1), we have obviously (6.9) Here,

r

t

'pi,

...j k (t)=

?is

...jk-l(t)

Tik (

~dt.

(6.10)

i,

This is obtained from formula (6.2). Thus, in the case of (6.8), we have the integral matrix X ( f ) normalized at the point t -0 in the form of a series of compositions

which converges for arbitrary finite values of the matrices A, ,..., A,,, and of t in the interval 0 4 t 4p. We note now that the series (6.1) and (6.11) converge uniformly in the region D of the complexvariable t if the matrix P ( t ) is a continuous function of t in that (closed) region. Consider now the system -=

(6.12)

dt

k=O

where X and the Pk(t)are nth-order matrices, where the series P (f,

E)

=

Pk

(t) Ek

(6.13)

k=O

converges for all 0 4 t & b in the region l e I < r , where the Pk(t)are continuous in the interval 0 s t 5 6 , where IP(t, E)I 4 M (where in I

*lUs assertion remains in force if we assume that ing of the constant M in (6.3) is modified.

0

I P (t)1 dt

< W. Here,the mean-

CONSTRUCTION OF SOLUTIONS OF CERTAIN LINEAR SYSTEMS

47

turn M is a constant matrix with positive elements), and where 0 4 e 4 el < r. Theorem 6.1.* (Lyapunov). An integral matrix X normalized at the point t = 0 can be represented in the form of a series m

x=

x k (f)

e k , Xo (0)= I ,

x k

(0)=O,

k >1

(6.14)

k=O

that converges f o r I E I < r for all values o f t in the interval O < f 4 6 , where the & ( f ) are continuous matrix functions o f t in the interval 0 4 t Q 0 that are defined inductively by dxo ( f ) df

--_

dxk x, (f) Po ( t ) , = X,P0 + df

+ x&lp1+

...

or t

(x,+pi

x k(f)=

+ XOPk,

(6.15)

+...+ x p k ) , x,'

(T) d 7 X,(f).

(6.16)

n

Proof: Consider the auxiliary system**

dY -YP(t,

e)h,

dt where A is a numerical parameter. We have

(6.17)

m

(6.18) Yk(2,

e ) = f Y k - - l ( f l C)P(t, e ) d t , k > l ,

Yo(t, & ) = I ,

(6.19)

0

and the series (6.18) converges uniformly in the region 0 s t -Q 6, I< el < r, I)iI 4 R, where E, is an arbitrary positive number less than r . To see that this is true, note that, since I P ( t , €)I d M , we have the series

I8

which majorizes the series (6.18). This Y is a solution of the equation dY -YM)i. (6.20) dt

-

*See also [9], in which this theorem is formulated in a different way. *Were, we follow Lyapunov [26].

48

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

If we set A

=1

in (6.18), we obtain a solution of Eq. (6.12): (6.21) k=O

This series converges uniformly in the region 0 4 t G 6, I e I :: el < r , and the matrices Y k ( f ,E ) are holomorphic in the region 1 E \ < f . From this it follows that (6.14) holds. Remark 6.1. The series (6.22) converges for O < E~ .< r. Let us suppose* that IFl((t)lQ 11111~(t), where 11111denotesamatrix every element of which is unity and q ( t ) is a scalar positive funcb

cp (t)dt

tion** such that

< co.

Then, the s e r i e s (6.14) converges

0

uniformly in the internal 0 4 t 4 b f o r l e t < the series

51.

This is true because

€1

majorizes the series (6.13), that is,

The matrix

is a solution of the equation

dY -=Ydt

I11It cp (0 1 - 2

(6.23)

€1

Were we depart slightly from Lyapmov's line of reasoning [26]. b

**If

1 p', (f)1 dt < 0

00,

then I p', (f)1 = 11 Ill 'p (t), where, for example,cg (f)= z I Pnr (:)I-

SOLUTl ON OF THE POINCARE- LAPPO-DAN ILEVSKIY PROBLEM

49

and also amajorant for the series (6.14). This last assertion follows from the fact that

since I Po (t)I c 11 1 11 cp (t)and, in accordance with (6.16),/ X, ( f ) I & Y , rt), because

Here, the matrices 1

X,' (.).Xo (0 and exp II 1 II

cp(t)dt

are solutions (normalized at the point f = 5 ) of the first of Eqs. (6.15) and Eq. (6.23) fore = 0, respectively. Therefore, the reasoning followed with regard to Eqs. (6.17) and (6.20) is applicable to 'them. A somewhat different approach to these problems and other cases of the matrix P ( t , E ) are examined in [9] and [26, Chapter I11J e

7. Solution of the PoincarC-Lappo-Danilevskiy

Problem Now, we shall solve the Poincar6-Lappo-Danilevskiy problem in the analytic theory of linear systems of differential equations. Afterwards, we shall show that the solution of this problem is closely connected with the theory of linear systems of differential equations with periodic coefficients. Let us suppose that P in Eq. (4.5) is an analytic function of a complex variable z that is single-valued in a neighborhood of the point z = a . If P ( z ) is a regular function at the point z = a (that is, if all elements of the matrix P ( z ) are regular functions at the point z = a)

50

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

where the Pk a r e matrices that a r e constant with respect to z , then, as we know, the integral matrix X(z) with initial value So at the point z = a will also be holomorphic in a neighborhood of the point 7 = a ; that is, we have m

Bk(z - a)k ,

X (z) =

(7.2)

k=O

where the Bk a r e constant matrices and the series (7.2) converges for z- a I < r. Here, r is the distance from the point a to the closest singular point of the matrix P(z) [in other words, r is the distance from a to the closest singular point of the elements of the matrix P(z)]. This also foliows directly from the reasoning followed above with regard to the uniform convergence of the series (6.1) in the region of continuity of the matrix P(z). Here, the functions X, (z) will obviously be regular functions in the region of regularity of the function P (2). Now, let us assume that P ( z ) has a single-valued isolated singularity at the point z = a; that is, at the point z = a, the matrix P (z) has either a pole o r an essential singularity, so that the matrix P(z) can be expanded in a Lament series in a neighborhood of the point z = a. Suppose that X ( z ) is an integral matrix of Eq. (4.5) with initial value X, a t a point z = z, in aneighborhood of the point z = a. Let us assume that D ( X , ) # 0. Then in a region ofregularity of the matrix P (z) ,we also have D ( X (2)) # 0 according to a familiar property of a fundamental system of solutions of a linear system of differeiitial equations. Let us continue X ( z ) analytically along a curve L encircling the pointz == a and passing through z = z,. The curve L does not pass through a singular point of the matrix P (z) and does not encircle any singular points of P ( z ) other than the point z = a. In general, when we complete a circuit around the singular point z = a , we obtain at the point z, a value X (2,) == X different from X,. Thus, when we make the circuit around z = a remaining in a neighborhood of the point z = z,, we obtain an integral matrix X ( Z )different from X (2). But since X ( z ) is made-up of a fundamental system of solutions, the new integral matrix X ( z ) can be expressed in terms of X(z) by means of the equation

-

x

(2)

=

v (a)x (4,

(7.3)

where V(a) is a constant matrix defined by the equation

.

To= v (a)x,,v (a) = &xi-'

(7.4)

Since D (X(2)) f Oand since a fundamental system of solutions remains a fundamental system under arbitrary analytic continuations along a curve that does not pass through a singular point of the matrix P(z), it follows that D ( X ( z ) )# 0

.

SOLUTION OF THE POINCAR~-LAPPD-DANILEVSKlY PROBLEM

5I

The matrix V ( a ) is called an integral transformation around the point z = a. If X, = I, that is, if the integral matrix X ( z ) is normalized at the point I = % , , we- shall denote the integral substitution byV(a, 2,) = ??(z,). Here, X ( z ) is the value of the integral matrix X ( z ) at the point z = zo after encirclement of the point z = u. W e shall also assume that the matrix X ( z ) is normalized at the point z = z,, Let us define W by

.

so that

2x i W (a, z,) = In V (a, to),

(7.5)

exp 2~ i W (a, q,) = V (u, q,).

(7 6)

We introduce the function

N (2) (z -a)-" X (2) = e-wln(r--a) x (z). On the basis of (7.3),afterwegoaround the point z = u this function assumes the value I=

p ( z ) = e-Wln

(2-0)

- 2x IW V (a, 2 0 ) X (z) = N (2).

Thus, the function N ( z ) is single-valued in a neighborhood of z = a . It follows fsom this that

x (2) = (z - U ) W N (2-

a),

(7.7)

where N (z - a) is a single-valued (matrix-valued) function in a neighborhood of z = u. The factor ( z - a)' characterizes every multiple-valued singularity of the matrix X (2) in a neighborhood of the point z = a and N ( z - a) is a matrix that can be represented in a neighborhood of the point z = u a in the form of a Laurent series. Following Lappo-Danilevskiy, we shall call I the exponential transformation in a neighborhood of the point z = a. If the point z = a is a first-order pole of the matrix P (z), then z = a is called aregularsingularPointofthe systems (4.1) and (4.5). From the analytic theory of linear differential equations, we h o w that, in this case, N(z -a)may be assumed regular at the point 2 = a, that is, OD

N(z-a)= xNk(t-a)k, k 4

where the Nh are constant matrices. If the characteristic numb e r s of the matrix P-, = ( z - u ) P ( z ) I r ~ , ,donotdiffer from each other by integers, the representation of the solution in the form (7.7) is

52

LINEAR S Y S T E M S O F O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S

given in Lappo-Danilevskiy's book [l]. In the case in which some of the characteristic numbers of the matrix P-,differ by an integer, the question of the construction of an integral matrix of the form (7.7) is studied in the works of Donskaya [27, 281. Suppose that we have a system of differential equations of the form

where the U j are constant matrices (with respect to z ) and the a/ are simple poles of the coefficient matrix. Lappo-Danilevskiy first gave a general representation for matrices W j characterizing the multiple-valuedness of the integral matrix Y ( z ) in a neighborhood of the corresponding points t = at. Specifically, he showed that, for the case of Y (zo) = I, the matrices W,can be represented in the form

-

MI x=I j ,

...i.

Here, the Sk (U,) are polynomials of the elements of the matrix U i ; A ( U j ) is an entire function of the elements of the matrix U i ; the series of compositions of U , , . . ., U,,, converges for all finite values of the matrices U,, ..., urn;the quantities Qj(aj, . . .ajxI zo)a r e functions of a,, . . ., a,,, , and zo as calculated from the recursion formulas. From this it is clear that the W j a r e meromorphic functions of the matrices U,, . . . , Urn. Lappo-Danilevskiy not only gave an explici: and general representation for the functions wj (the Poincare problem*) but also characterized the wj in an exhaustive way as functions of the matrices U1,. . ., u,. We shall later be interested in constructing the exponential transformations also in the case in which the singular point z = a of the matrix P ( z ) is a pole of arbitrary order and hence the function N ( Z - a) ,although single-valued in a neighborhood of z = a, is not regular at the point z = a. We note that, for the system (7.8), the matrices W i a r e similar to the matrices U j , (which, following Lappo-Danilevskiy, we shall call differential transformations); that is, Wj = S j U j S r ' , where Sj is a matrix such that D ( S i ) + 0. In the case of a system

-.

.

dX dt

0

&=-I

X

(7.9)

W. Poincad posed the problem of factoring out a multiple-valued factor of the matrix

Lappo-aanilevskiy solved the problem of the general representation of the matrix W in terms of the parameters of the matrix P (t) and made a study of the nature of Was a function of these parameters. (2).

SOLUTION

OF THE POINCARE-LAPPO-DANILEVSKIY

PROBLEM

53

(where the U, are constant matrices), we a150 have W = Su-, s-1 (see 1291) in the representation X ( z ) = ( ~ - - a ) ~ X ( z )(where (2) is a single-valued matrix in a neighborhood of z = a and W is a matrix that is constant with respect t o t ) if the characteristic numbers of the matrix Url do not diffecby integral values. Donskaya [27, 281 has shown when this formula remains valid without satisfaction of the condition stated regarding the characteristic numbers of the matrix P-l., The fact that we do not always have W = SU-,S-l is pointed out in the book by Gantmakher [ 31, who proved Donskaya’s result by a different procedure. Consider a system of differential equations of the form

x

(7.10)

where the Tv a r e constant matrices with respect to z and Y is an integral matrix which we assume to be normalized at the point I = 6, that is,

In accordance with the above,

Y (z/b)= Z“

(7.11)

(z),

where W is a constant matrix with respect to z (though it is a function of the matrices T-s, . . ., TI)and 7 (z) is a single-valued matrix. We denote by V the integral transformation of the matrix (7.10) around the point z = 0, s o that V = k=‘W. From Lappo-Danilevskiy’s theorem, we have

r=l

P,...Pr

=--I

.pr

(2Xi)Z

.

(7.12)

Here, under the second summation sign areallpossible products of the v matrices T-s, . .., T i , and the a are rational numbers defined by the recursion formulas

54

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

Q P(O),

-1I

1

p1+ 1 .z- 0

p1+-

=

+ I = 0.

arbitrary

For example, we may set

In the case in which PI .(v) p,

...p.

+ . . . -I-pv -+

= 09

.(P) P I . ..Pr

v

+ 0, we have

-

- p1+

If we now set successively p = y - 1,

and, in general,

v

1

. . . + Pv +

X .4

- 2, . . ., 1, 0, we obtain

SOLUTION

OF THE

POINCARE-LAPPO-DANILEVSKIY

+ .. .+ (

x

In the case in which p1

+ . . . + pv +

v

-. I)"-p-'

PROBLEM

55

x

= 0, we have

are arbitrary. For the complete formulas see [l, and the p. 1881. The series* (7.12) for V is entire, that is, it converges for all finite values of the matricesT-,, . . ., T, and its coefficients do not depend on the order of these matrices. Lappo-Danilevskiy [l] also expressed U:' in the form of a series of compositions of the matrices T-s, . . .,. Tl that converge in a neighborhpod of the zero values of TdSl . . ., TI. Thus, he solved Poincare's problem of the representation of Was a function of T-s, . . ., TI in the case of an The simplest expression for W irregular singular point z=O. is given in [5]. Specifically, IV' is represented in the form

(7.13) !.=O

The general representation of W (that is, a representation that holds for all values of the matrices T,,. . TI ) and a study of the analytic properties of the functions W = W (T-s, T-s+l,. ..) T,,TI, . . .) for the case in which 1 = 00 appear in works by the author [29] and [5]. These investigations remain valid in general for a system of the form

:.

*Lappo-bnilevskiy also constructed the corresponding expressionfor Vin the case in which I = rn in Eqs. (7.10). This can be extended to the case in which s = L.O.

56

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

(7.14) In the case of the system ( 7 4 ,the function W i?, as we have seen, a meromorphic function of the matrices U1,. 1 ., 0,. In the case of an irregular singular point z =0, W is an infinite. .)of the system valued function of the parameters (T-s,. . ., To, TI,. (7.14) [5]. For the case in which T-$,. . ., Tl are second-order matrices, the general representation W is of the form [29]

For V we have the general representation (7.12) in the form of a - series. Lappo-Danilevskiy constructed a single-valued factor power Y (z) in (7.11) of the form

8. Formulation of Certain Problems of Linear Systems of Differential Equations with Real Periodic Coefficients In this section we shall consider a system of linear homogeneous differential equations with periodic coefficients

dx = XP (t). dt

Here, ' P ( t ) is a continuous periodic matrix with period 2a: P ( t +2r) = P(t).

FORMULATION O F CERTAIN P R O B L E M S OF LINEAR S Y S T E M S

57

We have already noted that an integral matrix (let us say, normalized at the point t = 0) can be represented by the series (6.1), which converges uniformly in an arbitrary finite interval 0 G t d p . Because of the periodicity of the matrix P ( t ) ,the matrix fc (f 4- 2x1 will also be an integral matrix. To see this, note that, for t = T + 2 r , we obtain from (8.1)

dX (T + 2 x ) dT

= X(T

+ 2 x ) P +2 4 (T

= X(T

+

2X)P(T),

from which the assertion follows. From a familiar property of fundamental systems of solutions of linear differential equations, X (t 2 x ) can be expressed in terms of X-(t) by the equation

+

x (t +2 x ) = vx (t),

(8.3)

where V is a constant matrix with nonzero determinant for t = 0. From this, we obtain

x ( 2 4 =v.

(8.4)

Thus, as we increase t by the period 2 x , the integral matrix X(t) is multiplied on the left by a constant matrix V, which is equal to the value of X ( t ) at f = 2x. In accordance with (6.1)* we have

k=O

where xk(t)is given by Eq. (6.2). Let us define a matrix W by

and a function N by the equation

N (f)

FJ:

e-wt X (f).

(8.7)

The function N (t) is periodic with period 2u. This is true because

N ( t + 2%)= t~w"-"" X (i + 2%) = e-wt

e-h"

VX (t)= N (t).

From this we see that the integral matrix of the system (8.1) normalized at the point t = 0 can be represented in the form X ( t ) = e W ' N (f),

(8.8)

58

LINEAR S Y S T E M S OF ORDINARY D I F F E R E N T I A L EQUATIONS

where the matrix W is defined by (8.6) and N ( t ) is a periodic matrix with period 2%. Let us now pose the foliowing question: When will Wand N ( t ) in Eq. (8.8) be real matrices, assuming that P ( t ) in the system (8.1) is real? Since the integral transformation is real, it follows, a s was shown in Sect. 1, that In V must be real (we recall that we a r e dealing here with the principle value of In V ) if none of the characteristic numbers A,, . . ., A, of the matrix V a r e negative.* In the case in which there are negative characteristic numbers (none of them can be zero since D ( V ) + 0), in accordance with (1.45) we have 1nV = V,+niV,, (8.9) where V , and V,.are real commutative matrices. Here, V, is of the form V , = SL (0.1) s-1.

Here, L(0,l)is a diagonal matrix the elements of which are equal to zero and unity. From this we see that if the matrix V has any negative characteristic numbers, then, in accordance with (8.6) and (8.8),

or

X (t)= eW1t N , (t). Here, W,

1

= -V,

2r

(8.10)

is a real matrix. Therefore,

is also a real matrix since the matrixX ( t ) is real. Let us show that the matrix N , (t)possesses the properties that N,(t.+ 2x)sr; N,(t) but N , (t + 4x) = N , (t).We have

2%

*We see from the equation D (V)= cxp a ( P (I))& that the number of negative charac,O

teristic numbers 'of the matrix V is always even (Lyapunov) Therefore, we can always take In v real. But this real value of In 'b will not be the principle value (cf. Remark L4), nor will it be a regular value.

59

FORMULATION OF C E R T A I N P R O B L E M S OF LINEAR S Y S T E M S

because D (N(t)) + 0 we would otherwise have

which is not the case. Furthermore, we have

N, (f + 4 A )

= N 1( t )e2nlSL(OSI)S-'= N,

( t )SL (exio

, C;L" ) S-'

=

-=N,( t )SL(1, 1)s-1= N, ( t ) .

since SL (1, 1) S-' = 1. Thus, if any of the characteristic numbers of the matrix V a r e negative, the integral matrix X ( t ) , which is normalized at the point t = 0, can be represented in the form (&lo), where W,is a constant real matrix and the matrix N , ( t ) is a real periodic matrix with period 45r. This assertion was proven in essence by Lyapunov in his remarkable dissertation. Here, we have only given it in a somewhat different form and have proven it in a different way. We note also 1

1

that W 1 is not equal to - In V, whereas we had W-1nV 2* 2r We note, however, that*

in (8.8).

Here, if we used formula (1.46), our reasoning would remain the same and the value obtained for N , (f)wouldbe the same (thoughN(t) would be different). We note also that the general value (cf. formula (1.46)) is

In X (4x) = V, + i:iVS,

(8.12)

where V , = SL (2m) S-' and L (2m) is a diagonal matrix with elements equal to even numbers (possibly 0) and where V1 is the principal (and regular) value of In X (4x) if the matrix X (4x) has no negative characteristic numbers. In formula (8.11), W, is a real matrix. However, the matrix X (4x)can have negative characteristic numbers (if the matrix X (2r) had purely imaginary characteristic numbers) since X (4s) = XZ ( 2 x ) . Therefore, InX(4r)may be anirregular value (cf. Remark 1.4). Taking the principle value of I n X (4x)in this case, we again obtain *From (8.10) since N1 (0)= N1 (4x) = 1.

60

LINEAR S Y S T E M 5 O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

In X (4x) = V , (4x)

+

iV, (419.

i ;

where V , (4%)is a matrix that commutes with V , ( 4 4 and has a purely diagonal canonical form with characteristic numbers equal to zero and unity. Then, we may write X ( t ) = exp W,t.N,(t), where 1 Lvz =-VV,(4x),

f l

.V,(t)=exp -VV?(4X)..V,(!I 4x

1;;

We also have 1 W a-- In X (Sic), N,(t f 8q = X2(t).

-

8:

It may happen that we finally end up with

(8.13) where InX(2kzj is a real and principal value* and . V k ( t )is a real periodic function with period 2 k r : (8.14)

9. Solution of the Problems Posed in Section 8 on the Basis of Real Functions Let us look at the problem of finding an expression for Win formula (8.8) or W, in formula (8.10). First of all, we obtain Win the form (8.6) in terms of V, which is a convergent series (8.5). By using the form of Lagrange's polynomial that we derived earlier, we can give 2 representation of W. Suppose, for example, that P ( t ) is a second-order matrix. Then, as we saw in (1.40), we can write 2x W = InV =

u In D - 2 M

4 0 - u2

'1 u (D- U

Vi-

uM+2D-a" 4 0 -an

+

-t I ) t 0 ' 2 0 -u 4 (D-U+ 1)t2+(u-2)t+ 1

0

a=u(V), *Also regular. obviously, such a

k always

D=D(V).

exists.

(9.1)

S O L U T I O N OF T H E PROBLEMS POSED I N S E C T I O N

61

8

In accordance with (8.5), we have m

(V) =

(9.2)

G (xk(2x)).

k=O

If the matrix P(1) is second-order and the characteristic numbers of the matrix v are negative,* then, in formula (8.10), we have -t1

N, ( t ) = e * N (t).

(9.3)

For a system of n equations, we obtain Wfrom formula (1.31) (or, more precisely, from (1.39)). On the other hand, if the matrix Vdoes have negative characteristic num’bers, then, in accordance with (8.11), we have n-I

(9.4) k=O

o r possibly (see (8.13)) 2 x W = In X (2k x) (principal value).

Example. Suppose that we have the system Kdt= X P ( t ) ,

(9.5)

where P ( t ) is a second-order matrix:

+

+

pll (t)= all c o d t %z sin2 t - (a, a2,)sin t cos t , 2 2 2 2 plr (t)= 1 al,cos2 t - a,, sin2t (all -- h) sin t cos t 2 2 2 2 2 1 t t (all q.*)x PZl (4= -. - arl cos2 - - als sin22 2 2

+

+

+

-

+ -

t X sin -cos2

t -!-a22cos2 t Paz (t)= a11 sin22 2

t 2 ’

t . + (a21$- u12)sin -t cos 2

2

*If there i s one negative characteristic number, the second w i l l also be negative.

62

LINEAR S Y S T E M S OF ORDINARY D I F F E R E N T I A L EQUATIONS

The integral matrix X normalized at the point t = 0 is of the form

Here, X(2x)

= -e

:;1: :: II **

=

v.

Let us find the characteristic numbers of the matrix I/. Suppose that

where S is a matrix and A, and A, are real numbers. Then,

and, consequently, the characteristic numbers PIand V are equal: p1 = - exp 2 4 , pLa=

of the matrix

- e x p 2xi, .

If

I

l I l

a11

Ql2

a,,

Q22

0

1

=

s-I,

j.

we have

-exp 2 d - exp 2xh - exp 2nk and , I = p2 =

IJ,

A

exp 2ai.

Thus, the period of thematrix P(1) is in‘this case equal to 2r: and, in the representation of the matrix

SOLUTION OF T H E P R O B L E M S P O S E D IN S E C T I O N 8

X

63

= expAt-N(f),

the matrix

is of period 45: and the characteristic numbers of the integral substitution V a r e negative. Consider now the system

where P(t) is a periodic real nth-order matrix with period 2%. In connection with it, consider the system dX = XP(f)j.. (9.7) dt where A is a real parameter. Let us seek a solution of the system (9.7) in the form

-

-

X = exp At 2 (t),

(9.8) where A is a real constant matrix and Z (t) is a real periodic matrix. We shall use the method expounded in [ 141. In accordance with (6.5), we have, for the system (9.7), m

x (f) = Y Xk (f) isk?

XO = I

.

(9.9)

k=O

This series converges for all finite values of we have, in accordance with (8.5),

A.

For the matrix V ,

(9.10) This series also converges for all finite i . In accordance with formula (8.6), 1

5' '

in^= A ~ A ~ . m 21: k= I

(9.11)

This series converges for sufficiently small values of i. in accordance with a theorem of Lappo-Danilevskiy [l] or Theorem 2.1 of the present book. From the relations

64

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L EQUATIONS

we see that Z ( t ) can be representedinthe form of a series W

z (f) = I +

z,(t)Ak

(9.12)

k= I

If we substitute (9.8) into (9.7) and multiply on the left by exp ( - A t ) , we obtain dZ = ZP).dt

AZ

.

(9.13)

If we substitute (9.11) and (9.12) into this equation and equate coefficients of like powers of A , we obtain dZk -= dt

ZkYlP - A,

-

,

(9.14)

1- I

(9.15)

Since P ( t ) and 2; (t) a r e periodic matrices, we have

1 A, ,= A = L2% J P ((t) t )dt d t,, 2%

(9.17)

0

(9.18)

From this, we see that the matrix Z, is of period 2%. The matrix Z, can be found from the equation

dZ,= ZIP - A , - AJl

.

df

Since the matrix ZIP -AZ, is periodic, we have

(9.19)

S O L U T I O N OF THE P R O B L E M S POSED IN S E C T I O N

8

65

2x

A, =

2.rr

(Z1P-A1ZJ dt

(9.20)

Z, = J (ZIP- AJJ dt - Aft.

(9.21)

0

and t

0

In general, we have

(9.22) I

k-l

(9.23)

Thus, the coefficients of the series (9.11) and (9.12) will be found and these series will converge for sufficiently small values of h . If they converge (see Sect. 2) for h = 1, we obtain the solution of the system (9.6) by setting h = 1 in (9.11) and (9.12). We call the reader's attention to the following fact. We saw above that, in formula (9.8), A and Z(2) may at times be real only under the condition that 2 (f) is of period4x and not 2x. Here, on the other hand, A and 2, in (9.11) and (9.12) a r e always real and Z ( t ) is of period 2n. The apparent contradiction is explained by the fact that, for sufficiently small values of h , the matrix V given by the series (9.11) is always close to the unit matrix. Consequently, its characteristic numbers a r e always close to unity. In other words, for small values of A , the characteristic numbers of the matrix V will not be negative. Therefore, in accordance with what was said earlier, for small'values of A , the matrices A and Z ( t ) in formula (9.8) will be real and Z ( t ) will be of period 2x. These considerations show that the series (9.11) and (9.12) cannot converge for values of A at which V has negative characteristic numbers. In the example of (9.5), the matrix P is such that, for the system (9.7), the series (9.11) and (9.12) diverge for A = 1 since the integral substitution V for the system (9.5) has, as we have seen, negative characteristic numbers and A and Z ( t ) will be real only when the period of Z(f) is 4G

This phenomenon restricts the applicability of the method described. For some systems, the series (9.11) and (9.12) can, of course, converge for h = 1 and may even be entire (as was shown in Sect. 2).

66

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

In [14] it is shown that if a second-order matrix P f t ) is of period w = 1, if I pSl(t)l 4 u l , and if ! p 1 2 ( t ) la %, then the series (9.11) In 2 and (9.12) converge for I A I < -Consequently, when ulu,< Ina2,

V G *

these series also converge for A = 1 (see Sect. 11 of the presentbook). In stability questions we often need only to have a representation of the magnitude of the characteristic number of the matrix A . It follows from Remark 2.2 and Theorem 2.2 that, if we find the invariants of the matrix A by using the series (9.11), that is, if we find D(A)and o (A), then the series representing these quantities will converge in the region 1 )i I < )il, in which there is no more than one branch point* A, of the roots of the characteristic equation of the matrix (9.10). It follows** from this that the series (7.13) and also the series of the invariants a (W) and D (W) of the exponential substitution W (given by (7.13)) constructed by Lappo-Danilevskiy will converge at least in a region of the matrices T-,, ..., T,,where the discriminant of the characteristic equation of the matrix (7.12) does not vanish except when T-, = ... = T, = 0. It should be noted that the invariants of the matrix W coincide with the invariants of the matrix H constructed by Lappo-Danilevskiy in the form

v=l

where

-

p, ... p - - *

~

...

is the Kronecker delta and the

are given by the PI. ..P, formulas of Sect. 7. Here, H is the exponential substitution of the so-called metacanonical integral matrix [ 11 of the system (7.14) 8;O)

~ ( 1 )

Z(2) = Z H Z ( Z ) ,

where

The matrix H is similar to the matrix W. This enables us to find approximative expressions for the characteristic numbers of the matrix A by use of the series (9.11), where the A, a r e given by formulas (9.17), (9.20), and (9.22) for i. =- I ; ifaa(V)-440(V)#Ofor I k l < I , A =#= 0, that is, if Were, we need to take that branch D (Althat is single-valued in a neighborhood of the branch point A, (see Remark 2.2). *Since A and ware defined analogously. In this section we shall establish the exact equation showing the relationship between A and Fy!

SOLUTION OF THE PROBLEMS POSED I N SECTION

8

67

m o"(v)

- 4 exp L Ja

0

0

for IAI..; I , A + O . The characteristic equation for the second-order matrix A is of the form ka-a(A)L

+ D(A)= 0.

(9.25)

On the basis of Jacobi's formula,

D (X(2%))= D (exp 2a A) D (2 (2x)) = 2%

= e x p a ( 2 a A ) . ~ ( ~ ( 2 x= ) )e x p

Ja(P)dt. n

Here, the matrix Z ( t ) t 4 , which is periodic with period 2x, is equal to I because X ( 0 ) = 1 (see (9.9)). Therefore, D(Z(2r)) = 1. Consequently, a

expa(2xA)= e x p J a ( P ) d t . 0

(9.26)

We note that the following cases may arise: Case I.

Then, the characteristic numbers of the matrix A are positive. Case IL a(A)aO,

D(A) < O

o r a ( A ) < O , D(A)
aa(A)-4D(A)>0.

(9.28)

In this case only one characteristic number is positive, the other negative. Case 111. a(A)

D(A)> 0; a4(A)-40(A)>

0.

(9.29)

68

LINEAR S Y S T E M S O F O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S

Here, the characteristc numbers are negative. Case IV.

Here, the characteristic numbers a r e purely imaginary. Case V. G~

(A) - 4 0 ( A ) = 0

(9.31)

Here, the characteristic numbers coincide and they vanish for 3

(A) = 0

Case VI. O2(A)

- - 4 0 ( A )
.

(9.32)

Here, the characteristic numbers a r e complex with real part equal to-

5

(A) 2

.

Later, we shall bring up certain cases in which the real nth-order matrix A and all its characteristic numbers a r e purely imaginary (see Sects. 1 6 and 44).

10. Expansion of an Exponential Matrix in a Series of Powers of a Parameter Consider now a system of the form dX dt

m

(10.1) k=O

Here, the Pk(t) a r e nth-order matrices that are continuous and periodic with period 2x. The series (10.1) converges for I e l < r . In accordance with Theorem 6.1, the integral matrix of Eq. (10.1) normalized at the point t = 0 can be expressed as a series of the form W

x ( t ) = x X k ( t ) 6 k , xO(o)cI,

xk(o)=op

k > 1,

!

(6.14)

k=O

that converges for I E ~< r. Consequently, we have the integral substitution in the form

69

EXPANSION OF AN EXPONENTIAL MATRIX

v (e) = x (2%)=

~k(2s)Ck.

'€1 < r

.

(10.2)

k=O

It was shown in Sect. 8 that the integral matrix (6.14) can be represented in the form (8.8)

X(t,

s)=

e s p ( W ( r ) t ) . Z ( t ,H ) ,

(10.3)

where W is the real constant matrix defined by Eq. (8.6), where 2x W (E)

=:

In V (e)

(10.4)

(principal and regular value), and where Z ( t , E ) is a periodic real matrix with period 2x if the matrix V ( e ) has no negative characteristic numbers. On the other hand, if the matrix V (e)does have negative characteristic numbers, then W ( E ) , as defined by Eq. (10.4), will not be real* (cf. (8.9)). If we wish W to be real in (10.3), we need to take (see (8.11)) 4a W (z)

= In X (44.

(10.5)

Here, N ( t , C) will be of period 4n. But here, lnX(4x)may be a nonprincipal (and nonregular) value if we wish this quantity to be real. However, we can always take (see (8.13)) (10.6) ) be a real, (where k is a positive integer), so that In X ( 2 . k ~will regular, and principal value. Here, Z(t, E ) in the formula

X(f,

E)

= e x p W ( e ) t . Z k ( t ,e)

(10.7)

will be periodic with period 2k T . On the basis of the theorems in Sect. 2, W ( E and ) Z ( t , E) can be represented in the form of series in positive powers of C [ if In X (2.k x ) in (10.6) is the principal (or a regular) value].** Let us suppose that the integral matrix X , ( t ) of the limiting system

dX,= xo(t)Po (t), x,(0)= I dt

(10.8)

*If we take the principal or regular value of In V (r). *Wowever, we can sometimes choose an irregular vdue for In x ( 2 k x ) (see Remark 2.3).

70

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

has been obtained in the form

X,(t) = exp (At).2,(t), 2, (0) = I, 2, (t 42r) = 2,(1).

(10.9)

If the matrix V, = X o (2x1has no negative characteristic numbers, this is possible. In this case, the characteristic numbers of the matrix 'V(e) obviously will not be negative for small values of E and W (E) will be real in formula (10.3), but Z(t, s)will be of period 2x. The matrix A, in (10.9) is given by the equation (see (8.6))

2x A,

= In X,

(24,

(10.10)

and we may assume here that In X0(2n)is the principal value. Then, the characteristic numbers a, (for k = 1, ..., nj of the matrix& obviously satisfy the condition uR-. a, f im (where m is an integer), o r In X , ( 2 d will be a regular value. According to Theorem 2.1, we have W ( e ) i n the form of a series 0

W(Z) =

VIV,E~, 2 c w 0 = I ~ x , ( ~ A ) 2x4,. = Y

(10.11)

k=O

which converges at least (see Remark 2.1) in a circle I P I c R < r, in which there are no zeros of the discriminant A(E) of the characteristic equation of the matrix (10.2) An

+V1(E))."-I

+...+V,,-l(~)k+V,,(e)=O

(10.12)

for E # 0. Here, the V , @)areseries that converge in the same region as the series (10.2). If the characteristic numbers of the matrix X , ( 2 x ) a r e not only nonnegative but also distinct, the series (10.11) also converges in the circle I s I < R < r, in which there is no more than one zero of the discriminant A(e). Here, lnV(a)must be taken in such a way that it will be single-valued in a neighborhood of the point E =r E* at which the discriminant 4 ( E ) vanishes. In the region of convergence of the series (lO.ll), the matrices W (z) and 2 (t, E ) remain real and 2 (t, E) is of period 25t. Let us suppose now that the matrix X, ( 2 4 also has negative characteristic numbers. Then, we again find the series (lO.ll), but it will not yield a real function if we require that 2 (t + 217, e) = 2 (t, E). On the other hand, if the matrix X, (4x)=X; (2;r)has no negative characteristic numbers and we assume that the function Z ( t , a ) i s of period 4x, then we can again find W (s) in the form (10.11) and it will be real. However, in this last case, the function 2, (t) in (10.9) will also be of period 4n and W(Z) can be found in accordance with (10.5), where X (4x)has no negative characteristic numbers. Whenever the series (10.11) converges, so will the series

71

E X P A N S I O N OF A N E X P O N E N T I A L M A T R I X

(10 13) k=O

which is obvious from (6.14)and (10.3). A particular case of the system (10.1) is the one in which the matrix P,, (t)= Po is constant. Then, X o( t ) = exp P o t . If the charac-

teristic numbers p"k (for k = 1, ..., n) of the matrix Po a r e such that p"4 -q + im (where m is an integer), we may, in accordance with (10.9), assume that &, = Po and 2, (t)= I. Let us suppose now that* p"k = im, but that the matrix exp 221Po has no negative characteristic numbers. Then, 2x Po= In exp 2x Po is not a regular value. But we can write

-e

Xo (4 = exp

&fzo(4,

(10.14)

where 2xAe == lnexp 2xP0 is the principal (and regular) value and Zo ( t ) = exp (- &t) exp Pot.

Here, & is a real matrix (since the matrix exp 2%P,, has no negative characteristic numbers) and 2,( t ) is real and periodic with period 221. To see this, note that Zo(t +2x) = exp [-2x&].exp2aPo.exp t= exp [-&t].exp

A,,tl-espPet =

Pot,

since exp t--22rAo].exp[2xP0]= = exp I- 2%Aol.exp [Inexp 2rP0] = 1.

because 2%4= In exp 2 x Po and the matrices em[- &t,] and exp Pets commute.** Thus, 2r& in (10.14) is the principal value of In e x p 2 x P o . We could have proceeded in this case in a different manner. Specifically, since 2%Po = In exp 2x Po is not a regular value, we may write (see (1.46))

*.2x A. = In exp 2x Pois theprincipal (andregular)value. Therefore, it is a polynomial (Lagrange's) in Po.

72

L I N E A R S Y S T E M S OF O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S

Po = Al

1 + iA, = lnexp2aP0, 2a

(10.15)

where the matrices Al and A, commute, where 2aAl = Inexp2aA1 is the principal (and regular) value, where A, iu a real matrix, and where the characteristic numbers of the matrix A, are equal to 0 and/or other integers. Then, X o ( t ) = expP,t where the matrix

= expAlf.expiA,f = expAlt.Zo(t),

2,

(t)= exp i Ad

(10.151)

is periodic with period 2r; and real since the matrices Po and Al a r e real. We have obtained (10.14). If the matrix exp 2xP, has negative* characteristic numbers, then the principal (and regular) value In exp 2xP0 will, in accordance with (1.45), be complex; also, A, in (10.15) has characteristic numbers equal to 1/2 if they correspond to negative characteristic numbers. Therefore, the function (10.15 ) is of period 4%. Consequently, we have exp 4aP0 = exp 4%A, instead of exp 2a Po = exp 2r A,. But if the matrix exp 4 r Po does not have negative characteristic numbers, we may write

(10.16) exp Pot = expBot-Zo( t ) , where, for 4xE0, we must take the principal value of In exp 4 x P , ( i t will be regular and real) o r follow the second procedure, that is, use (10.15). Specifically, we need to write Xo( t )= exp Po t = exp A,t.exp iA& where the second factor is a periodic function with period 4a and 4xAl = lnexp 4 x 4 (the principal value) is regular and real (if the matrix exp 4aP0 has no negative characteristic numbers). If we do not require W , (and along with it W )to be real in (lO.ll), we can always (that is, even when the matrix exp 4nPo has negative characteristic numbers) take for 4aA0 in (10.14) the principal and regular value of In exp 4 r Po. However, in many cases, it is necess a r y to take A, real. A,, can always be chosen real on the basis of (8.13). Here,Z,(f)wili be of period 2%. Thus, in all cases, we have (10.3), where W ( ~ ) a n d Z ( ta)are , real and representable in the form oftheseries (10.11) and (10.13). Note the following facts: (1) if the matrix Xo(2n)does not have negative characteristic numbers, then A, is given by equation

73

E X P A N S I O N OF A N E X P O N E N T I A L M A T R I X

(10.10) (where the principal value of In X , ( 2 4 is meant) and the matrix Z,(t) is real andperiodic withperiod 2x; (2) on the other hand, if the matrix X0(2x) has negative characteristic numbers but the matrix X 0 ( 4 r ) does not, then the real matrix A, is determined by the principal value of 4% A, = In X , ( 4 4 ,

(10.17)

but the matrix Z,(t) will be real and periodic with period 4x. Suppose that Po is a constant matrix such that exp 2rP, has no = mi. negative characteristic numbers and suppose that Pk--*q Then, we may proceed a s follows: If we introduce into (10.15)* a new unknown matrix** Y defined by X = Y exp iAd and substitute it into (lO.l), we obtain YiA,exp iA,t

dY e x p i A t = Y exp id&. (Po + Pl (t) + ...I. +dt E

Since the matrices P, = A, + jA, and A, commute with Al, we obtain by multiplying the above equation on the right by exp (- A&) dI' = Y [A, + e x p id&(P1( t ) e + ...) .exp ( --iA&)] . dt Since the matrix exp iAd is periodic with period 2 x , we have

..

where the Fk ( t ) (for k = 1 , 2 , .) are periodic with period 2 x and lnexp 2 x 4 = 2 x 4 is the principal value. Instead of making the transformation X = Y exp id&,we could have made the transformation X = Yexp(-dot)-expPot,

where 2 x A 0 = In exp 2 x Po is the principal value. From what was said above, this amounts to the same thing. An analogous transformation can be made in the case in which the matrix exp 2 x Po has negative characteristic numbers but the matrix exp 4 x P , does not. Specifically, we introduce Y defined by X

= Y e x p (--

A,t).exp Pot,

*We recall that, since the matrix exp 271Po has no negative characteristic numbers, i t follows that the characteristic numbers of the matrix A2 in (10.15) are integers.

*+Such a transformation was first used in [7], where the question of series expansion of the characteristic numbers of the matrix (10.11) was studied. See also [30].

74

LINEAR SYSTEMS O F ORDINARY D I F F E R E N T I A L EQUATIONS

where the real matrix 4 r A, = l n e x p 4 x P 0 is the principal and regular value. Or, we may write in this case Xo( t ) = e x p A,t.exp (- Af).exp Pot =: e x p Ad.2, ( f ) ,

where 4 r A, = In e x p 4x Po is the principal value and the matrix Zo(t)= exp(--A$).expP,f

is periodic with period 4%. Specifically,

2,(t

+ 4 4 = e x p (-. 4A0 - e x p 4a P,.exp ( x)

= e x p ( --

-

-

A, t) e x p P,t

=

4 )e x p Pot,

since e x p (-4xA,).exp4xP0 = = exp(-44nA,)-exp[lnexp4rP0] = I. Now, in the expansions (10.11) and (10.13), we need to set respectivel y 4x Wo= 4%A,

= In e x p X,

(4x) = In e x p 4%Pa

and Z , ( t ) = e x p ( - - &t).expP,t.

However, in this case (see (1.44), (1.45), and (1.46)) we shall have Po = Al iA,, where the real matrix Al commutes with the matrlx A,, which has a purely diagonal canonical form, the characteristic numbers of which a r e equal to integers and numbers of the form k l/*, where k is an integer. Consequently,

+

+

e x p Pot = e x p A d . e x p i A j -

Therefore, we may either set

W,= A, and 2, (t) = e x p iAd in (10.11) and (10.13) or introduce Y as defined by X=Yexp

iA&

75

DETERMINATION OF T H E COEFFICIENTS IN T H E SERIES EXPANSION

= im, then, in accordance We note also that if we have P i with Theorem 2.3, Remark 2.3, and the example following it,

in certain cases we can set 2a Wo -- In Xo(2x) =r In exp 2x Po

+ So[2m, x i lp,, ..., 2m, x i I,

] S;],

in (lO.ll), where In exp 2x Po is the principal value (see (2.11)). But here, Womay be complex.

11. Determination of the Coefficients in the Series Expansion of an Exponential Matrix Let us find the coefficients in theexpansions (10.11) and (10.13).

If we substitute (10.3) into (10.1) and multiply on the left by exp (- W ( 0 ) t), we obtain

(11.1) If wesubstitute the expansions (10.11) and (10.13) into this and equate coefficients of like powers of E , we obtain

k- I

k

v=

1

v=

0

Let us assume that Xo (2r)in (10.10) has no negative characteristic numbers. Then, (11.3) 2 0 (0)= 1, ( t f 2r) = z k ( t ) and, in accordance with (lO.ll), 2%W,,= 2%

= In Xo (2n).

Let us write the system (11.2) more briefly in the form

where the periodic matrix Fk(t) with period 2~ is of the form k

k-I

(11.5)

76

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

The general solution of the system

dY = YPo (t) dt

doY

(11.6)

is

Y

(1.17)

= exp(- Aot).CXo( f ) ,

where C is an arbitrary constant matrix and Xo(t)is the solution of Eq. (10.8) given by formula (10.14). We may seek a solution of the system (11.4) in the form Z,(t)= exp(-Aot)-C(t)X,,(f).

(11.8)

If we substitute this into (11.4), on the basis of (10.8) and (10.14) we obtain

dC-- e x p (dot)-[ Fk (t) - W,Zo (l)l ~

(t).exp (- - Aot).

T I

dt

From this, we obtain t

1 it

C (t) = e x p dotIFk(!) 2 ';

(t) - W J . e x p (- dot)dt .

If we substitute this into (11.8), we obtain zk

(t) = CX p. (- - dot)

ex p (dot) [Fk (t) 2;'

(t) -

w,]><

0

X exp (- dot) dt .e x p dot.2, ( t ) .

(11.9)

If we impose* on . Z , ( f ) the requirement that

z, ( t + 2 r )

L :

Z,(t), 1 -- 0 ,

we obtain 2%

f C X P dof [ F k (t)2 ~(1t )- W,j . e x p (-/lot)

dt

y-

0,

(11.10)

Q from which we find W,.

*Essentially, Artem'yev was the first to use this method for determining Zk (f)and

wk [9]. Independently of him, this method appeared afterwards in [I41 (October, 1942)

and [101 (in this case, for systems of an even more general form).

DETERMINATION OF THE COEFFICIENTS IN THE SERIES EXPANBION

77

Formula (11.9) gives a solution of Eq. (11.4) that satisfies the condition z k (0) = 0. We obviously have a solution satisfying the condition ,Z, (0) go, in the form t

2,

exp (-4,f)df.exp$t.$

(f)

+ exp ( -'Aof)*Zk (0) Xo(t).

(11.11)

From the condition that

(11.11 1) (0)= z&(2x) we find a W,, such that z k (f) will be periodic for arbitrary z k (0). Conversely, we may choose Wk arbitrarily, for example, zk

and find 2, (0)from (11.111). Then Z k ( t ) will again be periodic with period 2 x . For Zn (t) = I and 2, (0) = 0, we find the matrixwkfrom the equation

r

2s

exp d o t .[F,(f) - W k ] exp ( -dot) dt

: :

0

.

(11.11 2)

i,

Let us examine in greater detail the particular case in which the matrix Po (f) = Po in the system (10.1) is constant. Suppose that the matrix exp 2x Po has no negative characteristic numbers and that 2 r P 0 = In exp2xPnis the principal value. Thus, we may set Wo = Po and 2, (f) = 1. We shall study in greater detail the equation (11.4), in which we now set 2, ( t ) = 1 and Po (f)==Po-do.We denote by km the characteristic numbers of the matrix Po. Let J be the canonical matrix: J = S-lP,S and suppose that IV = S-lW,S, Z = S-lZ,S and F = S-'FkS. . If we multiply Eq. (11.4) on the left byS-Iand on the right by S , we obtain an equation of the form (11.4), where we only need to replace Po with J . WkWith W ,Zk with 2, and Fk With F: dZ

- = Z J - J Z + , F (t) - W.

(11.12)

lit

If in general we denote by 6kl the elements of the matrix B, we can write (11.12) in an expanded form:

78

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

(11.13) -wk,l

Here, the numbers a,, a r e equal to zero and unity. They are equal to zero if the elementary divisor of the root A,, is a prime o r if Z k . , , at the vertex of the Jordan cell on the principal diagonal corresponding to the root A,, is defined. If. A,#A,+mi

the

Zk,

(m=0, f 1 ,

+ 2 . +...),

are found from (11.13) to be periodic for arbitrary*

Wk, .:

1

Zk, 1

= e x p [( A,-

Ak)

t ] .[C t [ f (t)eXp (- (Al - -. i.n) t )dt],

(11.14)

b

where

f ( t ) = - ' k Zk-1.

1

+

1+1

-k F k . I - w k p 1

is a periodic function and the constant c is determined by the equation 21:

( e x p [ ( A 1 - & ) 2 x ]- 1)c-

~ f ( t ) e x p ( - ( ( E , - A & ) f ) d t =0. (11.15) b

,

If we set c = 0 and find W k , from Eq. (11.15), then we obtain z k , ( t ) periodic and Z k , (0) = 0. We can take c arbitrarily in (11.14) and we can take Wk.1 so that Z k , (f) will be periodic. Finally, we can take

,

and we can choose c so that zk, I will be periodic. For Ah = I., = A, the element 4. corresponding to the vertex of the Jordan cell of the nonprincipal diagonal corresponding to the root ). is found from Eq. (11.13)with 6, = 6, = 0 : that is,

where

Ck,

is an arbitrary constant. 2K

[

I F,,,df +Including the case in which wk, I = 2x.

0

.

DETERMINATION O F T H E COEFFICIENTS IN T H E SERIES EXPANSION

For ckS1=O, the function

is periodic and

L-

9v

U.

79

The

remaining elements zk, of this square are found from Eq. (11.13) in the form (for h& = A l ) : I

where c,, is an arbitrary constant. For c,, = 0, the element z,.! is periodic with average value equal to 0. Thus, in (11.12), we can always take* Wav and can find a C such that Z ( t ) is periodic with period 2x o r we can set** C = 0 and can find a W so that Z ( t ) will be periodic. Or,finally, we can take 2 (0) arbitrarily (choosing C accordingly) and can find a W such that Z (t) will be periodic. This Wwill be unique. However, it should be noted that, when we set Z, (0) = 0 in (11.4) (that is, when we set c k = 0) and determine W, in such a way that the Z, (t)will be periodic with period 2 x , we obtain convergent series (10.11) and (10.13) (in the region mentioned in Theorem 2.1 and Remark 2.2). If X ( t , €)isa normalized integral matrix, that Remark 11.1. is, if X (0, e) = I o r Z,(O) = I and Z, (0) = 0 (fork > l), then, as was shown by Artem'yev [ 91,

Y

=

CA,Ek

. X(t,

E)

R=O

is an integral matrix***[9,7,12,13] such that Z, (0) = A, (for k > l ) , m

where A,

= I.

We assume that the series A

==

x A , ~ kconverges. k=O

The exponential matrix mof the integral matrix

is determined by the equation

If 2, (0) = 0 (for k > I), the series for Wand Z will, in the case of a periodic matrix P ( t ) , converge when the conditions stated above

*%at is, Z (0) = 0. ***The analysis given below was carried out in [I31

80

LINEAR SYSTEMS O F ORDINARY DIFFERENTIAL EQUATIONS

with regard to Po a r e satisfied. However, if we choose another value for Z , (0) (fixing w k in some way as we do so), there is no guarantee that the series (10.11) and (10.13) will converge. If

we take -4 -=-

9

Ak

k=O

sk

.that is, a polynomial in a, then the s e r i e s cor-

-

responding to W (E) will converge and z k ( 0 ) = 0 for k > m $. 1.In other words, if we take for wk (where k = 1, ..., m ) the mean* values of the corresponding functions and then take wk (fork m t 1. ...) corresponding to the values of z k (0) = o (for k - -k 1, ...), then the series (10.11) and (10.13) will still converge. However, let us suppose that nonzero values a r e chosen for Z k (0) (for k = 1, 2, ...) (and that the W, a r e therefore unique) or,forexample, let us choose for all the W , (fork = 1.2....) the corresponding average values (in which case, the & ( O ) will be unique and, in general, nonzero). Then, the series 2

m

k=O

may diverge. The series W ( Z )may also diverge. However, the invariants of the characteristic equation of the matrix (E) will, by virtue of the' formula (*), coincide with the invariants of the characteristic equation of the matrix ! V ( E )and therefore they will be convergent series in terms of c.. However, 111

m

l:=O

k=O

will now obviously be a poor approximation** of the value of the integral mstrix Y. In fact, there may not be an integral matrix Y with such value of & (C) (if the s e r i e s (**) diverges). Thus, we have obtained the following rule. Suppose that 2r Po = In exp 2 x Po (principal value) and suppose that the matrix exp 2* Po has no negative characteristic numbers. Then, we have the integral matrix X of Eq. (10.1) that is normalized at the point t =; 0 in the form (10.3) *In [lo], U7,+ was originally defined as the average value, but then convergence of the series (10.11) was not required See Sect. 20 of the present book **As we have noted, these are s u l l segments of convergent series if we takeZh (O)-O (where&I= tn -:-I , m $- 2, ...).

DETERMINATION O F T H E COEFFICIENTS IN T H E SERIES EXPANSION

81

where W ( E ) is a real constant matrix that can be represented by the convergent s e r i e s

(10.11) k=O

and a real periodic matrix Z (t. vergent series

e)

that can be represented by a con-

ca

Z(t,

E)

=

c Z,(t)

Ek.

(10.13)

k=O

Here, Wo = Po, Zo = I, and W,and Zk(t)(fork >, 1) a r e obtained from Eqs. (11.4) (or with the aid of (11.12)) under the condition that the &(t) must be periodic with period 2n and that Z,(O) = 0. Suppose that 2aPo = Inexp2aPo is the principal value but that the matrix exp 2nP0 has negative characteristic numbers. Then, the matrix ekp 4nP0 does not have negative characteristic numbers and, if 4x Po = In exp 4nP0

is the principal value, then, in (lO.ll),we should take 4xW0 = Inexp45:Po = 4xP0, W o= Po,

and find the W, (fork z 1) from (10.5) (or with the aid of (11.12)) under the condition that the 2, (t)are periodic with period 4n. On the other hand, if 2xP0 is not the principal value of In exp 2cP0, we need to proceed as was shown above (see Sect. 10). We now note that the expansions (10.11) and (10.13) represent the matrices W (2) and Z (t, 2) in the above-mentioned neighborhood of := 0. Formulas (1.40), (1.41), and (1.39) enable us to represent\V(E) throughout the entire region in which E exists (thus providing ourselves with a representation of Z ( f , E ) for all possible values of :), where the characteristic numbers of the matrix X(27, o) are nonnegative. On the other hand, if the matrix X (2z, E ) has negative characteristic numbers, we again obtain (10.3), (10.11) and (10.13) with the aid of formula (1.39) for all values of E at which the coefficient matrix in (10.9) is given, butZ(t, 2) will be of period4a and W ( 5 ) can be constructed in accordance with (1.31) on the basis of the formula 4aW(:) = lnX(4n,:).

(11.18)

82

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Here, it is assumed that the matrix X ( 4 x . E ) has no negative characteristic numbers. For those values of E at which the charac, correspondingly, X (4x, E ) teristic numbers of the matrix X ( ~ T EC) or, coincide, we need to take the limiting value of Lagrange's formula. We can obtain this limiting form of Lagrange's formula both from (1.31) and with the aid of the minimum polynomial (1.7), where it is shown below formula (1.7) how to find vk (A,, ... , I,,,). Also, a way of finding rpk (Il,... , 1,) is given in Sect. 2, based on (2.4). Sometimes it is possible to find a boundfor the radius of convergence of the series (10.11) on the basis of the bound for the coefficient matrix of Eq. (10.1). Thus, for example, (see p. 88 of [14])for the system

dX-- X P ( t ) ,

dt where the characteristic numbers p k ( t ) of the matrix P(t)satisfy the inequality 1 &(t)I 4 a, the series (10.11) will definitely converge* In 2 =E, for I E I <2x a -

. For 2r a

t In 2 , we have the region of convergence

of the series (10.11): l a l < ~ , > I . On the other hand, if

then the series (10.11) will converge for In 2 l o ' < 25;7/(11121

For 2x=
we have I E I < E ~ > 1.

12. Approximate Integration of Equation (10.1) For Eq. (lO.l), we have obtained** a solution of the form .?his follows from the fact that the series In X (2=, c)converges i f the maximum absolute value of any of the characteristic numbers of the matrix X (2%. I ) I I is less than unity. This maximum absolute value does not exceed the maximum absolute value of the characteristic numbers of1 exp 2%P I I ,where I P (1, L ) 1 6 Pand the matrix Pis constant. Let a denote the characteristic number of the matrix Pwith the greatest absolute value. ?he< the characteristic number of the matrix I exp 2x1 p I lwith greatest absolute value does not exceed (exp 2xr u I). From this, we have the region of convergence In 2 I 1< since, here, (exp 2x a E - 1) < 1.

-

-

-

2x u

-

...

*.?he continuity of the characteristic numbers of the matrix I at the point = 0, , c, = 0 is proven i n [a] for the case in H3lich the coefficient matrix of the linear sys,..., t,)is continuous at the pointEl = ... = c m = Oand is periodic with respect temp(t, to t.

APPROXIMATE INTEGRATION OF EQUATION (10. I )

-

X (t, 8) = exp (W ( E ) t ) Z (t, el,.

a3

(10.3)

where W ( E )and Z(t, C) a r e given in the form of series (12.1)

4)

z (t, e) = 2’ z, (t)E L ,

(12.2)

k=O

that converge for We define

lei

<: R. (12.3) rn zm

( t ,E,

zk(t) Ek

= k=O

. (12.4)

Then (see [4]),

will be an approximate value of the solution (10.3). Here, we construct W(E) and Z ( t , E ) in such a way that they will be real. It may happen that the given system

(12.7) so that, for E = 0, we obtain (12.8)

the integral matrix of which is known: (12.9)

84

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

and, for E = 1, we obtain (12.6). Then, we have the integral matrix of the system (12.7) in the form of a series (12.10)

which converges for all finite E. Now, we can find X (t, E ) in the form (10.3). If the series (12.1) and (12.2) converge for E= 1, we shall obtain an approximate solution in the form (12.5). As we have seen, it is sometimes possible to establish the convergence of the series (12.1) and (12.2) on the basis of bounds of the elements of the matrix P ( t ) . In particular, these series will converge for ~ = l if (see Sect. 9)P(t)isasecond-order matrix (see p. 86 of [14])and if 2x

( ~ ( 2 rE ,) )

0%

-4 exp J u

(p0(t)j

+

a (p1(t)) E]

dt

+o

for 0 < 1.1 4 1 since, when this condition holds in the circle 1~141, the matrix X ( 2 q E ) has no multiple characteristic numbers. Here, we assume that the matrix Xo(2n)has no negative characteristic numbers. For Pa(t), we may take one of the matrices for which the integral matrix has been found in theprecedingsections. O r , in the case of a second-order matrix P ( t ) , wemaywrite the system (12.6) in the form (12.11) In particular, it is convenient to introduce this system in the case in which the average integral values (over the period) of the functions pll (f) and pea@)are nonzero but those of the functions pla and 2 1 are zero. In general, for any matrix P,, ( t )such that the integral matrix Y of the system -= dY dt

YP, ( t )

(12.12)

is known, we obtain the system (12.7) in the form (see (31.30))

that is, here, P l ( t ) = P ( t ) -Po(&

THE CASE OF CONSTANT

Po(t), Pl(t), ..., Pm(t)

85

For Po(t) it is convenient to take a matrix such that the matrix Y is already a good approximation of the matrix X. For example, for Y , we may take the matrix

Y = X m ( t , 1) = esp

2

A k f '

2,. (t, I),

(12.14)

k=O

where X m ( t , 1) is the approximate value of the integral matrix of a system of the form (12.7) o r that of a system of the form

dX

-=

dt

XP (t)E for

E

=1

(12.15)

If we choose Y this way in advance, we easily find the corresponding periodic solution

or, we may choose Po(t)in such a way that P (t)-Po (t)is small over a large region of variation of t. If we choose Po(t) in one way o r another and then evaluate some terms of the seriesforW(o), andZ(t, E)for E = 1,we evaluate a partial sum of the series representingw (E)andZ(t, E ) for the system (12.15) for E = 1 if these series converge and, of course, if the series representing W ( e )andZ(t, c)for the system (12.13) converge. On the other hand, if they diverge but if the series representing W (E) and Z(t. €)for the system (12.13) converge, then the approximate values of W m (E) and Z m ( t , ~for ) (12.13) provide, for E = 1 approximatevalues of W ( E ) and Z (t, E ) for their analytic continuation corresponding to the system (12.15) with E = 1.

13. The Case in Which P,(t), Pl(f), ..., Pm(t)in Equation (10.1) Are Constants Let us now consider the particular case of the system (10.1) (13.1)

where (see [31]) the matrices Po( f ) , PI (t),..., Pm (t) a r e constants. Let us suppose also that the matrix Po has no characteristic numbers P k , p t such that P k - p f = Y i (where Y is an integer) or, in other words,

86

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

2 x P 0 = lnexp2xP0 is the principal and regular value. Consequently, in the expansions (10.9) and (lO.ll), we can set W o = Po and Zo = 1. To determine w k and z k (for k > l ; ) , we have equations (11.2):

We need to find Z k that are periodic with period 2 x such that We see from (13.2) that& = 0 and w k = P k (for k = I , .... m). Thus, we have

Z k ( 0 )= 0.

(13.3) k=O

k=m+l

z ( t , e) = I +

C Z&(t)

m

ek

.

(13.4)

k=m+l

We may also proceed this way: Let us write the system (13.1) in the form

rn

m

and let us look at the auxiliary system

-dX df

-

x IP (4 + PI (t,41k

(13.6)

where k is a parameter. We seek a solution of this system in the form

X

k= I zk

(h) t . Z ( f , h),

A; k k , Z ( f , h) = I

A (A) =:

To determine

= exp A

+2Z k ( f )A' . k=l

and A k , we have the equations

(13.7) (13.8)

T H E CASE OF CONSTANT

dt

- P (e)

a7

k-1

-dZk - Zk-1 [ P ( e ) + PI (t,e)] dt -dZ1

(t)I ...( Prn(t)

Po(&

- Aa -

2 A&-,,

(13.9)

l=I

+ PI(t,e) - A,.

From them, we obtain t

?c

A , = - J1P , ( f ,

2X

e ) d t + P ( E ) , Z l = ~ P l ( ~ , ~ ) d ft - - ~ P l (e t) d, t .

2r

2r 0

0

0

We see that 2, (t),Z,(t), ...,andalsoA,, As,... are infinitesimals of order m I with respect to E . If it turns out that the matrix P (e), for small E $; 0, has distinct characteristic numbers whose order of smallness

+

OD

Q m,

then the matrix A ( E )

AL 1k, will not have multiple charac-

= k=L

if 11 J .s I (or, in general, if p of this matrix will be for small values of E from the equation -

teristic numbers for small values of

E

1 k I < M ) since the characteristic numbers determined up to

~m

IXP(e)-ppl=O,

IP(e)-pJ=O.

p=pX-l,

But under these conditions, the series (13.8) will, in accordance with Theorem 2.1 and Remark 2.2 following it, converge also for 1 = 1. Consequently, for small values of E , we have for the system

(13.5)

Here, Zk (t, E) = Zk (t) in accordance with (13.8). In our reasoning here, P l ( t , e) is an infinitesimal of order ~ m f l and is not necessarily a series in terms of e. Of course, we can also proceed in the following way: If we write Eq. (13.5) in the form

we find

X

= exp

(W(X)t).Z (t, 1).

(13.12)

88

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L EQUATIONS

If we now assume that the matrix P ( E )does not have characteristic

numbers

where

v

P k (E), p t ( 2 ) . such

that

is an integer, we may set

w

P

.-_ ( E ) ,

0

2, (t) :--I.

(13.15)

We find Zh ( t )and W kfrom the equations

(13.16)

3= Z,P dt

( E ) -- -

P (2) 2, + PI(t, E )

w,.

(13.17)

If we define W , from (13.17), we find, on the basis of (11.112),

To determine the elements of the matrixWl, we obtain from this a system of linear equations with nonzero determinant since the elements of the matrixW,are, as we have seen, uniquely determined. The elements of the matrix Pl (t, e ) are infinitesimals of order m 1, o r greater with respect to E . Therefore, theelements of the matrix W , are also infinitesimals of order m + 1 o r greater with respect to E , and the same is true of the elements of the matrix 2,(f). We see from Eq. (13.16) that the elements of the matrices wk and z k ( t ) a r e in general infinitesimals of order k(m + 1)or greater with respect toe. Consequently, we have

+

m

w(1) =

w k (m+l) ( E )

hk -k P(€),

(13.18)

k= I

where the index k(m+ 1) indicates the order of smallness with respect to E . This series also converges for 1 k 1 4 I and for sufficiently small e if the matrix P ( E ) has distinct characteristic

THE CASE OF CONSTANT

P0

AND PERIODIC

exp P0 t

89

numbers for z + 0 that are of an order of smallness ..,;;. m. Here, for. the system (13.5), we have W (z) in the form of a convergent series 00

\\!'(s) = P (s)

+ ~Wk <m+ll (a),

(13.19)

k=l

and Z (t) in the form Z (t)

"'

=~

Zk<m+I) (e, t),

Z0

= I.

(13.20)

k=O

In these considerations, again P 1 (t, e)is an infinitesimal of order no greater than m + I with respect to s and is not necessarily holomorphic in the region Ia I 4 r. We note again that the series (13.19) and (13.20) converge for those e for which the characteristic numbers of the matrix W (I.) (see (13.18)) Pk (e)= P~m) (e)+ pkm+I) (e) + pkm+l) (e, 1.)

are distinct for all 1 I. I ~ I. Here, Pkm+ll (e) is an infinitesimal of order m + l generated by P (e) and Pkm+l! (e, 1-) is a quantity generated by the expression

..

~

wk

<m+l) (e)

).k.

k=l

The quantity p~m) (e), although infinitesimal, of of order ,;;_ rn. If the p~m> (e) (for k = I, ... , n) are distinct and if iP(k)(e) />IP1m+ll(e)/+1Pkm+ll(e,A)j, 1'-l<.l,

then the series (13.19) and (13.20) converge (for small s ).

14. The Case in Which P0 is Constant and exp Pot is a Periodic Matrix in Equation (10.1) Consider the system

d~ = XP (t),

P (t)

"'

= ~ Pk (t) ek = P0 + R (t, e), k=O

(14.1)

90

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

where the matrices P k are periodic withperiod2 xand Po is constant. Suppose that the matrix exp Pot is periodic with period2r. Then, before finding the series (10.9) and (lO.ll), we may proceed as follows. We introduce* [31,Sect. 51 a new unknown matrix

X

= YexpP,,t=

YX,.

(14.2)

If we substitute this into (14.1), we obtain

Here, the matrix

is periodic with period 2 x .

Now, we may seek Yin the form

which simplifies the calculations in finding w k and z k since Po =W,-0 in Eq. (11.2).

15. An Example Illustrating Section 14 Example. Consider the system of two equations

dY -- Y tPo+EP1(t)]=YQ,Po=

1I-y :[I, P1(t)=Asint+Bsin2t, (15.1) dt where A and B are constant second-order matrices. Let us find a representation of the matrix Y = exp [W ( E ) t ].Z (t, e),

Yo(0)= I,

(15.2)

*Reference [31] also considers the more general case in which Po (1) is a matrix such that the integral matrix of the sysre&

= x p , (1)is of the form (8.8), where the matrix

W has purely imaginary characteristic numbers with prime elementary divisors.

AN EXAMPLE ILLUSTRATING SECTION 14

91

where

(15.3)

(15.4) Here, the matrix Pohas the characteristic numbers p1 = i, pa = - i , pl-pe= 2 i , and 2 x Pa 3 lnexp 2 x Po is not a regular value; that is, we may set Wo= Po..

First of all, let us find W

(15.5) k==O

To determine Yo and Y,,we have the equations f l 0 --

dt

-YoPo,

dyl = YIP

-

Yo

dt

+ YOP,,

exp Pot.

(15.6) (15.7)

We seek Y, in the form

Y , = c (t)Yo. If we substitute this into (15.6), we obtain 1 dc =Y O P 1 ~ c ' ,= jYOP1YP'dt. dt 0 On the basis of (1.141), Yo= sin faPo cost, Y;' = - sin t -p0+ cost.

+

(15.8)

(15.9)

(15.10)

On the basis of (15.7) and (17.8), we have (15.11)

92

LINEAR SYSTEMS

OF ORDINARY DIFFERENTIAL EQUATIONS

where the bn are the elements of the matrixB. Consequently, 1

Y(24=

I

E

+ €c2 + ba) $ ( 4 2

(b22

blJ

P E-

2

1-

(bas-b11)

( 6 1 2 + bad 2 PE

1+ I

c2yz

+... . (15.13)

and, on the basis of (10.9), (15.14) and (15.7), 2, ( t ) = exp Pot.

(15.16)

To determine the matrices Wk and Zk (f) (for k >/ 1), we have the equations (see (1,1.2), where W, =-0)

Therefore,

dZ1 = ZIP,+ ZOP, - W,ZO. dt

(15.18)

(15.20)

s

c =0

[ZJJ,Zi-*

- W,]df.

Since Y o= Zo, we have, on the basis of (15.10),

(15.21)

AN EXAMPLE ILLUSTRATING SECTION 14 t

c = j [-

pop,p0sin2t

93

+ (popl- plP0)sin t cos t + PItoss t - W,]dt.

0

Thus, we have found 2, in the form (15.19). To find W, we need to require that 2,satisfy

z1 (t + 2 iT) = 2,(t).

(15.22)

Then [(popl-plpo)sin t cos t

W1=-1

2iT

+

0

+ PIcosat -P,,PIPosin*t ] dt. Keeping the value of PIin mind, we obtain from (15.1)

w1= 1 [POB-BP,]. 4

(15.23)

Similarly, we can easily find w& and z k (for k ' k 2 ) from Eqs. (15.17). Specifically, to determine zk and . v k # we have

(15.25) (15.26)

(15.27) On the -asis of Theorem 6.1, the series (15. ) converges for all finite values of E . The series (15.3) and (15.4) converge at least in the region 1 e I < R in which the discriminant of the characteristic equation of the matrix (15.11)

A(&) = a2(Y( 2 ~ -4 )) has no more than one zero. Here, it is convenient to keep in mind that D(Y(2n))=exp

r

a(Q(t))dt= 1

94

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

We may proceed in a different way, following the lines shown in Sect. 14. Specifically, we introduce the matrix Y defined by

X

= Y exp Pot = YY,.

(15.28)

If we substitute this into (15.1), we obtain

dY

= YYoPl (t)Yr' E = Y R ( t ) E, df . R (t) = YOPI(t)6'.

(15.29) (15.30)

If we use (15.10), we shall have

R (f) = -PoPIPosinzt

+(POPl- PIPo)sin t .cos t + P, cos21.

(15.31)

Now, we may seek Y in the form OD

m

(15.32) k= I

k-0

In Eq. (15.24) we shall have Po = 0 and 2, = I, which simplifies the calculations. We also obtain the integral matrix (15.5) in the form

(15.33)

on the basis of Theorem 2.3. Let us first represent the matrix (15.13) in the form (15.34) We easily find

Suppose that M + 0. Then,

AN EXAMPLE ILLUSTRATING SECTION 14

95

(15.36)

+

where A = M - (blz bal), b = bl - bll, and sl, and are arbitrary nonzero numbers. In accordance with the example following Theorem 2.3, we have 0

N, Ek,

In Y (2 x, 8 ) =

(15.38)

k=O

where we may set

No= So[2k z i , -2 k x i ] SF' , k being an integer.

(15.39)

Keeping the value of So.in mind, we obtain

It is easy to see that b2- A' = 2(bla Therefore, we have

+ bzl)A, bg + A'=

2 MA.

(15.41)

(15.42)

96

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L EQUATIONS

We recall that our assumption is that M Z 0. Consequently, on the basis of the example following Theorem 2.3, we have (15.33) for the matrix (15.5), where 1 2x

W , = - InY (2 x , 0)=

sin9

COST

cosy

-sincp

I

.

(15.43)

To find Zo(t), we set k = 0 in (15.17). We obtain dZ0 = ZOP, dt

-

WOZO, 2, (0) = I,

2, = e x p (- Wot).exp Pot.

(15.44) (15.45)

To determine 2, and W,, we set k = 1 in (15.17). We have

+ Z,P,-W&, 2, (t + 2x) = 2, (t).

Z,P0 - W,Z,

g 1 =

dt

2, (0) = 0,

(15.46)

(15.461) This equation is not an equation of the type (11.13) that we studied above since, in this case, the characteristic numbers of the matrix Po are k, = i and X1=-i. Therefore, they donot possess the property that k1 - A, + mi for integral m since A1 -A, = 2i. However, Theorem 2.3 assures us that, in this case, a solution satisfying condition (15.461) exists for a suitable choice of W,. Let us seek a solution of Eq. (15.46) in the form (11.7) 2, = e x p (- Wot).Ce x p (Pot). (15.47) If we substitute this into (15.46), we obtain

dC = e x p Wot .[ZoPl- WJ,]e x p (-- Pot),

(15.48)

dt

C=

s

u

e x p W o ~ ~ [ Z o P l - W I Z o ] e x p ( - Pot)dt.

Since necessarily Z1(0) = 2, ( 2 4 = 0 and e x p (- Wo 2 4 = e x p 2%Po = I,

we have

C (2s) = e x p Wot.[Z,Pl0

W,Z,] e x p ( - Pot)dt = 0.

(15.49)

97

AN E X A M P L E I L L U S T R A T I N G S E C T I O N 14

But

Therefore,

7

~(?x)=

[expPot.Pl-exp(-Pot) -

(15.50)

0

- exp W o t - W l e x p ( - W,t)] dt = 0. In accordance with (15.31), we have e x p P o t - P l e x p ( - Pot) = -PoP,Posinat +(P,P,-PIP,,)sint~cost+ P,cos*t,

or, substituting the value PI given by (15.1), we obtain exp Pot.P,.exp(-Pot) = -PodPosinsf -PoBP,sin2t.sin*t + (P,A - dPo)sin2t .cos t (PoB- BP,) sin t .cos t 1sin 21 + A sin t .cos2t B sin 2t -cos2t ,

+

+

+

+

On the basis of (1.141), we also have expWot.Wl.exp (- Wot)= - WoW,Wo

4-(WOW,- WlWO)

sin kt .cos kt

~

sinW kz

t

+ W,cos2kt.

If we substitute this into (15.50), we obtain PoB - BPo 2

W"WIW0 + --w,=o. ka

Here, W , is given by formula (15.43) and Po by (15.1). Let us rewrite the last equation in the form

-

PoB -BPo -w,-- wow,wo 2 ks

(15.51)

and substitute the values of Po, W,,and

We obtain (15.52)

98

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

where M is given by formula (15.42) and 711=w11(1 +sinacp)+ w12sincp~coscp+wz,sincp~coscp+

+ w22 cos=cp,

rIZ = wl,

sin cp-cos +w12 cosa0

+w21cos2q-wB

sin 9 . ~ 0q, s

+ wz,coszcp - wBsincp-coscp,

(15.53)

rsl = wl,sincp.coscp+w12cosPcp

wllcosPcp-wl,sincp.coscp= w,,sin(p.coscpf f wz2( 1 sin2cp). Consequently, we have T

~

+

M -sin 2

cp

=:

T,,,

M M cos cp = T~~~ -cos q = T

2

2

- 32 sin cp =

~ ~ .

ra

(15.54) (15.5 5)

If we add the first and fourth of Eqs. (15.54), we obtain

(15.56)

w,1+ wa = 0.

If we substitute w,, = we have M -sincp 2

-w,, into the first and second of Eqs.

= 2wllsinzcp

(15.54),

+ w,,sincp.coscp + ~ ~ ~ s i n p . c o s c p (15.57) . +

+

coscp = 2wl,sincp.coscp w1,cos2cp wzlcoszrp. (15.58) 2 Here, the following cases are possible: 1) cos cp = 0; 2) sin cp = 0; 3) sincpXcoscpz0. In case l ) , we have 4w1, = 2 M = f 161, 621 1, and w12 and wu a r e arbitrary. In case 21, 2 (w12 wzl) = i M = f I bE -6x11, and to,, and tu, a r e arbitrary. In case 3), Eqs. (15.57) and (15.58) become

+

+

M = 2wllsinp +(wlz 2

+ wz,)coscp,-M2 = 2w1,sincp+

(wI2+ w2,)cos(p.(15.581)

Consequently, wlz and w21 may be considered arbitrary but w,, is uniquely determined. 2, is found in accordance with formula (15.47) :

2, = exp(- Wot).C(t)exp(Pot), t

c = J [exp Pot-P,exp (-

-

pot)

0

- expWof-Wlexp(-

Waf)] dt.

99

AN E X A M P L E I L L U S T R A T I N G S E C T I O N 14

Now, we can determine 2,and W,from (15.17) fork 2 2 : -= dz8

df

+

230 - WOZ, ZlP, -W14- WJa.

(15.59)

.Z2 can be sought in the form

2, = exp (- Wot).C(t)exp (Pot).

To determine C, we obtain the equations dC = exp Wof.[Z,P, - WIZl - WJol exp (-

df

Pot),

-

C = exp Wof [Z,P, -WJ,- W,ZO]exp (- Pof)df. 0

If we require that Z, ( 2 4 = 2, (0) = 0, we obtain C (24 = 0. Then to determine the elements wk, of the matrix W,, we obtain equations whose right-hand members coincide with the right-hand members of Eqs. (15.53), but whose left-hand members are different and contain two arbitrary elements of the matrixw,: A,

= w,,(l

+ sinscp) + w,,sincp.coscp + w,,sincp.coscp +

+

A, = wl, sin 'p .cos cp

Wz-2 coszcp,

+ w,, cos2cp +w21 cos, cp -

-w=sincp.coscp,

A, = w,, sin cp .cos cp A,

= wll cose 7

(15.60)

+ w,, cos2cp + wZ1

COS, 'p

- wzzsin cp - cos 9.

- wls sin 'p .cos cp -q,sin 'p .cos cp +

+w28 (1 + sin8cp).

In case 1) (where cos cp = 0), we have A,=A,= 0, from which we find the arbitrary elements of the matrix W,. In case 2) (where sin cp =0) ,we have A, = A,, A, = w,, + w i and A, = w,, + w, from which we obtain A, =A,. From these two equations, we again find the arbitrary elements of the matrixW,. In case 3) (where sin 'p-cos'p + 0) we have

-A, + A, - w,, + w,,. 2

If we make the substitution

100

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

in the first and second of Eqs. (15.60), this gives us M, = Pw,,sin 'p (w,, w2,)cos cp, M, = 2w,, sin cp

+ + + + @I2

WZl)

+

cos '9,

from which we get M,= M,. Also, A, is obviously equal to A,. From these two equations, we again find arbitrary elements of the matrix W,. This follows from Theorem 2.3, which ensures uniqueness of the determination of the matrices W,and Zk. Thus, in all cases, wefinallyfindW,and 2,. It is easy to see that we can also find W,and z k (for k > 2). We shall not complete the evaluation of W, and 2,. We note now that the system (15.1) can be first replaced with the system

I1 - i

For the solution X of the system (15.61), we easily find the representation m

m

x = e x p W ( E ) f ' Z ( t ,c),

x ( 0 ) = 1,

w(E) =

wke',

z ( f 2), =

k=O

ZkEk,

k=O

after which we obtain such a representation for Y with Y (0) = I since Y = N-lXN

= exp

[N-lW(c) Nt]-N-'ZN.

(15.63)

Remark 15.1. In accordance with Theorem 2.3 and the example following it, we may take, instead of (15.39),

N o = S , [ 2 m x i , 2nailSC' , (15.64) where m and n a r e arbitrary integers. Correspondingly, instead of (15.43), we then need to set W, = 5 = S, [mi, nil SF' = 2K

-

m sin2'9

+ ncos2cp

( m-n) sin 'p coscp (15.65)

101

CANONICAL SYSTEMS

where

and the values of A and b a r e given in formula (15.37). After this, we would again find wk (for k > 1) and 2, (for k 2 b 0) that appear in (15.33).

16. Canonical Systems [8, 9, 12, 13, 31, 33, 34, 67, 681 Remark 16.1. Suppose that the characteristic equation of a real matrix K ( E )of order 2n is of the form

D (A,

E)

= A2n

+ a, A*

(-1)

+...+ a,,+ h2 +a,, + @ (A,

E)

= 0,

(16.1)

where the coefficients a,, ... , a,, a r e independent of h and e. and where CP (A, E) is a polynomial in Aa of degree not exceeding n - 1 with coefficients that a r e holomorphic functions of E in a neighborhood of E = 0 such that @ 0.. E ) + 0 as 2 + 0. Then, if the roots of the equation 1131 D(p) = pn

+ alpn-'

t ... + un-1 4 an = 0

(16.2)

a r e negative anddistinct, the roots of Eq. (16.1) will be purely imaginary for real sufficiently small* values of E; that is, A = pk i + i v k ( E ) (for k = 1, ..., n ) , where the pk are the roots of Eq. (16.2) and T ~ ( E ) is a real function that approaches 0 as 8-0. We note also that if the coefficients a,, ..., a, a r e infinitesimals of lower order than the coefficients of @(A, E ) for small values of E , then, in the case of distinct p k .< 0 , the roots of Eq. (16.1) will also be purely imaginary for small values of E . We write the expression (16.1) inthe form

... + A , ( E ) = O , (k = 1, ..., n).

D(p(E))'= pn+A,(E)pn-l+

Ak (0) = ak

(16.11)

Remark 16.2. Suppose that the roots pl, ..., p, of Eq. (16.2) a r e negative but that some of them are multiple roots. Now, for all "roots of (16.1) to be purely imaginary for small values *And of course for I E I < r . , where r is the distance to the closest root of the equation A (K(L)) = 0. Here, A (K(c)) is the discriminant of 4.(16.1). This follows from the fact that the complex roots of (16.1) arise immediately, in complex conjugate pairs, close to the multiple root if L is real.

102

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

of e , it is necessary and sufficient that all roots of equation'(l6.11) be negative* for small values of E . But this will be the case if and only if all roots of (16.11) can be represented, for small values of E , in the form m

1 -

p k ( ~ ) = p k + x a j k ) e a P( k = 1,

... ,n), p = 1 o r p = 2

(16.21)

equation with real** coefficients a ! & ) . If the discriminant A (K(~))of (16.11) is not identically*** equal to 0, then the pk (€)willbe distinct. On the other hand, if there exist pk ( E ) , where p is an integer exceeding 2, some of the roots )i ( E ) will be such that R (1(€)) > 0 and others will be such that R ( ~ ( E ) ) < Osince, in this case, there a r e complex pk

(€1

Suppose now that we a r e given the canonical system of 20 differential equations (16.3) where the Pa ( t ) a r e real continuous periodic matrices with period 2x, where E is a numerical parameter and where Po is a real constant matrix. Thus, the system (16.3) corresponds to a system of the form

_ _ - dH

dH

dy,=(s = 1, 2, ..., n ) . dy,' dt dx, .. where H is a quadratic**** form in the variables x ,,..., x,,, yI,..., yn, the coefficients of which a r e periodic functions off. Let us suppose that 2a Po = In exp 2%Po is a regular value. Then,***** dxs dt

*For the general solution of this problem, see [32] and Sect 47 of the present book. m

m

, then pk ( 6 ) will obvious~ya ~ s obe complex **+That is, if at least one coefficient in the expansion of the discriminant A ( K ( E ) ) =

ajk) m

hkck

is nonzero.

k=O

k

k

s-l ,==I *****It

is shown in [33] that the system

is canonical.

103

CANONICAL SYSTEMS

According to Lyapunov's theorem [26,p. 2091,the characteristic

2 m

equation of the matrix

W k c k is of the form

k=O

p"

+ a,

(e) pa-,

$-

*..

+ a,,

(E)

p

+ an

(E)

= 0,

(16.5)

where p = La, wherea, (0), ..., a,, (0)arethe coefficients in the characteristic equation of the matrix P o , and where the series a,,, (E) converge (for m = 1, ..., n ) at least in the region I e I < R c r in which the discriminant of the characteristic equation of the matrix

X& ( 2 x ) Ek,

x,=

(16.6)

k=O

has no more than one zero. T h e o r e m 16.1 (See [8]and [131.)

If the roots of the equation

are vegative and distinct, then allsolutions of Eq. (16.1) are bounded but do not approach zero; that is, they are bounded and oscillatory functions. This theorem follows from Remark 16.1. In [8], Artem'yev, studying canonical systems of the form (16.3) carried out detailed investigations* for a specific system of four equations. Remark 16.3. If some of the roots of Eq. (16.7) a r e 0 (though none of them are positive) ,it is sometimes sufficient to examine the matrix W,. Specifically, if the characteristic equation of the matrix PO WlE

+

*Artem'yev's investigations [8, 91 were unknown to me at the time of completion of [14], which is a portion of the studies made in October-December 1942, when I had no access to mathematical literature (except for one book by Lyapunov, Obshchaya zadacha ob ustoychivosti dvizheniya [The General Problem of Stability of Motion]). The book [I41 includes almost my complete doctoral dissertation, which was defended in July of 1943 at the University of Kazan. (A few sections of the dissertation were not included in that book and were w i s h e d separately.)

104

LINEAR S Y S T E M S OF ORDINARY D I F F E R E N T I A L EQUATIONS

is such that all p
characteristic numbers of the matrix W ==

2

WkSk

will be purely

k=O

imaginary (and distinct) for small values of e. Remark 16.4. If the roots pl, ..., p i of Eq. (16.7) a r e all negative but some of them coincide, then the solutions of the system (16.1) will be bounded but oscillatory if the p k are of the form (16.2 1) and A ( K ( E ) + ) 0 (that is, if A

m

Aksk and atleast one of the Ak # 0)

= k=O

since, in this case,

p1( E ) ,

..., pn ( E ) are negative and distinct (for

small values of E ) . Remark 16.5. On the basis of the preceding sections, we can also easily consider the case in which 2 x P 0 is not a regular value of ln exp 2~ Po. For example, we canpas a preliminary, transform the system (16.1) into a system for which 2%Po is a regular value of In exp 2x Po. But we do not need to do this. Instead, we may simply take 1 2R

Wo = -In exp 2 r Po

which is a principal value. However, we need to keep in mind that if we take the irregular value 1

Wo = 2 x In exp 2 x p0

in accordance with Theorem 2.3, then the characteristic equation of the matrix

2

W,ok may fail to be of the form (16.5).

k=O

For example, if (15.1) is a canonical system and we take W,in the form (15.65), where m # --n, we do not obtain an equation of the form (16.5) since the roots of the characteristic equation of 0

the matrix

C

wkEk

obviously do not come in pairs of opposite

k=O

sign. However, if we take 1

W o = -In exp 2x Po 3r

105

THE S Y S T E M (16. 3)

which is the principal value and is actually irregular, of the form (15.43) [on the basis of (15.39)], we obtain (16.5).

17. The System (16.3) With pO=Pl=...= P,_,=O Let us suppose that, in the system (16.3), Po=Pl--...=Pm_..l==U and that P m ( t ) # 0. Then, in accordance with (13.3) and (13.4), we have

where

-_

dZm

dt

E

is real, and P, ( t ) -w,,

w, = -

2;:

P,,,(t)dt, Zm-:

Sr

0

0

Pm(t)-WW,]dt.

We have the characteristic equation for the matrix M=.W&-m in the form p z n - k a , p j ( n - - l ) -I-..

/-p2a,-,+an -tPn-l(pa, 8 ) ==O,

(17.1)

where ul, ..., an a r e the coefficients of the characteristic equation for the matrix W,, and the coefficients of the (n- 1)st polynomial Pn-l (p2, E) converge to 0 as E 0. Theorem 17.1. I f the roots p& of the equation -+

p"

-ta@-'

-1-

... -tpun+ + a, = 0,

p

(17.2)

= pz

are distinct and negative, that i s , i f pk = -b i , where each bk ( f o ~ k= 1, ..., n) i s positive, then the characteristic numbers of the matrix W are purely imaginary and distinct f o r small values of E. Proof: Fortherootsp(o)ofEq. (17.1),we haveh(E)= - 6 ~ - i - ~ k ( t ) (where k = 1, ..., n ) , where the real function y k ( € ) -+0 as E + a. Consequently, for characteristic numbers A, and L,(for Y == 1, ..., n) of the matrix W,we have A, = i (b,

+ +,

(E))

E",

L, = -i (6,

+ +,

(2))

r m (v =

1,

..., n).

where the real function q, (E) 0 as E -.0. This result is obtained in [34] by a different procedure for the system --f

dX dt

-=

XP(t)5

(i.e., Po = Pz = P3 = ... = 0).

(17.3)

106

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Re m a r k 17.1. Suppose that the roots of Eq. (17.2) are nonpositive numbers. Then, by considering Wm+l, we obtain the characteristic numbers of the matrix W,,, + W m + l ~If. i t turns out that the Pk (for k = 1 , .... n ) are distinct and negative, we again see that the characteristic numbers of the matrix Ware purely imaginary for small values of C. They remain purely imaginary for I 8 J < r , where r is the smallest of the absolute values of the number el that are roots of the discriminant A ( E ~ = ) 0 of the equation(- l ) , D ( X ( 2 a , E ) -11) = 0.

In [34] a general study is made of those intervals of values of a in the system (17.3) for which the characteristic numbers of the matrix Ware distinct and purely imaginary. It is also assumed that the characteristic numbers of the matrix Wl are distinct and purely imaginary.

18. Artem’yev’s Problem In [a] Artem’yev examines the following problem. Consider the canonical system

(18.1)

where the matrix of order 2n

and pl, .... pv,

....

are parameters. Find those relationships between . under thecharacteristic numbers of matrix system (18.1) purely imaginary. note that, in E

which the W of the will be We accordance with Lyapunov’s theorem, only in this case can the general solution of the system (18.1) be bounded. Artem’yev proposes to solve this problem thus: According to Lyapunov’s theorem, ~3each characteristic number u of the matrix W,there exists a characteristic number-;. Therefore, we have the following set of characteristic numbers ak ( E ) and ~ ~ ( of 0 the ) exponential matrix W: pl,

pv,

E

ARTEM'YEV'S

107

PROBLEM

From this, we obtain the relationships between p,, ..., pv, and from the equations

E

(18.2) the satisfaction of which makes it possible for the zero solution of the given linear system (18.1) to be stable. (We still require that the elementary divisors of the exponential matrix W be primes.) In many specific problems, Artem'yev points out, it is possible to obtain the first few terms of the expansion of the quantities X,, ..., k,, in terms of a parameter €.andthen calculate approximately the roots of Eqs. (18.2). In line with this, note that, in this problem, wedonothave equation& stating the relationships between the parametersp,, ..., p v E . Let us show this. The characteristic equation of the matrix W of the system (18.1) is of the form

where p = Ca and C is a characteristic number of the matrix W. From this we see that the characteristic numbers i of the matrix Ware purely imaginary if and only if all roots of Eq. (18.3) a r e negative. But this will be the case if and only if Hurwitz' inequalities are satisfied [3, 35-37]. Let us agree to write these inequalities of Hurwitz in the form ~ ( P I ,..., ~

v E ,)

> 0, ...* bn(tr1, ...

pv,

E)

> 0.

(18.4)

Thus, the region of values of the parametersp,, ..., pv, E in which the characteristic numbers of the matrix Ware purely imaginary is given by inequalities (18.4). If these inequalities a r e satisfied, then all the characteristic numbers of the matrix W will be purely imaginary. If they a r e also distinct (that is, if all negative roots of Eq. (18.3) a r e distinct),then the general solution of the system (18.1) will indeed be bounded and oscillatory (that is, it will not approach 0 asymptotically a s f -1 ). But if some of them a r e multiple, we need to show that all elementary divisors of the matrix Ware primes since it is only when this condition is satisfied that the general solution will again be bounded and oscillatory. of values of the paramFrom this it follows that the region (0) eters pl,. .., pv, 6 to which the bounded solutions of the system Q,

YThis is explained by the fact that the system (18.1) is not arbitrary but canonical. Therefore, certain equalities for the integral substitution X ( 2 x . p1, ..., pv ,el of this system are already satisfied (The characteristic equation for the matrix X ( 2 x . p,. ..., pv is, according to Lyapunov's theorem, reciprocal.)

I08

LINEAR S Y S T E M S OF ORDINARY D I F F E R E N T I A L E Q U A T I O N S

(18.1) correspond is obtained by deleting from the region (18.4) that portion of it in which the characteristic numbers of the matrix W a r e multiple with nonprime elementary divisors. The region of values of the parameters pl, ..., pv, E to which multiple characteristic numbers of the matrix W correspond is given by the equation A(PI, ..., pv,

€1= 0, (A)

(18.5)

where A is the discriminant of Eq. (18.3). Thus, the region ( A ) ofvalues of the parameters pl, ...,pv, €,which generates bounded solutions of the system (18.1) can be written in the form

(4 = (b)-(4.

(18.6)

The boundary (&-(A), where (2) denotes the closure of(A), of this region will be the set of points (pl, ..., pvr E ) to which the zero and multiple characteristic numbers of the matrix W correspond. However, this set @ ) - ( A ) contains the set of points (C) corresponding to bounded solutions. These a r e those points that generate the matrix W with prime elementary divisors. corresponding to a bounded general Thus, the entire region (0) solution can be written

+

(Q = (4 (C).

(18.7)

The boundary (D)-(D) of this region will be the set of points ..., pv, 4 corresponding to the matrix Wwithnonprime elementary divisors. In this connection, the following problems arise: I. Suppose that Hurwitz’ conditions are satisfied for Eq. (18.3), that is, that all roots of this equation are negative. By using procedures familiar to us from algebra, we can find an interval (-w, W) containing all the negative roots of Eq. (18.3). The question then arises: what are the conditions that the matrix P ( t , pl. ..., pv, $of the system (18.1) must satisfy to ensure that the roots of Eq. (18.3) do not lie outside the given interval (-w, 0). If we denote these roots by - of,..., - W; , we obtain the frequencies 0, of the oscillatory solutions of the system (18.1). Specifically, if there is a frequency om, then there exists a solution of the form (pl,

s = z1(t)cos w, t

+z, ( t )sin

W,

t,

where 21(t).and z, (t)are periodic vectors with period 2a.

T H E THEORY OF R E D U C I B L E S Y S T E M S

109

11. Suppose that the point($. ..., $, ~ O ) i ssuch that the matrix W’ has multiple roots, whether with prime o r nonprime elementary divisors. Then, the question may be asked, will the general solution of the system (18.1) be bounded in a neighborhood of this point? To answer this question, we need to express all the parameters (pl, ..., pv, E ) as functions of some one parameter = such that

and

In particular, we may choose E ,for example, for the parameter T or we may simply fix all values of the parameters but one and leave some one of them variable. For example, the functions (18.8) may be such that they satisfy the following equations (on the boundary of the region (18.4)):

..., pv, 0 ) = 0, ..., b,(p1, ...,

(18.9) For simplicity, suppose that the functions (18.8) are holomorphic in a neighborhood of T = 0. Then the coefficients of ‘Eq. (18.3) will be holomorphic in a neighborhood of T -= 0, and, for = 0 , Eq. (18.3) has multiple roots. If it now turns out that all of the roots of Eq. (18.3) in a neighborhood of the point T = 0 are negative and simple, then, in a neighborhood of the point T = 0, we shall again obtain a system (18.1) whose general solution is bounded. It is possible to show, by using Hurwitz’ inequalities and [32], in which methods are expounded enabling us to tell whether the roots will be simple and real or not, whether, with given functions (18.8), Eq. (18.3) has only simple negative roots. In this way, we can investigate the behavior of the general solution of the system (18.1) in a neighborhood of the point (py, ..., p:, @)and, in particular, along the curve (18.8). bl(pl,

pvr c) = 0.

19. The Theory of Reducible S y s t e m s Consider a reducible [ 141 system of n linear differential equations XP(t) dt and the corresponding reduced system

(19.1)

YB,

(19.2)

-= dx

-=

dt

110

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

where P ( t ) is a real matrix that is continuous and bounded in the region t >/ 0 and Bis a real constant nth-order matrix. According to Remrirk 3 in [14], we may assume that B is canonical. According to Theorem 1 of [14], the system (19.1) has the solution

X

= expBt.2 (t).

(19.3)

where 2 ( t ) and 2-l ( t )are bounded matrices. The matrix Z ( t ) maps the system (19.1) into the system (19.2) according to the formula

x = YZ(t).

(19.4)

T h e o r e m 19.1. There exist real boundedmatrices Z(t),such that Z-l(t) i s also bounded, that mat, the system (19.1) into the system (19.2) according to formula (19.4). [We shall find all these matrices 2(t).J Proof: Suppose that Z ( f ) in (19.3) is complex,%(t)= Z,(f) iZ,(th where &(t) and Z,(t) a r e bounded real matrices. Then, we have a real integral matrix of the system (19.1)

+

X ( t , a ) = e x p B t . ( Z , ( t ) $- aZ,(t))

=:

exp Bt.%:(t,a),

where a is an arbitrary numerical parameter. There exists a real value a such thatD(X (t, a ) ) # 0 where D is the symbol for a determinant. To see this, note that, since D(Z-'(t. i)) is bounded, it follows that

where the & ( t ) a r e real bounded functions. Let us take some a a t which D (2(to,a)) # 0. Then, in accordance with a familiar property of a fundamental system of solutions of the linear system (19.1), we have

D(Z(t,

0))#

(19.5)

0.

From this i t follows that there a r e no more than n values of the parameter a at which

D (2 ( t , I))

= 0.

(19.6)

In particular, (19.5) is satisfied for all sufficiently large and sufficiently small absolute values of D (Zl ) = 0).

2

(except possibly

: I

0 if

111

THE THEORY OF REDUCIBLE S Y S T E M S

Thus, we have a real integral matrix of the system (19.1) in the form X

= expBt-Z(t, a),

D ( Z ( t , a)) # 0,

09.7)

where a is an arbitrary real number not equal to a root of Eq. (19.6) for any t = to. Let us show that the matrix [ Z ( t , a)]-1 is bounded in the region .t >/ 0. From the Ostrogradskiy-Jacobi theorem for the solution of (19.7), we have D ( X ) =exp(a(B)t).D(Z(t, z))=

c(a)exp ( f a ( P ( t ) ) d t ) , 1.

where G (v) is the trace of the matrix Y and Therefore,

D (Z(t,

x ) ) = c (a) exp

(/

a (P

c(a) = D ( X (to))# 0.

- B ) dt - a ( B )4,)

.

Since the function D(Z-l(t, a)) is bounded for a = i s it will also be bounded for all other values of a for which D (2 (t,a)) # 0 since the function

is independent of a. The boundedness of the function D (2-1 (t.a))also follows directly from Theorem 3 of [14]. We in effect repeated the proof of that theorem. Thus, we have shown that the real matrix Z ( t , a) is, together with Z-'(t, a),bounded and that it maps the system (19.1) into (19.2) for arbitrary real a not equal to a root of Eq. (19.6) for any t = to. According to (24) of [14], the general form of a real matrix m a p ping the system (19.1) into (19.2) can be written Z = Z ( t , a)exp(-Bt). Cexp(Bt),

where C is an arbitrary real matrix such that the matrix exp (-B) .Cexp(Bt) is bounded and D ( C ) # 0. If the matrix Pin the system (19.1) is constant, then, for a solution of (19.3), we may take X = expBt-exp ([PI., ..., p,] i t ) S , that is, we may have Z = exp ( I F ,

..., B,]

i f )S = 2, (t)t iZ,(t),

(19.8)

where P = S-IJS and B is the real part of the canonical Jordan form of the matrix P (i.e., the real part of the matrix J , where J = B ..., pnli.). Consequently, thematricesZ,(t)andZ,(t)are easily found. Here, [PI, ..., ($1 is a real diagonal matrix.

[el,

+

112

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

Re m a r k 19.1. we see from (19.8) that the real matrices Z ( t , a), which transform the system (19.1) with constant matrix Pinto the system (19.2), are of the form

where 5 is a constant and the pk are the imaginary parts of the characteristic numbers of the matrix P.

20. Shtokalo’s Method [lo, 381 Consider the system (20.1) where Po is a real constant nth-order matrix and the real matrices p k ( f ) are of the form (20.2)

Here, the Cki) are constant nth-order matrices and the P k are real numbers assuming a finite number of values. The series representing the bounded matrix P ( t ) converges uniformly for 1.1
=

xz,

(20.3)

where the elements of the real matrix Z, which, together with Z-l is bounded, are of the form 6 exp (i p t). Here, E is a constant number and 3 denotes the imaginary parts of the characteristic numbers of the matrix Po, Z-l=Z,(f,

a)==Z,(t)+aZz(f),

(20.4)

and Z, a r e defined by E q . (19.8) for P == P o , and a is defined by the condition (19.5). We denote by J the real part of the canonical form of @e matrix Po; that is, Po = S-l& where 1 is a canonical matrix,J = J 4- Pi, where in turn, p is a diagonal real matrix, p == [p,, ...,pal. %,

113

SHTOKALO'S METHOD

For the matrix Y , we obtain the equation

But obviously, in accordance with (20.3),

since Z ( t , a ) is chosen on the basis of. Remark 19.1. Since the of the matrix Z are of the form 6 exp (i i3 t), the matrices Pk(f)= Z-lPk (t)Zare again of the form (20.2)withvariable tLk. Thus, we may write

-elements

(20.5) Here, we again denote the matrices & ( t ) by P k (t). These matrices are of the form (20.2) and they are real. For the system (20.5), we find the formal solution (20.6)

k> 1, where the w k are constant matrices independent of E also and the z k (t) are bounded matrices. If we substitute (20.6) into (20.5) and multiply on the left by Zk(o)=o,

If we now equate the coefficients of like powers of

E,

we obtain (20.7)

114

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS k

k

-dZk dt

1zvpk-v

-

"= 0

(20.8)

wv Zk--v.

*=O

Let us set Wo = J , Zo = I.

(20.9)

Shtokalo did not transform the system (20.1) into the system (20.5) but sought directly a formal solution (20.6) for the given system (20.1). If we seek a formal solution (20.6) for the system (20.1), then, instead of (20.7), we obtain

dZ0 = ZOPO- WOZ,.

(20.10)

dt

Let us look at this equation. Suppose that Po = S-'TS = S-lJS

+ S-l p Si.

If we set 2, = exp (i p t ) . S into (20.10), we obtain

- s = e x p (i p t ) .S .S-l7S - Wo exp (i p t ) . S .

i p e x p (i t )

From this, we obtain W , = [exp (i t )rSS - i f~exp (i p t ) . S ].S-'exp (- i p t ) = = e x p (i Pt).Texp (--i

pt) - i p .

Obviously, the matrices p and 7 commute. Therefore, W,= J . Thus, we have again obtained for the system (20.1) W,= J but 2, = exp (i pt).s instead of Zo = I, as was the case with the system (20.5) in accordance with (20.9). This value of 2, may be complex. Let us examine the system (20.5) further. We have (20.9). Let us find W , and Z, from Eq. (20.8) for k- 1:

-dZ1 dt

- ZJ

-JZ,

+ P,

---

W,.

(20.11)

Suppose that the characteristic numbers (the real parts of the characteristic numbers of the matrix P o ) of the v a t r i x J are A, (for k = 1, ..., m). Then, when k& # ;ir, the element zh.i of the matrix 2, can be found from formula (11.14): t

= e x p (kl - k k ) t.

J f ( t )exp (0

( A l --

t ) dt,

(20.12)

SHTOKALO'S METHOD

f (t)

(1)

81 zk.l+l

-8k

z621.1

+ pi!{ -wk!),

1 IS

(20.13)

where the Pi!) and the wf,! are elements of the matrices PI and W,. Let us choose wplh! in such a way that zh!! will be of the form 8 exp (i f3 t), For this, it is necessary and sufficient that

a s t + 0. In other words, the second factor in the expression (20.12), which is of the form

cannot have a free term M. On the other hand, if kk = i l , we find zk.1 either from formula (11.6):

or from formula (11.17),

(20.17) as the case requires. Wecanalsofinda solution to Eq. (20.11) from the general formula (11.9):

where-W,is defined by

116

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Obviously, we also find Wkand z k (fork 2.2). Thus, we have a formal solution (20.6), where Wo = J and Z8 =I. W e note that we could also have found & and Wk (for k >, 1) if P k (t) (for k > 1) had infinitely many terms in the sum (20.2), that is, if it were quasi-periodic. We may also assume that the Pk(t)are uniformly periodic funcQ),and we can make tions [39] with exponents* A!k)--+ OJ as 1 various other assumptions. Re m a r k 20.1. We can find the coefficients W,and 2, in a very simple manner from the general formulas (20.18) and (20.19) when the matrixJ = 6 . Specifically,

-

(20.20)

and (20.21) The W k and z k (fork > 2) can also be found in this case. Also, if the matrix Po in the system (20.1) has purely imaginary simple characteristic numbers, then we do have the case J = O . For example, if the system (20.1) is canonical, then Po has purely imaginary characteristic numbers. If they a r e simple, we have J = 0. Let us now find the coefficients of the two series

R=O

by another method, proposed by Shtokalo.

21. Determination of the Coefficients of the Series (20.22) and (20.23) by Shtokalo's Method [lo,381 Consider an equation of the form (20.11)

.-!?-

dt

0

ZJ

-JZ

+ P ( t ) -w,

(21.1)

*For simplicity, we assume that the set of exponents{ Alk)}E1of the matrices Pk(t)

are independent of b.

DETERMINATION OF THE COEFFICIENTS OF THE SERIES

117

where J has its former value (a canonical real matrix), P ( t ) is an expression of the form (20.2) ,and W is a constant matrix to be determined in such a way that 2 will be of the form (20.2). Suppose that (21.2)

where the P , are constant matrices. Following Shtokalo, let us seek Z in the form 2 = C b P exp (ip t),

where b, is a constant matrix.

Following Shtokalo, let us set

W = lim L f P ( t ) d t = Po. 1-4)

(21.3)

t

(21.4)

Here, Po. is the free term in (21.2). Then, (21.1) can be rewritten dZ = ZJ - J Z dt

+x

P , exp (ip ti.

(21.5)

P+O

If we substitute (21.3) into (21.5) and equate coefficients of like powers of e i p t , we. obtain i p 6,= b p J w-6,

+ P,.

(21.6)

From this, we can find the matrix of the 6,. Shtokalo proved this in the general case. The given expression was also studied in the book by Lappo-Danilevskiy [ 11, who not only proved the solvability of (21.6) but also gave several forms of the solution in the case in which the matrix J has distinct characteristicnumbers. Equation (21.6) is studied in detail in the book by Gantmakher [3]. Following essentially the same lines as Shtokalo, let us prove that Eq. (21.6) can be solved. If we multiply (21.6) by exp (i p f)and define 0 = 6, exp ( i p t),

(21.7)

we obtain

-dU dt

- UJ - JU -l-P , exp(i p t).

(21.8)

118

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

We have the general solution of Eq. (21.8) for P,

=0

U = exp (- J t ) C exp ( J t ) .

in the form

(21.9)

where C i s an arbitrary constant matrix. If Eq. (21.6) has no solution o r has more than one solution, then this expression has a solution 6, +' 0 forf,, = 0. But then Eq. (21.8) for P, = 0 has a solution L: - h, exp (i p t ) with 6, s 0. Consequently, we obtain exp (- J t ) C e s p ( J t ) = 6, exp ( i p t). which is impossible since the matrix J is real. Thus, we have proven that equation (21.6) has a unique solution. Now, let us find it. Assuming that C is a function of t and substituting (21.9) into (21.8), we obtain dC dt

exp (-- J t ) -e s p ( J t ) =: P, e x p ( i p t).

(21.10)

From this, we obtain f

c=,Je x p ( J f ) P , e x p ( - J t ) . e x p ( i p t ) d t + K.

(21.11)

0

Here, K i s an arbitrary constant matrix. Let us set

In other words, K must cancel that constant matrix in (21.11) that is obtained by substitution of the lower limit of integration. We then have t u = e x p (-- J t ) . [ e s p ( J t ) . P , exp ( - J t ) x ij

X exp (i p t ) .dt

+K

and

1

(21.13)

exp ( J t )

(21.14)

6, = U e x p ( - i p t ) .

If we chose I( differently, there would be in the bracketed expression a free term that would generate a term different from Ae" in U and

we would obtain for

(Ithe

expression U = 6, e x p (i p t),

DETERMINATIONOF THE COEFFICIENTSOF THE SERIES

119

the existence and uniqueness of which we have proven. Thus, we can always find a solution of Eq. (21.1) in the form (21.3) by taking the matrices in (21.6) in the form (21.14). This enables us to find the Wk and 2, (for k> 1) from (20.8) by Shtokalo’s method, that is, by determining Wfrom equations of the form (21.1) with the aid of formula (21.4) and by determining 2 with the aid of (21.3) and (21.14). The matrix (21.14) can easily be written in expanded form. For example, if J = [hl, ..., hn],thatis, if the matrix Jis a diagonal matrix, then (exp ( ~ t P, ) . exp (- Jt))k, = p&) exp ( i k - i.,).t,

(21.15)

where the Pa’ are the elements of the matrix P,. From this and from (21.11), we have (c)k,

=

+

pi!? exp (hk - A, ip) t hk-AIfip

and, consequently, again on the basis of (21.15) (21.1 6)

Finally, we have obtained Lappo-Danilevskiy’s formula. The method of determining the coefficients in the series (20.22) and (20.23) is also applicable in the case in which the matrices pk (t) in (20.5) are periodic and can be represented in the form (20.2), where there may possibly be an infinite number of terms. Also, if the matrices Pk (t)in the system (20.1) are periodic, the transformation (20.3) may yield a system (20.5) in which the Pk (t) are not periodic. This will be the case when the matrix Po has characteristic numbers whose imaginary parts B are not commensurable with the frequency of the periodic matrices of the system (20.1). But then, we can transform (20.1) to a system that will contain not the matrix Po but a constant matrixnotpossessing characteristic numbers hk and hi such that kk - k, = 2xw k , where k is an integer and 2wx is the period of the matrices Pk(t)inthe system (20.1). We can then seek the coefficients of the series (20.22) and (20.23) by Shtokalo’s method since the solution of Eqs. (20.8) can be obtained by the given method even in this case in the form of periodic functions. [The numbers p are commensurable in (21.2) (see (21.3)).] We should, however, note the following. Indetermining W;;and by Shtokalo’s method, we find z k in the form (21.3). Consequently, generally speaking, we shall not have 2,(0)=0 (for k > l ) . In

120

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

particular, this is obvious from (21.11) and (21.12). But then, we shall obtain series (20.22) and (20.23) the convergence of which cannot be ensured even in the case in which the f k ( f ) in the system (20.1) are periodic mati.ices with a single period. But if we are concerned only with the fact of asymptotic stability of the zero solution of the system (20.1), the convergence of the series (20.22) and (20.23) is not required if we then follow Shtokalo's method. Thus, we proceed to further observations of Shtokalo after the series (20.22) and (20.23) are obtained.

22. A p p r o x i m a t e S o l u t i o n s O b t a i n e d by Shtokalo's Method Consider the system (20.5). We introduce the new unknown matrix X defined by m

(22.1)

where the second factor is a segment of the formal series (20.23) and x satisfies the equation m

(22.2)

in which the sum is a segment of the series (20.22) and R,(t, s)is a holomorphic function in the region I el < R. Let us find R, (t, E). If we substitute (22.1) into (20.5) and then multiply the resulting equation on the left by P w e obtain m

[

W#

:cm+lR,,,

(f,

t)

(22.4)

A P P R O X I M A T E SOLUTIONS O B T A I N E D BY S H T O K A L O ' S METHOD

by virtue of (20.8). Therefore, if we divide (22.4) by

12 1

&-*, we obtain

W e have m

111

(22.6)

and this series converges (see Sect. 1) for

where IZ is the order of the matrices 2,. To find the Mk, we multiply Eq. (22.6) on the left by

We obtain

k=O

From this, we have

and, consequently,

'I=O

I

122

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Now, we may rewrite R,(t,

c)in

the form (22.7)

k=O

*=O

Here,

The series (22.7) converges for

i g z k e k l

1 4 ,and

I iI < r[the region

of convergence of the series in (20.1)]. We have proven formula (22.2). We see from (22.1) that, if the matrix X i s bounded, then the matrix Y is also bounded. Thus, a study of the question of stability of the zero solution of the system (20.5) leads to a study of this same question for the system (22.2). By omitting infinitesimals of order emin the right-hand side of (22.2), we obtain

This gives an approximate value of Y ,

23. Inequalities Following from Shtokalo’e Method Consider the first system

-dY -YQ, dt

(23.1)

where Q is an nth-order matrix with elements {Q)ki

= @I*

Let us suppose that the real parts of the characteristic numbers of the matrix Q are negative. Then, there exists a positive quadratic form [26, Chap. 111 (23.2)

INEQUALITIES FOLLOWING F R O M SHTOKALO'S METHOD

123

such that

or, on the basis of the system (23.1), (23.4)

Consider now the system

-dY - Y [ Q + E R ( ~ , S . ) I , dt

(23.5)

where the matrix Q is as before but the matrix R (t, €)with elements

I R (t, e) )hi = Ru is bounded: I RklI <. r. Then, for every solution of the system (23.5), we have

(23.6) This equation can be represented in the form n

n

I=I

k,I=L

dU dt = - x Y : + e x

%Y&b

(23.7)

where the %I are constituted in the form of a sum of products of the quantitites I'M. and R,,.and are bounded. Since (23.8)

1 24

L I N E A R S Y S T E M S OF O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S

where 8 = i a max j Skl 1, it follows from (23.7) that, for sufficiently small E , we have (23.9) The asymptotic stability of the solutions of the system (23.5) follows from this. Now, we shall obtain an obvious inequality. Suppose that

k.1

where, for example,

Then, from (23.9), we have

dt

C ( e 8 - l ) A L I , U g U o e x p ( e 8 - l)A(t-f,)

(23.10)

and (23.11) I=I

Other and more precise inequalities regarding approximations of the solutions appear in Shtokalo’s works [lo, 381.

24. Shtokalo’s Theorem. Inequalities Involving Approximate Solutions Found by Shtokalo’s Method (for Linear and Nonlinear Systems). Particular Problems Now, we shall study the system (22.2). The coefficients T u of the quadratic form (23.2) are found from a linear nonhomogeneous system of algebraic equations that are obtained by equating the coefficients in the quadratic forms on the right andleft sides of Eq. (23.3). The determinant of this system, as Lyapunov showed [26,Chap. II], is L I = I I ( ~ ~ ~+nt,,An) . ~ + . ,(m,+ . ...+m,= 2, mk>0);

SHTOKALO'S THEOREM

125

where A,, ..., A,, are the characteristic numbers of the matrixQ. The determinant A, which will appear in thedenominator, is a symmetric function of Al, ..., k n . Consequently, it can be expressed as a rational combination of the coefficients of the characteristic equation of the matrix Q. We introduce the notation m

(24.1) k==O

and assume that the real parts of all the characteristic numbers of the matrix Q are negative for certain sufficiently large rn and O< e < e4, where E* is chosen sufficiently small for the given set m > N. This will, as Shtokalo showed in his article [ 10, Theorem 31 , be the case when the Hurwitz' determinants set up for the characteristic equation of the formal matrix (20.22)

have positive coefficients for low powers of E . The characteristic numbers Al, ..., A,, ofthematrixQare algebraic functions of the parameter E . Consequently, for small values of e , the hk can be represented in the form of series of fractional powers of E o r integral powers of el = EI/P ,where P is an integer. It is easy to see that the first nonzero coefficients in these expansions do not change with increasing m in formula (24.1), beginning, at least, with some m. From this it follows that if

converges to 0 as E 0, then the order of smallness of A in a neighborhood of e =O does not increase with increasing m in formula (24.1). From this it follows that if the coefficients Tki in the quadratic form (23.2) approach 00 as E - O, then the order of infinity does not increase with increasing m For the system (22.2) we obtain, instead of (23.6), the equation

.

I26

LINEAR S Y S T E M S OF ORDINARY D I F F E R E N T I A L EQUATIONS

instead of (23.7), we obtain (24.4) k. l=I

I=1

and, instead of (23.9), we obtain (24.5) 1=1

Since the order of increase of Tk1 as E -+ Odoes not increase, the order of increase of 6 does not increase either. (The quantities TM and hence 5 may fail to be bounded.) Therefore, for sufficiently large m ,we have ~ m f l 6- 1<0, *om which follows the asymptotic stability of the zero solution of the system (22.2). Of course, we also obtain an inequality of the form (23.11):

x n

y:

< U0Aexp I(em+l 6 -1)

I= I

A(t - to)].

(24.6)

-.

But here, it is possible that Uo oo and A +Oas E -0. Also, as was shown above, it is possible that 5 + 03. However, since the order of increase of 6 does not increase with increasing m, it follows that ~ m f l 6-1 <0 for sufficiently large m. All this proves Theorem 24.1 (Shtokalo). Suppose that, for all m > N and 0 < E .= E * , the real parts of the characteristic numbers of the matrix (24.1) are negative, but the zero solution of the system (22.2) and hence of the system (20.5) are asymptotically stable. Keeping inequality (24.6) in mind, we see that (22.1) provides us with an approximate solution of the system (20.5): m

Y=X

x Z , E k .

(24.7)

k=O

Here, inequality (24.6) provides us with an approximation for Xi n

that is, in (24.6), we need to replacethe sum x y ; [in the notations I= 1 n

of (22.2) and (23.5)] with the sum

x;.

Here, x,, ..., x, are the

1=1

elements of an arbitrary row of the matrix X. Suppose that yl, ..., yn a r e the elements of a row (of a solution) of the matrix Y and that x,, ..., xn are the elements of the same row of the matrix X. Then, from (24.7), we have

127

SHTOKALO'S METHOD

(24.8)

and

2

x;

s U,A exp [(cm+l

6 -1) A (f

- f,,)],

1=I

(24.9)

Here, (24.10)

where the h i are determined from the system obtained on the basis of (23.3) in terms of the elements of the matrix (24.1) and 8 in terms of 7k1 and R w v ;A is given by formula (23.91). If the matrix (24.1) has characteristic numbers with positive real parts, the zero solution of the system (20.5) is unstable [ 10.301. Reference [30]takes up a nonlinear system of the form

where the matrix P ( t ) of the coefficients P v m is of the form of the matrix of the system (20.5) and the Rk satisfy the inequality

I

%(&7

-**)

!/n7

t)l < K(Y:

+

- 0 -

f y",).

Then, under the condition of Shtokalo's theorem, the zero solution of the system (24.6) is asymptotically stable, and the equation of the type (24.8) provides us with an approximate solution satisfying inequalities of the form (24.9). It is alsoproven that the zero solution of (22.11) is unstable when the matrix (24.1) has characteristic numbers with positive real parts. For the system (20.1), the following problems arise: I. When do the series (20.22) and (20.23) converge for small e?

11. What is the representation of the functions Wand 2 for arbitrary values of E (when these series converge for small values of E ) ? III. When do the series (20.22) and (20.23) converge and when is 2 a bounded matrix for It I >O? In this case (if the matrix 2-' is

128

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

also bounded), the system (20.1) will be reducible, by virtue of Theorem 3 in [ 141. IV. What can be said about the stability of the zero solution of the system (20.1) o r -its boundedness in the case in which, for all m and arbitrarily small E , the matrix (24.1) does not have characteristic numbers with positive real part but not all the real parts of the characteristic numbers are negative? For example, suppose that the characteristic numbers of the matrix (24.1) are distinct purely imaginary numbers for all rn and sufficiently small E . Even if the series (20.22) and (20.23) converge, this question can be answered only in the case in which the function (20.23) is bounded. V. Show that the system (20.1) will always be regular. Obviously, this is the case. The question of the reducibility of a system of equations of the form (20.1) was studied for special cases in [14]and for a system of two equations of a general type (20.1) in [40]. Specifically, Gel’man found two rather general sufficient conditions for reducibility of the system (24.12)

where (24.13)

converges uniformly,-

< t < co, (24.14)

and the B’s are constant second-order matrices. He also showed that reducibility can be violated by an arbitrarily small change in one of the frequencies w I . In particular, when studying the system dX = X P (t),

dt --+sinat P (t) =

coset

(24.15)

sinat 1 cos2at --

2

2

(24.16)

OTHER APPROXIMATE F O R M S OF S O L U T I O N S

129

where a and p are incommensurable and a > O , he showed on the basis of his general tests that the system (24.16) is reducible if liii

I r n , a + t n , ~ I -l/(lm,l+imzl)

< R.

tm,l+lmrl -*a

where R is the smallest positive root of the equation 1 1 y + 2 y s ~ + - y2==-a+--. 2 2

This system is also reducible when CL and 3 are algebraic incommensurable numbers and a >3. It is possible to exhibit examples in which the series (20.22) terminates, that is, becomes a polynomial in 8 . Then, the series (20.23) will converge for all E However, the questions of reducibility and boundedness of solutions and even the regularity of the system remain open. Shtelik [41] solves problems I-IV for certain systems of equations. See also [42-451.

.

25. Other Approximate Forms of Solutions That Arise From Shtokalo's and Bogolyubov's Methods With regard to the approximate solution (24.7) o r (24.8) in the case in which the matrix (24.1) has characteristic numbers with nonnegative real part for every in, the following can be said: Consider Eq. (22.2), which was

(25.1)

Here, the simplest case is that in which W , Here, Eq. (25.1) is rewritten in the form

dX - = XR (t, 2) 5 , R (f. 2) dt

m

=

Wk sk-'

=

0 (see Remark 20.1).

+

grn

R m (t,E ) .

(25.2)

k= I

Along with (25.1) ,consider the equation

Then, in accordance with a theorem of Bogolyubov [46,p. 3701, for arbitrarily small P, and 'I and arbitrarily large L there exists

130

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

a positive number e0 such that, if y = y ( t ) is a vector-valued solution of Eq. (25.3), the inequality 0 < t < 3, in the interval 0 4 t < L/Eimplies the inequality

I x (0 - I/ (0 I < '1,

(25.4)

where x ( t ) is a vector-valued* solution of Eq. (25.2) that coincides with y ( t ) for t = 0. This enables us to obtain an approximate solution of the system (20.1) in the form (24.8) by taking instead of X the solution of Eq. (25.3). We may proceed in a different manner. Let W,in (25.1) be arbitrary, not necessarily equal to 0. Consider the equation

obtained from (25.1) for R, by

= 0.

Consider also thematrix Y defined

x = YX,,

(25.6)

where the matrix X is a solution of Eq. (25.1). Then, for I', we obtain the equation (see (14.3))

(25.7)

Let us suppose that the matrix wk Ck characteristic numbers with %pie 0 3 < E:i:a Then, the limit . 1 I im t - m

(25.8) has only purely imaginary elementary divisors for

R(r, E) dt = R,(E)

t

(25.9)

n

exists. Instead of the solution of Eq. (25.7), let us take its approximate value (using Bogolyubov's theorem), namely, the solution of the system

*But by x ( t ) , y (t), and 7 we may also understand respectively the matrices (25.2), y ( 1 ) in (25.3) andT =I1 .c I]where all elements of the matrix 7 = c.

Y

( t )in

OTHER A P P R O X I M A T E F O R M S O F SOLUTIONS

131

We obtain instead of (25.6) the approximation

x = YOX,.

(25.11)

We can substitute this into (24.8). Now, let us suppose that the canonical form of the matrix rn

C

wk

2'

is of a general form. We note first of all that, for every M

k=O

and T, there exists a positive constant M,,, such that

I Rrn(t, ~ ) 6j M m

(25.12)

where I f I
-

u1.

Consequently, it is possible to exhibit an c*: such that IA (e) I can be arbitrarily small for all I f 1 < T simultaneously. Therefore, we have an approximation for the matrix X for small values of E in the form (25.11), where Y ois the constant matrix ih (25.13). Finally, we have the matrix (24.7) in the form rn

rn

k=O

k=O

(25.14)

that is, we may, as an approximation, take

132

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

(25.15) 181 Q e::: and I t ( .< T. Thus, we have Theorem 25.1 In a given interval I tl g T, the solution of the system (20.5) with initial conditions Y (0) = Yois given approximately in the form (25.15) for sufficiently small E*. Here, the e r r o r A does not exceed

for

(25.16)

where M, is given by inequality (25.12). In certain cases, in finding the matrix M, in (25.12), we may use the inequalities of Academician I. M. Vinogradov (for example, [47, 481). Here, we need to set A k t =F(&)(for k = 1,2,..., n), where

m=l

and the

bk

k=l

a r e rational numbers. If the

.Ak

=k

are integers, then

a,= landa,=Ofor l > 1 .

26. Demidovich's Problem Consider the system of n linear homogeneous differential equations (26.1)

where P ( t ) is an nth-order matrix that is continuous and periodic with period W. Suppose that the limit

lim

0-0

's

P ( t ) dt

Q)

= A4

(26.2)

0

exists. The problem is to answer the question of the stability of the zero solutian of the system (26.1) for small values of o. Demidovich obtained the following result:

DEM I DOV ICH' S PRO6 LEM

133

Theorem (Demidovich). The characteristic indices A = h ( w ) of the system (26.1) approach the characteristic indices of the matrix M f o r suitable choice of imaginary part of A(w). The author concludes from this that if the real parts of the characteristic indices ho are negative, then, for sufficiently small values of a, the solution of the system (26.1) is asymptotically stable, but if there is a purely imaginary number A,, we can say nothing regarding the stability. Let us note first that this assertion regarding the stability of the zero solution can be obtained by using Bogolyubov's theorem [46, p. 4191. We shall, however, solve Demidovich's problem by a different method, one based on the theory expounded in the present book. Let us make the substitutiont = or in Eq. (26.1). Then, (26.1) can be rewritten dX dr

-= X P ( w r )w.

(26.3)

The matrix X can, for small values of W, be represented in the form* (26.4)

Here, & = I (see (9.15))

and we can determine 2, and W , from the equation

The matrix P ( w r ) is of period 1 with respect to r . Therefore, in accordance with (9.17), I

(26.5)

w 0

Demidovich's results follow. 'By considering the system

and then set E = a,

dX

= x p ( 0 7 ) E we can obtain

134

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

If, for arbitrarily small w, the matrix W l has characteristic numbers with real parts equal to 0 (but none with positive real parts), then, by taking, in the expansion of the matrix (26.6)

the following terms, one may again obtain a solution of Demidovich’s problem as was shown in the preceding section. Here, the problem is solved by the limiting values of the matrices W,,W,,W,,... as w 0. ;We shall not repeat the process.

27. Another Formulation of Certain Problems and Consequences of Them Consider a system of the form

1 (0

dX dt

x

-: P,

( t ,5) Ek,

(27.1)

k-0

+

where Pk (t 2*, 6) = P , (t;a). and where the matrices P k (t, 8) are uniformly continuo& with respect tQ 6 at the point 8 =Oand are absolutely integrable with respect to t in the interval(O,2r), where the series P ( t , a,

a) =

5

P , (t, a) Ek

k=O

converges in the interval 0 C t 4 2n, 0 Q 6 < a,, 0 Q 2 < +, and where l P ( t , 6, E ) 14 M,(where M is aconstantmatrixwithpositive elements). We obtain an integral matrix of the system (27.1) in the form m

x (t, a,

0)

=

1 x, ( t ,

3)

tk,

(27.2)

k-0

X o (0,s) = I, X , (0, 8)

by the equations

= 0.

for k > 1, where X o (t, ;)andX, ( t , %)are defined

ANOTHER FORMULATION OF CERTAIN PROBLEMS

135

By Theorem 6.1, the series (27.2) converges for 0 Q t 4 2 x , I E 1 < E ~ and is a continuous function of 6 at the point 8 = 0. If the matrix , P ( t , 6, €)is holomorphic with respect to 6 and 3 in the neighborhood 181 < s,,, I % I < 6, , then the matrix X ( t , 8, &) will also be holomorphic with respect to 6 and E in that region. The matrices x k ( t , 6)will be holomorphic with respect to 6. We seek an integral substitution of the matrix(27.2) in the form

181 < 6,

c W

x ( 2 x , 8,

8)

=

xk(2x,6)Ek.

(27.3)

k-0

Let us write the characteristic equation of this matrix X ( 2 x , 8, E )

where 0

(k

=

1, ..., n)

k=O

and these series converge in the interval I E 1 < a,. If the matrix P ( t , 8, z) is holomorphic in the region I E I 4 E ~ ,16 I < a0, thenthe ak(6, E ) will also be holomorphic in that region. To answer the question of the stability of the zero solution of the system (27.1) o r the boundedness of the matrix (27.2), we may, in accordance with Sects. 10 and 11, represent the matrix (27.2) in the form

where the matrices &(t, 6) are periodic with respect to 1. If, for all sufficiently small values of 6 and c, the real parts of the characteristic numbers of the matrix are negative, then the matrix (27.2) will possess t'ie property that X ( t , 8,

E)

+ .

11 011 as t

---L

00.

(27.6)

If the real parts of all the characteristic numbers of the matrix Wi(0) are negative, then (27.6) holds for all sufficiently small values of E and E ; if the zeros are also negative, then (27.6) is also valid for all sufficiently small values of 6 and E provided the real parts of all the characteristic numbers of the matrix

x w, m

w,(8,

E)

=

(6) &k

k=O

(27.7)

,

136

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

are negative for all sufficiently small values of b and E. We obtain this on the basis of Shtokalo’s method even in the case in which the matrices pk(t, 8 ) are only quasi-periodic o r some of them are almost periodic [ 131.On the basis of Theorem 24.1 (Shtokalo), we have the following result. If, for some 8 and sufficiently large m , the real parts of the characteristic numbers of the matrix (27.7) are negative, then the real parts of the characteristic numbers of the matrix (27.5) will also be negative for that 6 and for all sufficiently small E . We recall that the real parts of all characteristic numbers of the matrix (27.5) are negative for all sufficiently small E if the Hurwitz determinants of that matrix have positive coefficients for the lower powers of e (that is, if the coefficients are positive only for sufficiently large m in the matrix (27.7) [lo, Theorem 31. We see how the question of the boundedness of the matrices (27.2) is answered when the system (27.1) is canonical. We have also considered Artem’yev’s problem in this case for the system (18.1), a particular case of which is the system (27.1) if it is canonical. We shall return to these problems in Sect. 46, solving them on the basis of the matrix (27.3) and Eq. (27.4). Now, we shall touch on a problem that is considered in [49]and in various works referred to therein. Suppose that we have a set of periodic integrable matrices P ( t ) = P (t + 2 x ) . Let us take any one of these P(t) and let us examine the system of linear homogeneous equations ~

(27.8)

where the parameter w is positive. Let usalso consider the subset of this set that consists of matrices P,(t)possessing the properly 2n

f IP, (4- - P (t)I dt

&

t

11 1 11 ,

(27.9)

i,

where 11 111 is the matrix all of whose elements are equal to 1 and 1 P l ( t ) - P(t) I is the matrix whose elements are the absolute values of the differences of the elements of the matrices P l ( t ) and P(t). From (27.9), we see that the set of matrices P,(t)can be written Pi (0 = P (0 -t E Q (0, where the set of matrices Q(t) is such that

(27.lo)

(27.11)

ANOTHER FORMULATION OF CERTAIN

137

PROBLEMS

Along with the system (27.8), let us consider the system (27.12)

where 6 is a parameter. Let us suppose that the zero solution of the system (27.8) is stable.. The question* arises as to the conditions under which the zero solution of the system (27.12) will also be stable for sufficiently small values of ij and a. Let us introduce a new independent variable T definedby 'i = (W + 8)t into equation (27.12). We obtain [13] (27.13) Now, for the system (27.13),the problem becomes the following: For =-. 3 = 0 , the system has a bounded fundamental** integral matrix X(T). Under what conditions will this fundamental integral matrix be bounded for all sufficiently small e and 6 1 The system (27.13) is a particular case of the system (27.1). We have the obvious result: If the integral matrix of (27.13) X (T, 6, e) -t 11 0 11 as T + co when 6 = e = 0, then X (T, 8, a) 11 0 I/ as T for all sufficiently small 6 and s.. This follows from the fact that if X ( 5 , 0,O) I] 0 I1as 'i ,then the real parts of all characteristic numbers of the matrix (27.5) for ~ = = E = O will be negative. But then, the real parts of the characteristic numbers of the matrix (27.5) will also be negative for all sufficiently small*** values of 6 and s. On the other hand, if the real parts of the characteristic numbers of the exponential substitution (27.5) are nonpositive numbers when B = E = 0, we need to consider the characteristic numbers of the matrix (27.7). If, for all sufficiently small 6 and E , the real parts of the characteristic matrix (27.7) are negative, then X (T, 6, a) will again approach 11 0 11 as T -, for all sufficiently small 6 and e. Here, the matrices P ( s ) and Q ( ~ ) m a y be considered quasiperiodic o r almost periodic. If, for 6 = E = 0, the characteristic numbers of the exponential substitution (27.7) are nonpositive, we may consider the particular cases in which 6 = 6 (a) -+0 as E -+ 0 in accordance with what was said in Sect. 18. This is particularly convenient when (1) the real parts of all the characteristic numbers of the matrix W, (0) are equal to 0, (2)the matrices P (r)and Q (.c)are

-

--t

--f

--f

*Here, we pose a problem equivalent to the oneexaminedin[49] althoughina different formulation, **That is,- with D ( X (t)) # 0, where Dindicates a determinant. **%s was already pointea out in [9, p. 731, where it was assumed ody that the coefficient matrix of the system (27.1) was a continuous function of 8 and since the continuity of the characteristic numbers of the matrix (27.5) as functions of the 8 and D had been proven.

138

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

periodic, and (3) the system (27.13) is canonical. For example, we may consider the case of i = e . Then, the system (27.13) can be written in the form m

dX

- =x d.:

Pk(7)Ek.

(27.14)

k=O

where

(27.15) 0

Now, the problem for Eq. (27.14) becomes the following: for = 0, the integral matrix of the system (27.14) is bounded. Under

i

what conditions will this integral matrix be bounded for all sufficiently small e ? Obviously, if the real parts of the characteristic numbers of the system (27.8) (that is, therealparts of the characteristic numbers of the exponential matrixw) are negative, then for small values of 8 , the integral matrix X (0) of the system (27.14) possesses X

11 0 I1 as

(0) --t

5 ->

a.

(27.151)

On the other hand, if the system (27.8) has characteristic numbers whose real parts are equal to zero (and none of the characteristic numbers have positive real parts), then the question is answered by the coefficients in the expansion of the exponential matrix

w=

W,Sk.

(27.16)

k-0

Suppose that b = 0 and P ( w t ) = C is a constant matrix in the system (27.12). Then, we have the system

In essence, such a system was considered in [49],pp. 37-39. In this case, the system (27.13) is of the form z = . X [C+cQ(r)]--, 1 d.c

Q(%+2x)=Q(r).

0

(27.18) 2u c

Let us suppose that this system is canonical and that -= 2x c In exp -is a regular value.* Then, if the matrix C has distinct, (D

0

%at is, ____--2.r ( “ k -- w,) -j-

where m is an integer and

0

numbers of the matrix C. The values [34,49].

(,,

2.7 (~k Ill

:

04i

and q i are the characteristic

are called “critical values” in

ANOTHER FORMULATION O F CERTAIN PROBLEMS

139

purely imaginary characteristic numbers, the zero solution of the system (27.18) is stable for sufficiently small e [or the integral matrix of this system is bounded and oscillatory (i.e., does not approach 0 as T-+~)]. This follows from the Theorem 16.1 (Artem'yev's theorem). Suppose that 2x(ok -ao1)= o m .

(27.19)

Then, to solve the problem, we need to proceed as indicated in Sects. 10, 11, and 20-24. Let us construct the integral matrix of the system (27.18): m

X ( T ) = ~ X ~ X"( O~ ) =, I ,

CT

Xo(~)==exp- ,

(27.20)

0

k=O

and the integral substitution

Following the methods explained in Sects. 10 and 11, we can represent the matrix X(T)in the form (27.22) J k=O

Here, for example (see(10.14)),2x W,= In exp

(%) the principal is

2xc has nonegative characteristic numbers. value if the matrix exp Then, (D

Zo(T) = e x p (- W o z ) . e x p

(?)-

(27.23)

We showed in Sects. 10 and 11 how to calculate wk and under all hypotheses regarding the matrix C. In particular, we found above that it is sometimes possible (see Theorem 2.3 and the example following it), when condition (27.19) is satisfied, to take

140

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

C w,= s -

(27.24)

s-1,

IU

where S is a constant matrix that does not depend on 2 . In particular, this is possible in the case in which the characteristic numbers of the matrix (27.21) a r e holomorphic functions of E and the elementary divisors a r e prime in a neighborhood o f t = 0. Let us suppose that the system (27.18) is canonical (the set Q ( t ) is such aC). Then, for the exponential matrix W(E), we have the characteristic equation p"

+ a,

(E)

p"-'

+ ... +

an-1 ( E )

p

+ an (E)

;=

0,

(27.25)

where p = Aa, A being a characteristic number of the matrixW(a). I t may happen that (27.19) is verified but all the characteristic C numbers of the matrix - (or of the matrix W,) are purely imaginary (0

and distinct. Then, if Theorem 2.3 is applicable* , the roots of Eq. (27.25) will all be negative and distincl iora = 0. But then the roots of Eq. (27.25) will be negative and distinctfor all sufficiently small E From this it follows that the integral matrix of the system (27.28) is bounded and oscillatory for small E . For example, this will be the case if the system (27.18) iscanonicaland of the type (15.1). (In the system (15.1), the characteristic numbers of the matrix Po are i and 4.) It may happen that Eq. (27.25) also has zero roots for E = 0 but distinct negative roots for small E + 0. Then, the integral matrix of the system (27.13) will again be bounded and oscillatory. Here, i t is useful to recall Remark 16.3 and Theorem 24.1 (Shtokalo's theorem).

.

28. Solution of the Problems i n Section 8 by Use of the Method of Solving the Poincar6-Lappo-Danilevskiy Problem and Lyapunov's Contributions We shall now show that the complete solution of the Poincar6Lappo-Danilevskiy problem i s closely related to the problem of constructing the integral matrix X of the system dX

- = X P ( t ) , P ( t + 2x) dt

=P

(t)

(28.1)

in the form

x = e* N ( t ) , 'But, in applying Theorem 2.3, we need to keep Remark 16.5 in mind

(28.2)

SOLUTION OF THE PROBLEMS IN SECTION

8

14 1

where A is a constant matrix and N ( t ) is periodic with period 2n. The integral matrix (28.2) is in effect multiplied on the left by the matrix

when t is increased by 2 % . Suppose that the matrix* P ( t ) is of the form m

P ( f ) = 4 + x bkcoskt

+

k=l

where tions

b,,

aksinkt,

(28.4)

k-l

are constant matrices. If we make the substitu-

bk and

sin kt

w

@ti

=

-c k t i 2i

, cos kt =

&ti

+ e-kti 2

and z = &fs we can write Eq. (28.1) in the form zi

dX = x dz

m

PkZk,

(28.5)

k=-m

where

We can write the system (28.5) in the form

(28.6) k=-in-I

where

Tk - - i pk+lr T-,

= -ibo.

(28.7)

Thus, at a finite distance, the system (28.6) has one singular irregular point z = 0. Suppose that X ( t ) is an integral matrix of the system (28.6) that is normalized at the point z = 1. In accordance with (7.7), we have *In accordance with what was said above, our subsequent reasoning will remain valid m

form=coif the s e r i e s x Pkt"Cconvergesfor I2-1 c I. -m

142

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

X

( 2 ) =z W N (2).

(28.8)

where W is a constant matrix and N @)isa single-valued matrix in a neighborhood of the point z = 0. When we let the variable z move around the coordinate origin, the matrix X (z) is multiplied on the left by the matrix V = e2xiW , 2xiW = lnV, (28.9) which, as we have seen, can be represented by series of matrices T-m-1, ..., T m - 1 . which converge (with exponential speed) at all finite values of these matrices. The matrix W can be represented in accordance with formula (7.15) for all values of these matrices ..., Tm-1 if the matrix W is second-order, and it can be represented in accordance with (1.39) if it is of nth-order. By making the substitution z = 8' ,we may write X ( e i t ) = e'Wt N (&). (28.10) Here, the matrix N (dt) is periodic since N (2) is single-valued in a neighborhood of the point z =O with period 2x and iW=A (28.11) is the matrix A that appears in formula (28.2). From this we see that the integral matrix X ( f )is multiplied on the left by the matrix

when t is increased by the period 2%. Consequently, finding the matrix V by which the integral matrix of the system of differential equations with periodic coefficients is multiplied when the independent variable t is increased by an amount equal to the period of the coefficient matrix, leads us to seek a matrix by which the integral matrix of the system of differential equations with singular irregular point z=O is multiplied when the point z moves around the coordinate origin. Let us consider in greater detail the general representation of a second-order matrix A i n terms of matrices that appear inthe system of two differential equations with periodic coefficients. In accordance with (28.11) and (7.15), we have

where

SOLUTION OF THE PROBLEMS IN SECTION

8

143

On the basis of (28.7), we may also write

(28.16)

If (28.17)

then

To find A, we need to find Vand a(V), since, on the basis of (28.4), is given together with the system (28.1). To find V , we need to proceed as follows: Using the notations bo = U1, bl = U,, ... , b, = Urn+!, a, = Urn+,, ... a, = U2,+! we write the system (28.1) in the form a (60)

(28.19)

where cpl(t) = 1 ,

(Pk(t)

= cos(k-

(Pk(t)=sin ( k - m - 1 ) t

1)t

(k = 2. ..., m -+ l ) ,

( k = m + 2 , ..., 2 m + 1 ) .

(28.20)

Now, we can use formula (6.11) to find the matrix x(f)in the form of a series of compositions of the matrices U,,..., U,,+lthat converges ..., and arbitrary finite t. We can also obtain for all finite Ul, X ( t ) in the form (6.1), which amounts to the same thing. When we have done this, we can easily obtain V = x ( 2 4 , a (V) = a ( X ( 2 x ) )

in the form of convergent series.

(28.21)

144

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

However, we can also use the series (7.12), which represents V in the form of a series of compositions of TWs,..., T , The series (7.12) also converges for arbitrary finite values of these matrices. Here, we need to set 6 = 1. The coefficients in the series (7.12) are polynomials in x with are rational coefficients, where the rational numbers a and defined by Lappo-Danilevskiy’s recursion formulas given above. Since V is a real matrix, the coefficient of i in every sum

.

in (7.12) is equal to 0 when we substitute the values of T-m-l, ..., Tmv1 given by (28.7). A system of linear equations with periodic coefficients is a particular case of systems that can be reduced by Lyapunov’s method [14]. Z ( t ) is the matrix of the transformation of the given system with periodic coefficients into a system with constant coefficients and with coefficient matrix A, the general expression for which (as well as Z ( t ) ) , we shall now find. Let us consider the case in which thecharacteristic numbers pl, and pe of the matrix V a r e negative. Then, a(V) = pl +- p2 < 0 and, in accordance With (28.16),

From this i t follows* that t 4-1/ t 2 - 1 < 0 since1 t I > 1. Therefore, in accordance with (28.15), the matrix A is complex. Specifically, A=

ln(-t-v 2xv

*It follows from (28.22) that1 t

t2-

t2-

I > 1 since t -<:

1 )+xi 1

u1 +PZ

2 ) / K .

X

SOLUTION OF THE PROBLEMS I N SECTION

8

145

where

is a real matrix. The characteristic numbers

p1 and p2 are

p2 - a ( V ) p

roots of the equation

+ D (V)= 0.

In accordance with Jacobi's formula, we have, from Eq. (28.1), (28.22)

Therefore,

It follows from this that

.4

L = dl -k - S

2

[ I , - 11 S- I .

This enables us to write (28.2) in the form (28.24)

146

LINEAR S Y S T E M S OF ORDINARY D I F F E R E N T I A L E Q U A T I O N S

or

X = eA1t W , ( t ) . where 'r s [ I , N , ( t ) -. e *

-1)

s-'

N (4.

The matrix N, ( t )is real (since the matricesX and A, are real) and I

+[I,

is of period 4%since N (t)isof period 2a but the function e is of period 4n by virtue of the equation @ ~ S [ * - - I ~ S - = ' I. Earlier (see (9.3)), we had

4js-1

instead of (28.4), but, in accordance with Remarks 1.3 and 1.4, even there we could have obtained (28.24). Let us find the general representation of the matrix A for a system of n equations with the aid of formula (1.39). It should be noted that Lyapunov in his dissertation [26, Chap. III, 531 exhibited a case of a linear system of differential equations with periodic coefficients such that finding the characteristic numbers of the matrix A leads to a simple algebraic problem. This is the case in which, in a system of 2n equations with unknowns s,.. . . , X n , VI....~ YR, the system is broken into two systems for unknowns UI. ..., un, u1, ...vvn after we introduce the new variablesu, = x, iy, and us = x, &,(for s = 1, ..., n ) . The first of these systems is such that the substitution e" z leads to linear equations with regular singular point z = 0. Consequently, the characteristic numbers of the matrix A are found as the characteristic numbers of the matrix? that is the coefficient of 2-1. A second system of equations with unknowns u,, ..., 0, also leads to a system with regular singular point 2 - 0 by means of the substitution e-" = z. In the case of a system of two equations, this class of Lyapunov equations is an extremely simple particular case of a system in which the coefficient matrix P ( t ) possesses the property

+

-

-

t

t

P (1) [ P i t ) & = f P ( t ) dt P ( t ) . b

b

Thus, Lyapunov also turned his attention to the relationship between the theory of linear systems of differential equations with *?his is dear from the formula (see Sect. 7) W = SU-lS-l, in reference to Q. (7.9).

147

REMARKS ON BOUNDED A N D P E R I O D I C S O L U T I O N S OF A S Y S T E M

periodic coefficients and the analytic theory of linear systems of differential equations. We note that the invariants of the matrix W coincide with the invariants of the matrix H constructed by Lappo-Danilevskiy in the form I

m

-4p ,

...p"

=-s

where 8;) is the Kronecker delta and a::)... are defined by the formulas of Sect. 7. Here, H is the exponential substitution of the so-called metacanonical integral matrix [l]. The matrix H is similar to the matrix W. ~v

29. Remarks on Bounded and Periodic Solutions of a System of Two Differential Equations With Periodic Coeffficients In this section, we shall touch on the question of the existence of bounded and periodic solutions of a system of two linear homogeneous differential equations with periodic coefficient matrix p 0) dX , i X P ( t ) , P ( t + 2 4 = P(f).

(29.1)

dt

As we have seen, the integral matrix normalizedfor t

=0

of such a

system can be represented in the form

x =e M ( t ) ,

(29.2)

where A is a real constant second-order matrix and Z(t) is a periodic matrix with period either 27t o r 4r. If Z(f) is of period 4r, then 2 2 4 = - Z(1). When t is increased by an amount equal to the per od 25r, the integral matrix X ( t ) given by formula (29.2) is multiplied on the left by the matrix

't+

V = x ( 2 ~=) @ A ,

(29.3)

if Z(t) is of period 2n and by the matrix

V =X ( 2 ~ ) = - - e * ~ ~ , if the period of 2 (t) is 4n. Consequently, we have X ( t 2n n) = V"X (t),

+

where n is an integer.

(29.4) (29.5)

148

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

Let us suppose that the matrix V has the canonical form (29.6)

where pl,and

p2 are

roots of the equation

and, according to Jacobi's formula F a ( P Jd l

D ( V )= e o

The matrix X,

= S-'X

is also an integral matrix for the system (29.1) and is multiplied on the left by the matrix J =

/I I/ 0 PLZ

(29.9)

as t is increased by an amount 2x. This means that the solution in the first row of the integral matrix XI is multiplied by pl, and that the solution in the second row is multiplied by p2 as t increases by an amount 2x. This is true because (29.10)

where "kl (t), xRa(t) are the elements of the k th row of the matrix Xl(t). From a property of the roots of Eq. (29.7) and on the basis of (29.71)~ we have r a

PlPZ = e o

( P ) df

(29.12)

R E M A R K S ON BOUNDED AND P E R I O D I C SOLUTIONS OF A S Y S T E M

149

If

then p1p2 = I .

(29.13)

IF11 :.

(29.14)

We see from (29.11) that, if 1,

the system of linear equations corresponding to the matrix equation (29.1) has a one-parameter family* of solutions xl(t)and x,(t)with the property that

as

] x l ( t ) l -t I x z ( t ) I - - O

t+m.

(29.15)

If p1 and pz are less than unity in absolute value, then all solutions possess the property (29.15) and the zero solution x1 = x, = 0 is obviously asymptotically stable in the sense of Lyapunov. Let us suppose now that pl and p, are complex and that lpli =lp,l ,= 1 ,that is, that (29.1 51 ) e' , p2 = ,-i '. If, on the basis of (29.6) and (29.151), we write (29.5) in the form p1

Q

x (t + 2 x 4 = s [e'" ',

e-'" 'I S-'X (t).

(29.16)

we see that the matrix X (t) is bounded and oscillatory as t -,a. Since every other integral matrix of equation (29.1) is of the form. x1=

cx,

(29.17)

where C is a constant matrix, all solutions are bounded and oscillatory (except the zero solution). rf kcp = 2 ~ ,

(29.18)

where k is an integer, then all solutions will be periodic with period. 2%k.

.

%at

number.

is, the family xl ( t ) = C X I ~ (I). XI ( t ) = cx1a (t),where c is an arbitrary constant

150

L I N E A R S Y S T E M S OF ORDINARY D I F F E R E N T I A L E Q U A T I O N S

For cp = 2 x , we have p1 = p2 = 1. Therefore, all solutions are periodic with period2r. On the other hand, if cp = P ,then = pB= - 1 and all solutions will be of period 4%. Let us suppose now that the canonical form of the matrix V is of the form

(29.1 9) Then, we have

Since a p a - * O as IPI

u

' m

for Ip1 < 1, it follows thatX(t)-boas t-.mif

< 1.

c)n the basis of (29.17), we conclude that all solutions of the system (29.1) possess the property (29.15). If p = I, the integral matrix (10.8) is multiplied on the left by the matrix

(29.21) as t increases by 2 x . It follows from this that the solution x l l ( f ) , x l , ( t ) in the first row of the matix X1 is periodic with period 2r. The second solution property

x,,(t), xZ2(t),in

the second row, possesses the

s,,(t$-2xn) = xol(t)+nx?,(1),x,,(t+2xn).--x,?(t),

Thus , for p = 1,we have only a one-parameter family of periodic solutions with period 2;r; the remaining solutions are unbounded. For p = - I, we have a oneparameter family of periodic solutions with period 4x. If / p I > 1 in (29.19) o r if I p l / > 1 and I p21 > 1 in (29.10), then all solutions of the system (29.1) are unbounded as t 03. We note that when the condition

-

2%

(o(P)dt> 0 6

(29.21 1)

is satisfied, not all the solutions can be bounded since, in accordance with (29.12),we have I p1 I 7 1.

REMARKS ON BOUNDED AND PERIODIC SOLUTIONS OF A SY ST E M

151

Suppose now that (29.22)

that is, YZ

1

a

0

(Pldt

(29.23) ;. 1 .

Then, on the basis of (29.7), we conclude that, if ( V ) > 0. 2 ( V ) - 4 0 ( V ) > 0.

(29.24)

then the case 0
O
(29.25)

will occur if s ( V ) < 1 +D(V) <2.

(29.2 6)

~(1') < O . sz(V)-4D(V)> 0.

(29.27)

O
(29.28)

If we have

then, for

we have 1

-1

(29.29)

Consequently, when conditions (29.24) and (29.26) or (29.27) and (29.28) are satisfied, all solutions of the system (29.1) possess property (29.15). Suppose now that we have (29.23) and

-4 0( V ) = 0.

a2 (V)

Then, (29.31)

152

L I N E A R S Y S T E M S OF O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S

On the basis of what was said above and inequality (29.23), all solutions of the system (29.1) possess the property (29.15) independently of the canonical form of the matrix V. Let us suppose now that we have (29.23) and

."V) -4D(v) < 0.

(29.32)

Then, p1 and pr are complex and, since pIpr = D (V)<1, it follows that ! p1 I = I pe I < 1. Consequently, when conditions (29.23) and (29.32) are satisfied, all solutions of the system (29.1) possess the property (29.15). Let us now consider the case T P ( t ) d t = O , D(V)= 1.

(29.33)

0

Suppose also that G ( V )- 4 7 0.

(29.34)

a(V)7 0

(29.35)

Then ,when

we have PI =

. ( V ) + 2v w m ,

(29.36)

and (29.37)

and when .(V) -

1
(29.38) p2

< -1.

(29.39)

If (29.33) holds and a"V) - 4 4 0. (29.40) then p, and pe'are complex numbers of absolute value 1; that is, we have the case (29.151).

REMARKS ON BOUNDED AND PERIODIC SOLUTIONS OF A SY ST E M

153

Thus, when conditions (29.33) and (29.40) are satisfied, all solutions of the system (29.1) will be bounded and oscillatory. Suppose now that we have (29.33) and a2(V)--4 =O.

(29.41)

Then,

if condition (29.35) is satisfied; conversely, p1=

p2 =

-1

(29.43)

if condition (29.38) is satisfied. If we have (29.42), then there exists a one-parameter family of periodic solutions of the system (29.1) with period 2%. On the other hand, if (29.43) is satisfied, then there exists a one-parameter family of periodic solutions of the system (29.1) with period 4%. The question of the boundedness (and, in the present case, of the periodicity) of all solutions of the system (29.1) is answered, as we have seen, by the canonical form of the matrix V; Specifically, if V is of the form (29.19), where p is now* 1 ,then not all solutions of the system (29.1) will be bounded. Let us give a necessary condition for the existence of a periodic solution of the system (29.1) with period 2 ~ . Since the condition P I = 1,

(29.44)

is necessary and sufficient for the existence of a periodic solution, we have, by virtue of the property of the roots of a quadratic equation and (29.7), (29.45)

and (29.46)

If we eliminate pe from these equations, we obtain a necessary and sufficient condition for the existence of a solution

y. 1 +eo

( P ,dt

(29.47) = a(V)

154

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

that is periodic with period

2 x . To

1

see this, note that if

+D(V)

then, by substituting D ( V ) = g ( V ) -

= fJ(V).

1 into (29.7), we obtain =O.

p2-a(V)p+a(V)-l

From this we see that

p = 1 is

(29.471)

a root of Eq. (29.7). Consequently, if (29.472)

then the system (29.1) has no periodic solution with period 2z. If, in particular, condition (29.33) is satisfied, there is no such periodic solution for 2 i’ O W )but there is such a solution for 5

(V) = 2.

(29.48)

Suppose that

in the system (29.1), where p ( t ) and q ( t ) are periodic nonnegative functions with period 2r. In this case, as is easily seen, in formula (6.1) 9 (t))= 0

J

and a (X, (0) 0.

(29.49)

From this it follows that Q

and, consequently, there are no periodic functions of period 2 ~ . Remark 29.1. If we make the substitution t = k 7 , in (29.1), we obtain dX --

+

--Xp(r), p(7F)=P@r)k,

dr 2 x ) . If

(29.50)

we now write condition (29.47) for Eq. where p (T) = p ( 7 (29.50), we obtain a necessary and sufficient condition for the existence of periodic solutions of Eq. (29.1) of period 2 x k . There are no periodic solutions of the system (29.1) with period incommensurable with 2x (see Sect. 36).

P E R I O D I C A N D BOUNDED S O L U T I O N S OF T H E S Y S T E M S

155

30. Periodic and Bounded Solutions of the Systems of Equations Considered in Sections 3 and 4 Suppose that the periodic matrix P ( f ) in the system (29.1) possesses the property (4.6). Then, in accordance with (4.7), t

s'

(1) dt

(30.1)

~ ( t=) e0 or, since P ( f ) is of the form (4.11), we haw;

where

or (30.6)

Thus, the condition for the existence of aperiodic solution of the present system of two equations with period 2 x consists in the requirement that at least one of the two equations

156

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

be satisfied o r that, simultaneously,

=

4n2 a2

J

since we now have il (2a)= Bani and A, (2x) = - 2 x n i , where is an integer. If neither of equations (30.7) is satisfied, there will be no periodic solutions of period 2n. Let us now consider the system (5.1), where I / , and I/, a r e second-order matrices possessing properties (5.2) and (5.3) and where q1 ( t ) and qz(t)are periodic functions with period 2a. Suppose that the matrices CJ, and U, are ofthe form (5.6). Then, as we have seen, a(V) and D(V)can be obtained from formulas (5.8) and (5.9) for t = 21. If we substitute these values of a (V) and D(V)into (29.47), we easily find

Thus, there will be a periodic solution with period 2n only in the

case in which one of the two equations (30.9) or 2n

(30.10)

is satisfied. Let us now suppose that the matrices U,and .U,in the system (5.1) are of the form (5.11). Then, in accordance with (5.16) and (5.17),

and

157

Q U E S T I O N S INVOLVING BOUNDEDNESS AND PERIODICITY

If we substitute this into the condition for periodicity of the solutions (29.47), we find*

The system in question has a periodic solution with period 2 x only when one of the following equations is satisfied:

Thus, we have exhausted all possible cases of a system of two equations when the matrices U , and U , satisfy conditions (5.2) and (5.3).

31. Questions Involving the Boundedness and Periodicity of Solutions of a System of T w o Linear Differential Equations With the Aid of a Special Exponential Substitution Obtained by Lappo-Danilevskiy Let us now express the conditions for boundedness of solutions of the system of two equations (29.1) in terms of the elements of the matrix A that appears in formula (29.2). When we study the question of the existence of bounded and periodic solutions of period 2 x , we may assume that the matrix Z(t) in formula (29.2) is of period 2 x . In so doing, we allow the matrix A to be complex. On the basis of formulas (1.2) and (1.3), we draw the following conclusion: If (31.1) then

*Mu& more detailed systems, shown in Sects. 3, 4, and 32 (as well as other systems) are studied in the works by P. B. Gololwoschus [SO-551.

158

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

and if (31.2)

then

From this we see that, if where R (h) denotes the real part of i., then the characteristic number p1 of the matrix I/ will be less inabsolutevalue than unity; consequently, there is a one-parameter family of solutions possessing the property (29.15). In the case in which €?(A2) 20, all the solutions of the system (29.1) possess the property (29.15). If R(h)
A=O

or

Al = kli. le = k& ,

where k1 and k, are integers. If the matrix A proves to be real, we obviously can only have - kl = g k. If the matrix A has a multiple characteristic number h a 0,but is of the form

-

(31.5)

QUESTIONS INVOLVING THE BOUNDEDNESS AND PERIODICITY

159

there is only a one-parameter family of periodic solutions; the remaining solutions will not be bounded because

where n is an integer. Let us suppose now that the characteristicnumbers of the matrix A are (31.6) il = 'h i. L2 = - >. i , where 1. is a real positive nonintegralnumber. Obviously, this case corresponds to the conditions (29.33) and (29.40). If 1. is a rational number, then all solutions will be periodic with period T = 2k 11, where k is the denominator of the number A. On the other band, if A is an irrational number all the solutions will be bounded and oscillatory. A s can be seen fromwhat has been said earlier (see Sect. 8), the situation is possible in which k1 = k2 = i/2

(31.7)

(with the matrix A complex) o r 1, = i / 2 , h' ,

=

-

(31.8)

(i/2).

o r finally, in general,

a,=

% + ' i , &=2

-1

211,

2

i

(n. n,integers).

(31.81)

Then, all solutions of the system (29.1) willbe periodic with period 4%. We note that, in this case, (31.9) if we wish to have the matrix A real (n = - - n1). A necessary and sufficient condition for finding a solution that is periodic with period 2x under the condition that the matrix .4 is real is expressed by the equations ? ( A ) = 0. i.e.,

D(4

= ns (n -.

T

0

o(P(t))dt = 0

an integer

1)

I*

(31.10)

when we have two linearly independent periodic solutions o r by

160

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

D(A) =0,

o(A)

-arbitrary,

(31.101)

when we have a single periodic solution*. We recall that the general expression of the matrix A for the system (28.1), where P ( t ) is defined by (28.4), is given by formula (28.15), which becomes (28.18) when 2-

[ P ( t ) d t = 2xa (6,) = 0. b

Consequently, in the case (28.18), the characteristic equation of the matrix A is of the form )? - m2(a2 (V) - 4) 4

=

0

,

(31.102)

where

Thus, we have

On the basis of (31.10), the condition for periodicity of the solution may be written

A s before, we obtain the condition for periodicity (29.48) of the solution (29.47). On the basis of (28.13) and the reasoning followed in connection with formula (28.25), we have *If (31.4) is satisfied, then, as we have noted, there ar e two linearly independent periodic solutions with period 2 ~ .

QUESTIONS INVOLVING T H E BOUNDEDNESS A N D PERIODICITY

D ( A ) = - D ( W ) = -D(H).

16 1

(31.103)

Therefore, condition (31.101) for the existence of aperiodic solution may be written in the form D ( H ) = 0,

(31.104)

where H i s defined by the series (9.24). However, we need to keep in mind the fact that the series (9.24) converges for small values of Tp. Therefore, on the basis of (31.104), it is convenient to find the relationship between the parameters of the equation, a relationship that ensures existence of a periodic solution only for small values of the parameters, for example, for small values of thematricesT,. We note, however, that, in (31.104), we can sometimes obtain a series of infinitesimals of increasing orders even when some of the elements of the matrices T, (or even some of the matrices Tp themselves) are not small. We shall see this from some examples. The convergence of such series in infinitesimals of increasing orders (that is, when not all the matrices T, are sufficiently small) does not follow from LappoDanilveskiy's theorems but from the theorems proven in the present book (Sect. 29). Furthermore, a relationshipbetween the parameters that ensure existence of a periodic solution can also be found from Eq. (29.48), in which the series representing V converges for all finite values of the matrices T,:

A s was noted above (following formula (28.21)), the imaginary terms in this equation cancel since V is real. Of course, instead of this form of V , we could have taken its representation in the form (6.1), considering it as

k=O

for the system (28.19). Let us consider Mathieu's equation 2 q c o s 2 t ) y = 0,

(31.11)

where a and q are constants. This equation is equivalent to the system

162

dX

-=

df

LINEAR S Y S T E M S O F O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S

0 XJJ

2qcos2t - a

0

Here, the coefficient matrix is of period r. We introduce the new independent variable .c = 2t:

Consequently, this system is such that, in accordance with (28.4), we have in this case 0

0 *

(31.14)

If we write (31.13) in the form (28.6), we obtain

In accordance with (31.103), we haveD(A) = -D(ff) with H@ven by formula (9.24)

where

8p) is the Kronecker delta (31.17) 4 (1)

where the P ~ . . . P , are defined by the formulas in Sect.7, and where pl ,... , p , in formula (9.24) assume the values -2, -1, and 0. On the basis of formuls (31.16),

QUESTIONS INVOLVING THE BOUNDEDNESS AND PERIODICITY

163

It follows from this that all products of the matrices To,T-l,T-, , where Toand T, occur next to each other or one of these is squared, are equal to 0. We find that the sum of the terms containing the product TJ, ,...Tp,TP,forPi+ ... +pa 6 0 is equal to 0. Keeping (31.18) in mind as regards the remaining products, substituting the values of the coefficients a!!!.,~v,and leaving the products Tp,...Tp, in the sum (9.24) only for Y = 1, 2, ..., 7, we obtain

+

;=:

i --

0

H=

2

aq2 a+-+-+-+L 2 ( 2 2 92

2594 128

2

Therefore, 1 92 D(H)= ( a f*

4

2

25 + as" 2 + -@+ 128

From D(H)= 0, we have 25 64

q2a2+ (2 + 9') a + q2 +-94

= 0,

164

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L EQUATIONS

+-.-q2 2

790 128

+ 0 (9')

(31.19)

Here, O(9') denotes an infinitesimal of order # as q-+ 0. Thus,* in this case, a, is an approximate value of a root of the equation

-

7 D(H)= 0. If we substitute the value a = a, (9) = - 92 2 +r -4L I

128 into the general representation of the quantity A defined by (28.18), we obtain** an approximate solution of the system (31.12) in the form (28.2). These solutions differ only slightly from the corresponding periodic solutions over a large interval of variations of i To find a periodic solution of the system (31.13) with period 4x, let us apply the method presented in Sect. 10, treating the system (31.13) as a system of the form (lO.l)e Let us also set a = 1 - 2aand let us write the system (31.13) in the form

.

that is,

(1

Po = -!2 -1

(1

/I +

0 and PI = a

PRV) = II0 11,

0

qcost

2.

Let us seek a periodic solution of period 4~ such that

x = x (t, 9)

---t

ePof a s q+O.

We set (see (10.16))

*There is no justification for assuming that at is an approximate value of a since, for small 9, a, is great and it is not clear whether the sum of the discarded portlon of the series D (H)-0 is small in comparison with az. **If we set a = a1 in (13.13). we can obtain A and N in (28.2) both by use of the formulas of Sect. 11 and from Lappo-Danilevskiy's formulas determining the solution of the system (31.15)~ntheformQ.Il)(nameIy, formuIas(7.13),and(7.511), o r (9.24) and (9.241)). Here, we onlyneed to setz=exp if inaccordance with (28.5) The solution in the row corresponding to the zero characteristic number of the matrix A will be an approximate value of a periodic solution.

QUESTIONS INVOLVING THE BOUNDEDNESS AND PERIODICITY

1

and 00

(31.23)

00

A=xdkhk,

z=

k =I

Zk

I65

(t) hk .

k-0

Since the characteristic numbers of the matrix Po

are

i

Al = - , A2 =

2

--

=

-1

1 O 2 -1 0

1 , we have,

2

I(

inaccordance with formula (1.14),

(31.24)

According to formula (11.9)* I

[ZoPIZg'-- A,] dt Z,.

2, =

(31.2 5)

6

On the basis of (31.21) and (31.24), we find 2

+-Q

CQS 2t

4

- asint ~2

'

-- sin 2t 4

For 21to be periodic with period 4n, we must set

I

166

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

l o

1

2a

(31.251)

+9 4

as is clear from (31.25). 211

z1,

4 1

~3,

With such a choice of A,., we have

t

cos 2

sin

t -

cos

-

%, =

.

-sin

2

t

t

-

2

(31.26)

2

where a 2

z,, = -

+

q -a cost - 4 cos 2t, zle = 2 8

9 +sin 2t, 8 222"

---

a+

&, = 2

sin t

a -- 4--5 cost

8. 2 The first approximation* of the equation 2

5sin i +

+ -sin 4

2

2t,

8

+-48

cos 2t.

is q2

--

40:

D (A,) = -= 0. 16

Therefore, we have 2a, = q. 2a,

(31.27) Consequently, we obtain an approximate value of a in the system (31.13) at which it has a periodic solution, with period 41; ,in the form a1=

1

=

+ q, a, = 1

q.

-

q.

(31.28)

For a, = 1 +qwe obtain approximately a fundamental system of solutions of equations (31.12) in the form** *We set D = 1and assume q and a small. **The first row of X yields an approximate value of a periodic solution of pefiOd4X.

QUESTIONS INVOLVING T H E B O U N D E D N E S S A N D P E R I O D I C I T Y

167

(31.29)

where Zo and 2, are given by formulas (31.231) and (32.26), in which we need to set 2a = (7. It should be noted that the representation (31.29) is in general valid only for small values of 4. To obtain an approximate value of the solutions for u = 1 + q when q is large, we need to use the general representation of A given by formula (9.1). Then, we obtain Z ( f ) from the equation Z ( f )= X(f)e-A'. To find the following approximate value of a, we can find A, and 2, from formulas (11.9). But we can also proceed in a different way. Specifically, in accordance with formula (12.13), instead of writing the system (31.20), we can write the system

(31.30)

where Po(t) is such that the corresponding integral matrix of the system

-dY - Y P o ( f )

(31.31)

Y

(31.32)

dt

w i l l be of the form = eA1t

2,( f ) ,

where A, and Z , ( f ) are found from formulas (31.251) and (31.26). The corresponding value of P , ( f ) is easily found from the equation (31.33)

which is obtained by substituting the expression (31.32) into (31.31). In accordance with (12.13), we take the matrix P l ( f )in formula (31.30) in the form

168

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

Now,* we can seek A and Z ( t ) in the form (31.35)

where

and Zo is given by the expression (31.231).

32. Periodic Solutions of a System of Two Equations When Condition (3.6) is Satisfied Consider again the second-order system (32.1)

assuming that (4.6) holds. Then, we have (4.11) and t

X

P ( t )dt.

= exp

(32.2)

0

Since the functions 9, (t) and (Pk

'pz

(fjareperiodic with period

2nr. we

have

( t ) dt = akt -k $k (t),

0

where a,=

(PR (t)dt and 2% l j :

+k

(t)are periodic functions with period

0

2r. This enables us to write (32.2) in the form

X ( t )= e x p A t . e x p Z ( t ) , where A is a constant matrix

*We note that here, Po as defined by 4. (31.33) will be nonzero for a = q = 0 but PI(f)as defined by 4. (31.34) vanishes for a = q = 0.

that

169

LYAPUNOV'S EQUATION

and 2 ( t ) is a periodic matrix

with period 2n. The condition for the existence of a periodic solution (31.101) now takes the form (32.3) a: + bla14 -6,~: = 0. This condition is equivalent to the two conditions (30.7) since multiplication of equation (30.7) leads us to (32.3). Conditions (31.10) yield the equations 2a,

+

6 1 ~ = 2

0, a: -t. 6

-

2 1 ~ 1 ~ 2a262

= n2,

which coincide with (30.71). In particular, for a, = a, = 0. the system has a periodic solution with period 2x*. We can also easily establish conditions (30.9)-(30.11) by using conditions (31.10)**.

33. Lyapunov's Equation 2 4- p ( f ) x = o Let us stop to consider separately the equation (33.1)

+

where p ( t ) = p ( t 1) is continuous function. We are concerned with the existence of bounded solutions of this equation. Equation (33.1) is equivalent to the system (33.2)

In connection with (33.2), let us also look at the system (33.3)

corresponding to the equation

22- - E P ( t ) X = 0. dP

(33.4)

*We noted in Sect. 28 that Lyapunov studied a system that, in the case of two equations,

Is a particular case of the system (29.1) under the assumption that condition (3.6) is satisfied. k P. Gremyachenskiy [56]considered this system of 2n equations of Lyapunov and exhibited a case in which the matrix m has a zero characteristic number. *In Sect. 4, we noted that Fedorov [23] exhibited a more general case than system

(&I), one in which X ( t ) is obtained in closed form. This enables us to solve in an easy mannerthequestion of the existence of bounded and periodic solutions.

170

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

Suppose that x = f ( t ) and x = cp (t)are two linearly independent solutions of Eq. (33.4) satisfying initial conditions f ( 0 ) = 1, f ’ ( 0 ) = 0 and ‘p (0) =O, cp’(0) = J . If we determine these solutions by Lyapunov’s method, we obtain

f(O=

1 +E

.

f1(f)+g2f*(f)+...

‘P(t)=t+E(Pl(t)+82~I(t)+

..., (33.5)

where

(33.6) For e = -1, the system of functions (33.5) yields two linearly independent solutions of Eq. (33.1). The integral matrix of the system (33.3) is

(33.7) and the integral substitution is

(33.8) On the basis of formula (29.7), the characteristic numbers of the matrix (33.8) can be found from the equation ?‘-(f(I)+cp’(l))p+

1

=o,

1.

Pi

and

!.e

(33.9)

Here, .(V) = f ( l ) + y ’ ( l ) . Two cases a r e possible: I. The roots PI and pz are real and distinct. Then, one of them is greater in absolute value than unity and the other less than unity. 11. The roots PI and PZ are complex or equal to unity. Then, IPI!

= IPZI = 1

.

Let us look a t case I. We have a one-parameter family of solutions that approach zero as t a. According to formulas (29.33)(29.39), this will be the case when --f

LYAPUNOV~SEQUATION

171

From (33.9), we find pl. and pz:

On the basis of (33.5), we can write this in the form

(33.12)

From (33.6), we have, for IZ t

= 1, i

t

(33.14) and, consequently, (33.15)

If +1(1)#0, then, for small values of 5 , the roots p1 and pa can be represented in the form of series of positive powers of the On the other hand, if q l ( l ) = 0 and+,(l)#O., then quantity .I=E%. these roots can be represented in series of positive powers* of E . On the basis of formulas (33.6), we have (- l ) k $ k ( l )> 0 for p ( f ) .SO. But then inequality (33.10) is satisfied and p1 and pa are real and distinct for Eq. (33.1) and p1 pz= 1. We have obtained the well-known resultof Lyapunov. When condition (33.10) is satisfied, Eq. (33.4) has two linearly independent solutions xl= exp(-a

f).yl(f),

x g = exp(at).T,(t),

(33.16)

wherea=Inp,> Oand y k ( t + l ) . = y k ( t ) , k = 1 , 2. The set of solutions x ( t ) of Eq. (33.4) possessing the property that x (t) 0 as 1 43, is obviously given by the formula

- -

172

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L EQUATIONS

x = c a p ( - at).cf,(t),

(33.17)

where c is an arbitrary constant. Let u s find the set of initial values x (0) and x'-(O) of these equations. Obviously, these initial values are related by (33.18)

Without loss of generality, we can assume cp,(O) = 1 . Therefore, formula (33.18) can be rewritten x ' ( 0 ) = [ - a +cp;

(011 x(O),

Cpl(0) = 1 .

(33.19)

Keeping (33.16) in mind, we can write the integral matrixX of the system (33.3) as follows: X = exp([-a, where the matrix 2 (t)= 2 (t

a ] t).Z(t);

(33.20)

+ l), (33.21)

el

and the numbers q2and are constants to be determined; [-a, a ] is a diagonal matrix. In matrix form, the system (33.3) is written (33.22) To find the initial values x (0) and x' (0) of the bounded solutions, we must, as is clear from (33.19), find cf; (0). We can find the matrix (33.21) o r the matrix (33.20) as follows: We find the integral matrix X of Eq. (33.22) under the condition thatX(O)= I, in the form (8.8) o r (10.3):

X = exp At -2,(t), 2, (0)= I.

(33.23)

Here, A and 2, can be represented by the series (10.11) and (10.13) o r by formulas (8.6) and (8.7) [with 2 r replaced by unity]. Let us now find the canonical representation of the matrix A:

LYAPUNOV'S EQUATION

173

We can then rewrite (33.23) in the form

X = S exp ( [-a,

a] t) a s - '

2, (t).

(33.24)

The matrix

X = exp([-a,

a] t ) . Z ( t ) ,

Z ( t ) = S-lZ,(t)

(33.2 5)

will also be an integral matrix of (33.20), and the matrix (33.21) can be written in the form Z(0)

=

Ij q*' /

= s-1

(33.2 6)

$1

If we use formulas (8.6) and (8.7), then, in (33.23) we have A = InX(1). Z, (f) = exp ( [ - In X (l)] t ) X (f), andX (1)is given by the series (6.14) for t = 1

(33.27) (33.28)

m

x ( t )=

x k (t)ek,

(33.29)

k=O

The series (33.29) converges for all finite values of :and the general representation of the matrix (33.27) can be obtained from formula (28.18), where we need to set

174

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

Obviously, ' p l ( t ) in (33.16) is equal to the elements zll of the matrix Z in (33.20) o r (33.25). We note that 'pl (t) could also have been found as follows: If we make the substitution x ( t ) = exp (- at)-'pl(t), in accordance with (33.16), where (33.34)

in (33.4) and divide by em(- at), we obtain the following equation for determining 'pl (t) [here, we take 'pl (t) = 'p (t)]: d v + (as -e.p) 3 - 2a dt2 dt

'p = 0.

(33.35)

We noted right after formula (33.15) the conditions under which p2 (and hence a ) can be represented as a series in positive powers of 8

or E X . Consider the case in which m

k= I

Then, if we substitute the series m

m

&=I

k-1

(33.36)

into (33.35) and equate coefficients of like powers of 7 , we obtain equations from which we can find all the v k and ak, where the yk are subjected to the periodicity condition vk(t) = vk(t 1). In so doing, we establish the values of a,,whichcan also be found easily, directly from (33.34). Formula (33.34) provides an analytic continuation of the series (33.36) for a to all values of E since, f o r e - 1, we obtain a for Eq. (33.1). The analytic continuation of the series (33.36) for y is provided by the value of zll in (33.25), where the matrix &(t) is given by formula (33.28), in which X(1)is given by the series (33.29). This series converges for all finite values of e. We point out in addition that, in (33.28), exp [-InX(I).t] = Pa exp (- In PI 4 - P I exp (-- In P z - 0 +

+

Pa

+

exp(- lnp2-t)-exp(pa -P I

- P1

In pl.t)

x

(117

175

T H E FINDING OF PERIODIC SOLUTIONS

where pI and pz are given by formulas (33.11) and p1 pz -= 1. We have thus obtained Theorem 33.1. In the case (33.101, the set of initial values x(0) and x' (0) of the solutions x ( t ) of Eq, (33.4) [or equation (33.111 possessing the property that x ( t ) -,0 a s t -,ao, i s given by Eq. (33.19, where a is given by formula (33.34) and 'pl(t) is the element 211 of the matrix Z in formula (33.25). We have observed that a and 'p can be determined from Eq. (33.35) for small values of e. Specifically, it isclear from formula (33.34) just how small the values of E need to be since, if the series for a converges, it follows, as can beseenfrom (33.25), that the series for Z ( t ) also converges.

34. (33.1) The Case i n Which Equation ( 3 3 . 9 ) Has Roots l ~ ~ l = ~ ~ ~The l = lFinding . of Periodic Solutions. We have considered the case in which theroots p1 and pa of Eq. (33.9) are real and distinct. Now, assuming p > 0, we shall find conditions under which p1 and pz will be complex o r equal to unity. In this case, I p1 I = I pa I = 1 and all solutions of Eq. (33.4) o r of the system (33.3) are bounded and oscillatory (if we exclude the case in which pl = ps = 1 and a nonprime elementary divisor of the matrix (33.8) corresponds to this root). Let us suppose that e = - 1; that is, let us study Eq. (33.1). Lyapunov found an infinite number of inequalities [57] the satisfaction of any one of which ensures boundedness of all solutions of Eq. (33.1). These conditions exhaust all cases of boundedness of the solutions of Eq. (33.1) except for certain periodic solutions (cf. corollary to Theorem 34.2). Lyapunov first showed [26]that, for I

Spat we have

0

"I

.< 4

o r wJ p ( t )dt Q 4 if p ( t 0

lPll=

IPZ

I=

1.

+

0)

=p

(t)

(34.1) (34.2)

Then [ 571, he found all the remaining conditions under which (34.2) is satisfied. Let us elaborate on this question, using the a r t i c l e [58](which

rests on the results and ideas of Lyapunov's studies [26])and also Lyapunov's article [ 571. *Ibis work was completed in October, 1942, at a time when the author had no access to mathematical literature (except for possession of the book [26]). It was written on the

176

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

Since p > 0, it follows from (33.6) thatfk(t)and p ;< ( t ) are positive. Therefore (see (33.13)), q k ( t ) > 0 for t > 0. If we set E = - 1 in (33.12), we obtain

We see that, if

k=l

then PI and Pz are real and distinct (and PIPS = I).On the other hand, if

(34.5)

J (0,

then PI and Pa are complex (and I P i I a IPe 1 = 1 ) . Let us suppose that +k(l)-q&+I(l)>o

( k = 1, 2,*.*)-

(34.6)

Then ,

(34.7) k=l

since this is an alternating series (with terms decreasing in absolute value) and the first term is negative. If, in addition,

$ (-

k=l

l)k*k(l)

+ 4 7 0,

(34.8)

we have (34.5); that is, IPI I = I pz I = 1 , and all solutions of Eq. (33.1) are bounded and oscillatory. When recommendation of Academician V. 1. Smirnovwith the purpose of providing a generalization of condition(34. l b The author was unaware of [57] and other writings in that direction. During those days (October-November, 1942), the article [I41 was completed. ?his last article, which included the article [58] was defended by the author as a doctoral dissertation in July 1943 at the university of Kazan.

THE FINDING OF PERIODIC SOLUTIONS

177

(34.9) (34.4) is satisfied and not all the solutions of Eq. (33.1) are bounded. Inequality (34.8) will be satisfied if

since the sum of the discarded terms in (34.8) can, by virtue of (34.6), only intensify inequality (34.10). We note now that inequality (34.10) is equivalent to Lyapunov's inequality (34.1). Integrating by parts, we obtain

after which it is clear that inequality (34.10) is simply inequality (34.1). Lyapunov showed that, for all t and p > 0, the inequality [26, Chapt. 111, No. 48, formula (15)] (34.11) holds. Fort= 1, this formula yields

Therefore, for n>2, we have (34.6) in the strict-inequality form h-1(1) --h(l)

> 0,

(34.61)

if (34.1) is satisfied. Thus, (34.1) yields (34.6), (34.7) and (34.8). Therefore, all solutions of Eq. (33.1) are bounded. However, inequalities (34.6) may be satisfied when (34.1) is not satisfied. If, in addition,

1 78

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

we have (34.9). Consequently, when conditions (34.6) and (34.13) are satisfied, not all solutions of equation (33.1) are bounded. If (34.13) is not satisfied, but if

(34.14)

--1(1)+-2(1)--99(1)+42.0,

then we have (34.8) and all solutions of Eq. (33.1) are bounded. In general, as can be seen from (33.12), we have l e m m a 34.1. Suppose that *+&(I)-+gk+l(l) >/ 0, k

> 2m + 1 .

(34.15)

Then, f o r *+I

2m

(-

1)&+&(1) 40,

k= I

I(-

I)&+& ( 1 )

+ 4 2. 0

(34.16)

&=I

we have I PI 1 = I Pz 1 = 1; that i s , all solutions of Eq. (33.1) a r e b d e d but, f o r 2m

and also .for 2m+l

(34.18) h=l

not all solutions of Eq. (33.1) a r e bounded. For the moment, we exclude the case in which either

or m

If we have (34.16), then, on the basis of (34.15),

179

T H E FINDING OF PERIODIC SOLUTIONS

From this it follows (on the basis of (34.3)) that I?]!= I & I = If (34.17) is satisfied, then, on the basis of (34.15),

1.

(34.20) and a fortiori,

k= 1

Then, PI and PZ are obviously real. If (34.18) is satisfied, then, a fortiori m

m

and, consequently, P i and PZ are real. Remark 34.1. Lyapunov showed [57] by a complicated line of reasoning that

from which it follows that, if

(34.22) then

(34.23) Thus, inequalities (34.15) are satisfied if (34.22) is satisfied. l e m m a 34.2. (Lyapunov). Suppose that one of the inequalities (34.16), (34.177, o r (34.18) is satisfied. Then,

(34.24) (34.25) But then, on the basis of (34.21), we have

180

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

This contradicts inequalities (34.17) and (34.18) and the first of inequalities (34.16). Let us show that (34.26) contradicts the second of inequalities (34.16) also. On the basis of (34.26), we obtain from the second of inequalities (34.16) -41(1):-4>

0.

(34.261 )

since

But we have shown that (34.26) implies (34.24) [inequality (34.10) implies (34.6) when (34.12) is satisfied], and this means that (34.25) is not satisfied. This completes the proof of the lemma. Theorem 34.1. (Lyapunov). Z f (34.16) i s satisfied, then all solutions of Eq, (33.1) a r e bounded. Zf (34.17) o r (34.18) i s satisfied, not all solutions of Eq. (33.1) a r e bounded. Proof: Any one of the three inequalities (34.16), (34.17), (34.18) implies (34.24). Consequently, on the basis of Remark 34.1 and inequality (34.15), the conclusion of this theorem follows. We have shown that satisfaction of any one of the inequalities (34.16), (34.17),’(34.18) implies satisfaction of inequalities (34.15). However, inequality (34.15) can be satisfied when none of these three inequalities a r e satisfied. Let us give some other sufficient conditions under which (34.15) can be satisfied. Remark 34.2.When I

( pit

G

2 (2m -: 2),

(34.27)

b

inequalities (34.15) are satisfied. This follows from (34.12) with 11 > 2m 4-2 Remark 34.3. By a complicated line of reasoning, Lyapunov proved, instead of (34.15), the inequality [57, formula (26)]

(34.2 8) from which it follows that inequalities (34.15) a r e satisfied whenever

I’pdt b 1

4 2 (2m

+ a2.

(34.29)

T H E F I N D I N G OF P E R I O D I C S O L U T I O N S

181

Remark 34.4. Whenever [58]

we have + n ( l ) - +k+l(l)

>0

for k > n - 1 .

(34.31)

Proof: On the basis of (33.6), we obtain from (34.30)

f& (t)- fk-1

(f) 0,

(96( t )

-?;+I

(t) 0, k > n - 1,

from which it follows that

+

for k > n 1.. Remark 34.4 is contained in Remark 34.1, but it is much more easily proven. Theorem 34.2. Suppose that the inequalities h-1(1)

- +, (1) > 0, m

>/ n

(34.32)

aye satisfied and that, f o r every m , either

(--

l)k+&(l)

= 0,

p1= pz = 1

(34.35)

k= 1

in the first case and m

2 k=l

in the second.

(--- l ) k + k ( l )

+ 4 = 0,

pl = pz = - 1

(34.36)

182

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

We note that the quantitites in the first half of inequalities (34.33) and (34.34) decrease [by virtue of (34.32)] with increasing rn and that those in the second increase. Corollary. I f (34.35) is satisfied and cp(l)=f’(l) -0,

(34.3 7)

then, by virtue of (33.8), the general solution of Eq. (33.1) will be periodic with period o = 1. If (34.35) is satisfied but (34.37) is not, we have a one-parameter periodic solution, but thegeneral solution will not be periodic(or bounded). I f (34.36)and (34.37) are satisfied, then the general solution of Eq. (33.1) is periodic with periodo = 2. On the other hand, i f

=+0,

cp2(1) +f’2(1)

(34.38)

we have a one-parameter solution with period solution will not be periodic.

o = 2, but

the general

Suppose that (34.35) is satisfied and that Y (1)

=+

(34.39)

0.

Then, we have a one-parameter periodic solution to Eq. (33.1) with period u) = 1: x = cx1 (t),

(34.40)

where c is an arbitrary constant and x,(t)is a periodic solution of Eq. (33.1) with period o = 1. Let us find the initial values of x l ( t ) and the entire set of initial values of the periodic solutions (34.40). The general solution of Eq. (33.1) is of the form -I-( t ) = c1 f

( t ) -t

c2

93 (0.

(34.41)

Suppose that

(34.42)

x(t+ I)=x(t).

Then,

+ cz x‘ (0) = c, =cJ’(l) + s ( 0 ) = c, = c J ( 1)

‘p (l),

C,?’

(1).

From this, we have

(34.43)

I83

THE FINDING O F PERIODIC SOLUTIONS

The determinant of this system is A=f(l)8’(1)-8’(1)-ff(l)-8cp(l)f’(1)

+1

= 0.

since, in accordance with (33.8), (33.9) and (34.35),

Therefore, cl and (34.39),

c,

can be found from (34.43). On the basis of

Therefore, we write (34.41) in the form (34.44)

Obviously, the solution (34.45) has initial conditions

(34.46) Consequently, periodic solutions x ( f ) of (33.1) in the case (34.35) have initial conditions

(34.47) where c i s an arbitrary constant. In other words, these initial values are given by X’

1-f(l) . (0)= x (0)-

(34.48)

If (34.35) is satisfied and if ’

f’ ( 1 ) # 0,

(34.49) /

1 a4

L I N E A R SYSTEMS

OF ORDINARY DIFFERENTIAL EQUATIONS

then, instead of (34.47) and (34.48), we have x’

(0)=: c, x (0)= c 1 - cp‘(I)

(34.50)

f‘(1)

(34.5 1) and, instead of (34.44),

(34.511) If it turns out that f (1) == 1o r that cp‘ (1) = 1, we shall have respectively xl(t)= f ( t ) o r x l ( t ) = cp(t) and, instead of (34.48) o r (34.51), we shall have x(O)=c and x’(O)=Oor x(O)=O and x’(O)-c. Remark 34.5. Obviously, the conditions under which a periodic solution of (34.35) o r (34.36) exists and under which the general solution of (34.37) i s periodic are not connected with the condition p(t)>O. If conditions (34.35) and (34.37) are satisfied, then, on the basis of (33.9), we have cp‘(1) f ( 1 ) == 1 and, if conditions (34.36) and (34.37) are satisfied, we have [ ( I ) ~ ’ ( 1 )=- - 1. 7-=

35. Regions of Values of the Parameters Appearing in Equation (33.1) in Which There Are Bounded and Periodic Solutions Consider the equation d2x -++P(t)x=O dt2

(35.1)

(Eq. (33.1) repeated). Suppose that the function p ( t ) is a nonnegative periodic function with periodw = 1. If p ( t ) contains a parameter E , then Theorem 34.1 enables us to obtain those regions of values of B in which the general solution of Eq. (35.1) is bounded and those in which it is unbounded. Theorem 34.2 enables us to find those values of E (from Eqs. (34.35)) at whichthere is a periodic solution with period w = 1 (from (34.35)) or w = 2 (from (34.36)). Thecorollary to Theorem 34.2 gave conditions under which the general solution is periodic. If e satisfies Eqs. (34.35) and if cp(1) = f ’ ( l ) = 0,

185

R E G I O N S OF V A L U E S OF T H E P A R A M E T E R S

then the general solution is periodic with period satisfies Eqs. (34.36) and if

!#(I)

U)

: :

1. But if o

= f ' ( 1 ) = 0,

then the general solution is periodic with period 0=2. It may happen that p ( t ) contains parameters 2 and p. Then, from (34.35) and (34.36) , we also find the functions p c p(e) for which Eq. (33.1) has a periodic solution. If p ( t ) is an entire function of E and p, the function

will also be anentire function. It will then follow [32, Theorem 12, p. 471 that the function p = p ( ~ can ) have only algebraic singularities. It was shown in [32] how to find the region of convergence of the series p = p(e)

=;

k=U

.

Suppose now that 'p(1) is expressed in terms of parameters E , p, and A , and that it is an entire function of these parameters. Then, the functions (see Sect. 6)

&=I

will also be entire functions of these parameters. How can we find values of the parameters E , p, and ir for which Eq. (33.1) has a periodic solution? We can find the set of these values from (34.35) (then, there will be a periodic solution with period w = 1) or (34.36) (there will be a a periodic solution with period w = 2). To find those values of E., p, and ii for which the general solution of Eq. (33.1) will be periodic, we first find the functions p = p(e) and i, = k ( e ) from the equations Cp(1)

==

0 f'(1)= 0 .

(34.37)

If we substitute these functions into Eqs. (34.35) or (34.36), respectively, we obtain an equation from which we can, generally speaking, find the values of el, c2, ... (and hencep,, p2, ..., )ill A,, ... ) at which the general solution of (33.1) is periodic.

I86

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

We note also that, in satisfying conditions (34.37) and (34.35) o r (34.36). we a r e in effect (by virtue of (33.9)) satisfying the equations q(1) = f ’ ( 1 ) = 0, f ( 1 ) = # ( I ) = 1

or cp(1)

= f’(1) = 0,

f(1)

= rp’(1) =

-I.

Suppose, for example, that

P (0 = E Pl(t) Then, (34.35) will be of the form

x(-

+ P Pz (4.

P)

l ) ~ ~ k ( l ) = - ~ l ( l ) - r ~ ~ ( l ) - . . -0. .

k=l

where (see (33.14))

and the q k ( l ) (for(&> 1) will be infinitesimals of order k for small values of E and p (which is obvious from formulas (33.6) and (33.13)). On the basis of (34.35). we see that, if pa(t)dt+= 0. 0

there exists a function p = ale

4-azez + ...,

that is holomorphic in a neighborhood of e 0 and for which Eq. (35.1) has a periodic solution with periodw = 1. On the basis of the results obtainedinsect. 6 of [32],it is usually easy to ascertain whether, under all other assumptions regarding I

I

p d t , a function p - - p ( ~ ) exists that approaches 0 as

pldt and 0 E

0 --j

0 and for which Eq. (35.1) has a periodic solution.

REGIONS

OF

VALUES

OF THE

PARAMETERS

187

For a periodic solution to exist, it is necessary and sufficient that a real function 7 == q(e) exist, defined by the equation

that approaches zero as E - 0 .

Since the function

k- I

is an entire function with respect to : and q, the function 7,.=3(z) (which -4as I 0), if it exists, can, under analytic continuation, have only algebraic singularities. It is shown in 132, Theorem 121 how one may find the radius of convergence of the series +

However, we see that Eq. (35.1) cannot have a periodic solution with period 0=2 for small values of E andq since Eq. (34.36) is not satisfied for small values of E and q. If we assume that

in (35.1), then, ingeneral, (35.1) will not have a periodic solution for small values of E and q since the condition p ( t ) >/ Ocontradicts the equation

Actually, this follows from the more general hypothesis. Specifically, suppose (see (34.22)) that -ql(l)-l-?2(l)

40.

(35.2)

188

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

Then (see Remark 34.1),

and, hence, the equation (34.35)

is impossible. But if the *jl(l)a r e infinitesimals of order k for small values of E and :A, then, for sufficiently small 9 and p, (35.2) will be satisfied. It follows from this that a periodic solution with period w =-. 1 does not exist. But Eq. (34.36) is also impossible for small e and q. Remark 35.1. We have seen that if the condition

I f ( ] ) + g‘

-4 = 0,

is satisfied, then Eq. (33.1) does not have a solution that is periodic with period w = 1 or UI = 2. However, among the bounded solutions that do not approach zero (which can be found, for example, with the aid of Theorem(34.1)), there may stillbe periodic solutions with period w=n o r i = 2 n , where n is a positive integer. These periodic solutions exist if and only if

If (n)-;-

cp’

(n)]2-

4 = 0.

(35.3)

If

k=O

there i s a periodic solution with period w=n. On the other hand, if (35.5) k= 0

a periodic solution with period = in.will exist. For the general solution to be periodic with period n o r 2n, it is necessary and sufficient that, in addition to (35.3). the condition (8)

y (n)= cp (n)= o be satisfied.

(35.6)

189

REGIONS O F V A L U E S OF T H E P A R A M E T E R S

Just a s in the cases in which n = 1 orn = 2 , it is easy to find the relation between the parameters appearing in the expression for p ( f ) . that ensures the existence of a p'eriodic solution with period w = ft (on the basis of (35.4)) o r w = 2n (on the basis of (35.5)). In accordance with (34.45). it is easy to find this periodic solution: (35.7)

if q ( n ) + 0. Or, in accordance with (34.50,), we can find it from the formula (35.8)

if q ( n )= 0 but f' (n)# 0. Here, we need to set p = p ( E ) as found in (35.4), in (35.7) and (35.8). Or, in other words, the solution (35.7) is periodic by virtue of Eq. (35.4). The initial values of the periodic solution (35.7) are (35.9) We have shown that Eq. (35.1) can have periodic solutions with period = 1. 2. .... The question then arises a s to whether it can have periodic solutions with period w, where o is not an integer. We shall shortly prove Remark (35.2). With no loss of generality a s regards this question, we may assume that Eq. (35.1) does not have a periodic m

solution with rational period u) = - < 1, where m and It are integers n without a common factor. l e m m a 35.1. Eqzlation (35.1), where p ( t ) can have only integral periods n, cannot have a periodic solution with p e r i o d o + n . Proof: If x ( t ) is a periodic solution with period o, then x ( t .+ o) = x(f). Therefore from (35.1). we have

[P(t

+

0)

-P

(01 x (0 = 0.

from which the assertion of the lemma follows. m If p ( t ) has period m1 = 1 and rational period co, = - < 1, n where in and it are positive integers without common factor, then p ( t ) is of period m3=- 1 This follows from the fact that [61] 11 1 integers k and 1 exist such that kn t l m = 1. If - and -a r e

.

periods, then

-is also a period.

n(nS.1)

n

If p ( t ) has periods

n f l

190

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

then p ( t ) is constant. From this and from what was said above, we have l e m m a 35.2. I f p(t) i s a nonconstant periodic function with I rational periods, there exists a smallestperiod, 0) = -, where n i s a ri positive integer, such that all pem'ods can be obtained f r o m the k formula w = -, where k i s an integer. n We may assume that the smallest period is = 1. Then, all periods can be obtained from the formula w = n , where IZ ranges over the integers. This proves Remark 35.2. In Sect. 36, we shall show that Eq. (35.1) cannot have periodic solutions with irrational periods either. Now, let us look at the question of the radius of convergence of the series representing a periodic solution of Eq. (35.1). Suppose that. from (35.4) and Theorem 7 [or the results obtained in connection with Eq. (6.40)] of the book [32] (see Sect. 47 of the present book), we have a relation between p and E m

(35.10) k=O

where the pk a r e real and I E 1 Q r , under which (35.1) has a periodic solution (but not all solutions of this equation will be periodic). The radius of convergence of the series (35.10) is determined in [321. We note also that it is always possible to assume p represented in the form of a series (35.10) [in positive powers of E ] since we

can always represent

E

in the form

If substitution of the expression for 11givenby (35.10) into gives u s

p

(e, y,

t)

m

(35.11) k=O

then the solutions of Eq. (35.1) can be represented in the form of series of positive powers of E. The series (35.11) will, in general,

191

R E G I O N S OF V A L U E S OF T H E P A R A M E T E R S

converge in the region in which the series (35.10) converges. Therefore, the series representing the solutions of Eq. (35.1) also converge in the region of convergence of the series (35.10) if the initial values of these solutions are independent of c (see Theorem 6.1). For example, in this region, the solutions f ( f ) and q ( f )given by formulas (33.5) can be represented by such series. But the region of convergence of the series representing a periodic solution of (35.7) can also be determined by means of a series representing i ( 0 ) in accordance with (35.9). This region of convergence of a series (representing a solution of (35.7)) is determined by the inequality le!

where R is the smallest absolute value of the nonzero roots equation

E

(35.12) * of the

q ( n , 6 ) = 0. (35.13) But if (35.13) holds and f ’ ( n . :*) # 0, we obtain a periodic solution in accordance with formula (35.8) and the region of convergence of the series representing the periodic solution will be

(35.14) where R1 is the smallest absolute value of the roots equation f’(n, E )

= 0.

of the

E : ~

(35.15)

However, if E* is a root of Eq. (35.13), then the radius of convergence will again be lengthened. If, in addition, f‘(n, E * ) = 0

and

q(n, E*)

= 0,

(35.15 1)

then, for ~ = t : *, all solutions of Eq. (35.1) will be periodic with period w=n, including the solutions f ( t ) and q ( t ) given by formulas (33.5), for which the series in positive powers of E converge in the region of convergence of the series (35.10). We have obtained the following result: If Eq. (35.1) has a oneparameter periodic solutionwith period co=n., then the basic periodic solution of (35.7) or (35.8)canbe representedas a series in positive powers of E that converges in the regionin which the series (35.11) converges. On the other hand, all solutions of Eq. (35.1) a r e periodic, then the solutions f ( f ) and q ( t ) will also be periodic and representable in the form of series that converge in the region of convergence of the series (35.11). Thus, we have proven

192

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

Theorem 35.1. Periodic solutions of Eq. (35.1) can be represented as series in positive powers of s that converge in the region in which the series (35.11) converges.

The set of equations (35.7) and (35.4) provides the general representation (for all values of E and cc that are admissible for given p(t, E , p)) of a one-parameter solutionwithrespect to E and p. From (35.7), we can also obtain an expansion of this solution in a series of positive powers of e. However, let us show how we can obtain a periodic solution (35.7) in the form of a series of positive powers of t directly, without using the representation (35.7). Let us auppose that we have proven the existence of a solution (35.10) of Eq. (35.4). In so doing, we have proven that there is a periodic solution of Eq. = n. Let us suppose now that, on the basis of (35.1) with period (35.10). u)

Let us seek a periodic solution of Eq. (35.1) with period the form

Q) :=

n in

(35.16)

The initial conditions of the solution (35.7) [and hence, of (35.16)J will be (35.17) If we substitute (35.16) into (35.1), we obtain

k=O

k=l

k=O

If we equate the coefficients of like powers of b.

= 0, x1+ p1 (f.

PI) = 0,

t,

we obtain (35.18) (35.19)

REGIONS OF VALUES OF THE PARAMETERS

On the basis of (35.17), we take

x, = 1.

193

From (35.18), we obtain

where c1 is an arbitrary constant. Since this function is periodic with period m = n.,we need to choose p1 from the equation

f p1(t, pl) dt

= 0.

(35.21)

h

Obviously, the constant cl is the coefficient of the first power of in the expansion of x ( 0 ) given by Eq. (35.17):

E

(35.22)

From (35.20), we obtain (35.23)

Therefore, we find c1 in the form

Similarly, we find all the other equations

pk and ck

(for k = 2, 3,

...) from the

and

We have proven the existence of a periodic solution (35.16) under the assumption that the solution (35.10) of Eq. (35.4) exists.

I94

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

...

Therefore, we can find pl, pe, from Eqs. (35.21) and (35.35). In accordance with Theorem 35.1, the series (35.16) converges at least throughout the region in which the series (35.11) converges. Consider now a system of n linear differential equations (871 dX = X P ( t , p. E) (35.27) dt

.

where X is an nth-order integral matrix and P(f, p, E,) is an nthorder matrix that is continuous and periodic with period 2x with respect to t and that can be represented in the form of a series

that converges in the region (35.29)

Consider the integral matrix (35.30)

which can be represented a s series of the forms

or

these series converging uniformly in the region 1 p I < R , I E 1 < R , a < t < a, where a is an arbitrary positive number. Let us find a function (35.33)

such that the system (35.27) has a periodic solution with period o = 2xn. We need to seek this function from the equation

195

REGIONS O F VALUES OF THE PARAMETERS

since one of the characteristic numbers of the matrixX(2nn, p, E ) will be equal to unity when this equation is satisfied and this ensures the existence of a periodic solution of the system (35.27). We assume that Eq. (35.34) determines the function (35.33). where the series (35.33) converges in the region*

I

El

(35.35)

48.

Suppose that, for such p = p (E), the system (35.27) has a v-parameter family of periodic solutions. If we substitute (35.33) into (35.27), we obtain

-

dX -= XP (t,E ) dt

=

;I]Pk( t )

Ek

(35.36)

k=O

and this series, generally speaking, converges at least in the region (35.35). Suppose that

( I = 1,

..., n)

is a v-parameter family of periodic solutions the existence of which we have assumed. Let us suppose that the parameters a,, , a. a r e independent of E . Then,

...

12

(t) =

...

XcGkl

(f,

E)

(f!

= 1,

..., It).

(35.38)

k:=l

Here, clr , c, a r e constants and the X & ( t , E ) a r e the elements of the integral matrix X ( t , E) (where X ( 0 . 8 ) = I)of the system (35.36), which can be represented in the form of series (35.39)

that converge in the region (35.35). .?his region is found in accordance with Sects. 8 and 9 of [32]. See also Sect. 47 of the present book If Q. (35.34) defines a functlon

where p is a positive integer, then, instead of (35.36). we obtain a series in powers of and our subsequent reasontng is the same.

196

LINEAR S Y S T E M S OF ORDINARY D I F F E R E N T I A L EQUATIONS

We can find* the solutions of (35.37) by substituting (35.38) into the equations x,(2ffn)-x,(O)=O

(1-

1, ...) n ) .

(35.40)

Since, by assumption, we have a v-parameter familyof periodic solutions of (35.37). the rank m of the matrix A (E)

=X

( 2 C.~ E) - I,

(35.41)

corresponding to the coefficient matrix of the system (35.40) will be equal to Y , that is, rn = v. Consider all Y th-order minors A h ( € )(for k = 1, 2, , 9J of the , i q a r e the roots of smallest matrix (35.41). Suppose that E , , , 9). The absolute value of the equations A k ( 8 )= 0 (for k = 1, radius of convergence of the series representing the x, (f)(for I = 1, , n ) is not less than r = max { I el(, ..., l s q l ] since we can choose a s basic (in solving the system (35.40)) anyone of the minors A&(€). Suppose that r = 1 ~ ~ 1 .However, if E, is not a root of one of the minors A1(~), , A q ( E ) , then i E~ will not be a singular point of the series representing a periodic solution of (35.38). On the other , 9), then hand, if E; is a zero of all minors An(€) (for k = 1, the rank m of the determinant (35.34) will be less than V ,which contradicts the assumption. Consequently, the radius of convergence p of the series representing periodic solutions of (35.38) is less than r. Thus, we have proven Theorem 35.2. The series (35.37) representing periodic solutions of the system (35.27) withperiod o = 2rr1 converge (at least) in the region (35.35) of convergence of the sepies (35.36). In connection with the existence of periodic solutions of the system (35.27) with periods not equal to 2% n and methods of finding them, we point out the following facts: It follows from Lemma 35.2 that, by means of a transformation of the independent variable, it is possible toarrange for the matrix P in the system (35.27) to have periods only of the form o = 2 a n , where n is an integer. Furthermore, the following two facts follow immediately from the results obtained in Sect. 36: (1) if the system (35.27) is second-order, it cannot have periodic solutions with period different from the periods of the matrix P (cf. the reasoning following formula (36.2), where A can now be periodic with period 2 a n and where a = 2x6:); (2) if the system (35.27) is mth-order, it

...

... ...

...

...

z

...

.

CThat is, we find q. .... c, from Eqs. (35.40) such that (35.38) will be a periodic solution. Here X I (f)is found as the ratio of series in posidve powers of E, that converge in the region (35.35). The denominators of these series consist of auth-order minor of the matrix (35.41)

197

PERIODIC SOLUTIONS OF A LINEAR HOMOGENEOUS SY ST E M

can have periodic solutions with period different from the periods of the matrix P and even with period incommensurable with 2 ~ . Existence tests and methods of finding such periods a r e given in the following section.

36. Periodic Solutions of a Linear Homogeneous System of Differential Equations [94] We have found necessary and sufficient conditions for the existence of periodic solutions of Eq. (35.1) with period o = n and methods for constructing them. However, the question arises: have we found all periodic solutions? We know [60]that a nonlinear system can have a periodic solution with period that is incommensurable with the period of the right-hand members (as functions of t ) . We shall now show that this is impossible for a linear system of two differential equations. Theorem 36.1. Let dx dt

-=

xP(t), P ( f

+ w) = P ( t ) + const.

(36.1)

be a system of two differential equations, where x i s a twodimensional vector and P(t) is a second-order matrix the elements &(t) of which possess the PY@e&Y that pkl(tv)dpkI(T) as tv+T by some mode of approach. Here, T i s arbitrary and Pkl(t) i s continuous at the points t,. Then, the system (36.1) has no periodic solution with period a that i s incommensurable with W . Remark 36.1. Let cp(t) be a function such that, for every T. there exists a sequence tV-T such that q(t,)-+cp(T) as t,-T and suppose that q(t) is continuous a t the points t v . Then, if q ( t ) has two incommensurable periods a and 6, it is constant.* Proof. We have cp(t+an+bm) = q ( t ) , where rn and n are integers. Let 'c denote an arbitrary number. Then [61], M + 6m --., .E as Irn 1- 00 and In I 00 in some manner. From this it follows that 'p (t + T) = cp (t); that is, q ( t ) is a constant. Suppose that (36.1) is a single equation with a single unknown function. Then,

-

I

x = exp j P ( t ) d t 0

and x cannot have a period a different from o. Let us now suppose that (36.1) is a system of two equations. Then, if x = (q, xa)has a

*It follows from this on the basis of Remark 35.2 that Eq. (35.1) has no periodic solutions with period incommensurable with the period of the functionp (t).

198

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

period a that is incommensurable with [since x(t+a)=x(t)and x ( t + a ) = x ( f ] x (t) [P(t

o, we

obtained from (36.1)

+ a) - P (t)]= x (t)L (t. a) = 0.

(36.2)

Therefore. 2

(36.3)

where q!k'(t)= Pvk (f i a) - P v k (t)

9

that is, q!k)(t)a r e periodic functions with periodw. Since the matrix P ( t ) is nonconstant, it follows in accordance with Remark 36.1 that one of the quantities q!k)(t)$0. But then, we obtain from (36.3), for example, x2(tj = Ax, (t), where A is a constant since there are two incommensurable periods (w and a ) . Therefore, we have x1

= [Pldt)

+

AP21(41 X1*

Consequently, x1 cannot have a period distinct from 10. This completes the proof of Theorem 36.1. This theorem is not valid for a system of more than two equations. Example. The system

(36.4)

x , = A s i n f + B c o s t, x 2 = - A c o s t + B s i n t , -r3== (B --- A) cos t + ( B A) sin t,

+

where p l ( t ) , p 2 ( t ) and p 3 ( t ) a r e arbitrary functions and A and B a r e arbitrary constants. From this we see that, just as in the general case of nonlinear systems [60],a linear system can have periodic solutions even when the coefficient matrix P ( t ) is not periodic. This is also valid for a system of two linear homogeneous equations. We can find periodic solutions with period incommensurable with the period of the matrix P(t), or in the case in which the matrix

199

P E R I O D I C S O L U T I O N S OF A L I N E A R HOMOGENEOUS S Y S T E M

P (t) is nonperiodic. We then obtain the conditions under which there are no periodic solutions with period incommensurable with the period of the matrix P(t).. Specifically, we find the period a from the equation

D ( L( t , 4)= 0 ,

(36.5)

where D is the symbol for a determinant. If there is no number a incommensurable with u) that satisfies this equation, there is no periodic solution x ( t ) with period incommensurable with w. Suppose that Eq. (36.5) has a root incommensurable with w. Let us substitute this value a into Eq. (36.2). Suppose that the system (36.1) is a system of n equations. Then, (36.2) will be a system of n linear ,x,. homogeneous algebraic equations with unknowns x,, x2, Let us suppose that the rank of the matrix L(t, a) is equal to 12L Then, x k = a$l (for k = 2, 3, , n ) , where the ak a r e constants. We see from this that the system (36.1) cannot have a periodic solution with period a incommensurable with 0. Suppose now that the rank of the matrix L ( t , a) is equal to n - 2 Then, from Eqs. (36.2), we obtain

...

...

xk

= ak (f)

+ bk (f) x,

( k = 3, 4, .... n).

(36.6)

where the ak (t) and &k (t) are periodic solutions with period w . Let u s suppose that among the coefficients ak and bk there is one, let us say, b3(t), that is constant. Then, we have -%(f4- a)

- -%(t)= [a,(t4- a)-

a3

(61X i 4-Ib3 (f4- a) - b3 (t)]X,

+

= 0.

+

In accordance with Remark 36.1, we have b3(t a) - 4 ( t ) 0. Therefore, we have xz = Axl, where A is a constant. Consequently, , n). We again obfrom (36.6), we obtain X k ( t ) = A g , (for k = 2, tain i = a ( t ) x , . where a ( t ) is a periodic function with period w. Therefore, x; (and, consequently, x ( f ) ) cannot have a period incommensurable with w.. Suppose now that all the coefficients a k and 6k in Eqs. (36.6) are constants. Using these equations, we obtain the systemof equations

...

&=a(t)x1

+ b(t)xs,

& = ~ ( t ) ~ ~ + d ( t ) ~ ~ , (36.7)

where the coefficient matrix P (t) is periodic with period 10. If this matrix P ( t ) is nonconstant. then, according to Theorem 36.1, the system (36.7) cannot have a periodic solution (xl, );. with period a incommensurable with (0. But this matrix P ( t )may also be constant. If P ( t ) is a constant matrix, then the system (36.7) can have a periodic solution with period a incommensurable with OD. Suppose

200

L I N E A R S Y S T E M S OF ORDINARY D I F F E R E N T I A L E Q U A T I O N S

that such a solution (xl, x?) is found from the system (36.7). On the basis of (36.6). we obtain solutions xl, , x,, that are periodic with period a. If these x l , , x,, satisfy the system (36.1), we also obtain a periodic solution x ( [ ) of the system (36.1) with period a incommensurable with u' We consider those cases in which the rank of the matrix L f f , a) is equal to k c n - 3 in the same way. Here, we first of all obtain a system of ti -- k linear differential equations the coefficient matrix of which is of period Suppose that the system (36.1) is of the form

...

...

.

19).

(k

1.

.... 11).

(36.11)

Then, Eqs. (36.2) will

D

V

k (pk

(t i a ) - pk ( t ) )x,

-='

0, k = 1 . .. ., 11 .

.=.I

If a is not the period of the functions & ( t ) (for example, if the p,fl) a r e periodic functions with period 0 ) incommensurable with a ) , then, in accordance with Remark (36.1), we have

and these equations a r e of the form n

,= 1

If the determinant L) ( )/a,,.R ji j # 0, then the system (36.11) has no periodic solution with period a. (This is true both when the coefficient matrix of the system (36.11) is periodic with period 0 ) incommensurable with a and when this matrix is nonperiodic and Pk ( t

+a) -

Pk

!t) f 0.)

Let u s now suppose that

D ( It a,, jl ) = 0

(36.8)

and that the rank of the matrix it uvkj j is equal to n - m. Then, from (36.21), we have

c m ...

.vv =

[=I

C,,X!.

'

- constant = m -!-1, .... ti.

c,./ v

(36.9)

P E R I O D I C S O L U T I O N S OF A L I N E A R HOMOGENEOUS S Y S T E M

20 1

If we substitute the values given by (36.9) into the first m of equations (36.11), we obtain .i*k =

9

/Pok(t)

3-

-

(‘1

Prn+l, k

(’nr+l. v --

... -- pn. k (1) en, v l -v,

I

or, remembering the values of

Prk(t)

in Eqs. (36.11),

Here, it may happen that

where k = 1, form

..., nr and

v

= 1,

..., m.

Then, Eqs. (36.10) take the

This system with real constant coefficients can have distinct twoparameter* solutions (several at once) with period a incommensurable with a. If we have found a periodic solution ( x l , , xnl) of this system, then we shall also find from (36.9) xn1+,...., xn that are periodic. If these values found for q, , s,,satisfy Eqs. (36.11), then (36.11) has a periodic solution with period a incommensurable with 0). Let us suppose that Eqs. (36.11) are not satisfied. Then, Eqs. (36.10) have variable coefficients. Thus, we have arrived at a system of the form (36.11) with unknowns s,, , IC,.Again, we need to look at this system. Finally, either we shall obtain a system with constant coefficients or we shall arrive at a single equation with a single unknown function or we shall arrive at two equations withtwo unknown functions, and the question will be answered. Thus, we can subject the parameters of the system (36.11) to conditions that will ensure that this system has a periodic solution with prestated period a independent of the period of the functions PI.(0.

...

...

...

*In Sect 42 we shall find necessary and sufficient conditions for the existence of purely imaginary roots of the characteristic polynomial. These conditions ensure the existence of periodic solutions.

202

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

37. An Equation of the Form (33.1) With Variable-Sign Function P(t) Thus, thanks to the profound investigations of Lyapunov, the questions regarding the existence of bounded solutions of Eq. (33.1) have been entirely settled from a theoretical standpoint. Beginning with Zhukovskiy [ 621, many authors have found sufficient conditions for boundedness of the general solution of Eq. (33.1) (both for constant-sign and for variable-sign p ( t ) ) , using different devices. Examples of such studies are [63-651. Several surveys of these investigations appear in the articles [66-681. W e shall not stop for this. Lyapunov also considered the case in whichp (t) is a variablesign function. He exhibited various transformations that transform Eq. (33.1) with variable-sign p ( t ) into an equation of the same form with constant sign p ( t ) ([26],p. 332). Let us show how he did this. Consider the Eq. (33.1) S + p ( t ) x = 0, p ( t + w ) = p ( t ) .

(37.1)

We introduce a new unknown function 9 defined by .Y

(37.2)

= W!/

and a new independent variable T' defined by

+.

u,? dt

(37.3)

0

Here, is a real periodic function with period vanish and that has continuous derivatives zcl

o

that does not

We obtain (37.4)

where 41(T.)=Q(t)=d[icr+p(t)lplI

is a periodic function with period

(37.5)

AN EQUATION OF T H E FORM (33. I)

203

(37.5 1) that may prove to be sign-constant. Suppose that w > 0. If (37.6) then 9 & 0 and, consequently,* the general solution of Eq. (37.4) (and with it, (37.1)) will be unbounded. Lyapunov realizes the case (37.6) in the following manner: Let k denote any real number such that &J2a is not an integer. Then, the function W+Prord-O,

w ( t ) =_1 . -

'ko Cwk( 2k sin 2 O

which is periodic with period ri

m,

S-

$)p(s+l)ds

(37.7)

is asolution of the equation

+ kk = &* -- p ( t ) .

(37.8)

Suppose that k satisfies the condition P-ppO,

w-110.

(37.9)

Then, by virtue of (37.8), rer$puv=

-

( w - l)(P--p).

(37.10)

Therefore, the condition (37.6) is satisfied. Of course, it may turn out that, on the basis of (37.7), 4 (t)

a 0.

(37.11)

Then, we need to use Theorem 34.1 to solve the problem. With the aid of this device, Lyapunov studied the differential equation

where

U)

= 2c,

L

7

0, and n is a positive integer. For this equation,

*Cf.the reasoning following formula (33. IS).

204

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

formula (37.7) yields

Condition (37.9) will be satisfied if

k2 + h2 - i.28 7 0

J (t) =

s

2sinkz ka

(37.14)

c o s k ( ~ - - ) s i n ~ ( .+r t ) d t

(37.15)

0

Lyapunov noted that, for even n and negative p

=_

--

A2(1 --Esin"t)

2,

4 0,

that is, the general solution is unbounded and, for odd n, the sign of e, does not affect the answer to the question (since the left member of inequality (33.10) contains only even powers of E ) . Therefore, we may assume that E, 7 0. By using different devices, Lyapunov studied many particular cases of Eq. (37.12). We note that the integral in (37.15) can be calculated. If we find the maximum J (I) = M from the condition that J' ( t ) = 0, then, for M
we have (37.15). Let us define 2%

J , (t) = cos k (7_TC). sins (7

+ t ) d7.

(37.17)

0

It is easy to see that the equation

J ; (t)= 0

(37.18)

leads to the equation 7.7

0

sink (7 - x)sinn (T + t )d T = 0.

(37.19)

AN EQUATION OF T H E F O R M (33. I)

205

Thus, to show that condition (37.15) is satisfied, we need to show that ka J , (t) Psin k T:

J (t)=

(37.20)

1

on the basis of (37.19). Let us consider in greater detail the case in which n == 2. From (37.17), we have Jl(t)=

\ cosk(r-

2:

.rr).siii2((.:+t)d.c=

b

(37.2 1)

sinkx. ---___ ks-4

{

.

In accordance with (37.20), we obtain (37.22)

k2- 4

from which we get J'(t)

=

~

eka

k2 - 4

sin2t -=- 0.

Therefore, two extreme values of J ( t ) a r e J ( 1 ) =-

2E 4 ___ k2 '

J(2) = E

-.2 - k2 4 -k2

Thus, if (37.14) holds, if J ( I ) 4 1, and if J ( * ) 4 1, then conditions (37.9) will also be satisfied; that is, 4 ( t ) 4 0.

(37.23)

But, for ks < 4, we have J ( * ) L J ( I ) and, for k* > 1, we have J ( I ) < J ( 2 ) . Therefore, for condition (37.23) to be satisfied, it is necessary that 2e

-
XO'(E-

1)
if

k2<4

(37.24)

k 2 > 4.

(37.25)

and 5-

2 - k2 .=c 1, A2(&-4 k2

-

1) .c k2, if

206

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

Consequently, if we have [from (37.24)] Ae(€ - 1)

< 4,

(2

+ ke)(e - 1) < 2,

then the general solution of Eq. (37.12) will not be bounded for R = 2. The first and third of Eqs. (37.25) are impossible.

If 1

<E-

2 - k' and ke <4, 4 - ke

then 1

< J ( 2 ) <&/('I.

-

In this case, w 1 < 0. It is easy to exhibit* in addition inequalities that can be satisfied for k , A, and E such that w 7 0. Then, we shall have (37.11). Lyapunov noted that in determining the solution of Eq. (37.8) from formula (37.7) and requiring that w- 1 be nonnegative, we are in effect considering the case !.it

< 0.

(37.26)

0

since it follows'from (37.8) that m

OI

1) dt.

Therefore, for the case in which

(37.27) Lyapunov exhibited a different w obtained from the first by replacing kz with k? This second UJsometimes satisfies the inequality

-

*If we substitute the values of t from the equation W' = 0 into , I we obtain

Now s(')> 0 and oo(')>

0 are necessary conditions for w > 0.

AN EOUATION OF T H E FORM (33. 1)

207

(37.2 8)

W+puv>O,

which yields q(t) > 0. Let k denote a nonzero real number. Thenthe equation

..

W-k%=-kB-pp, (37.29) obtained from (37.8) by replacing P with .. k2 has the periodic solution

Here, we have from (37.29) G+pW=(k'+p)(w-

1).

(37.31)

+

For rks p > 0 and w - 1 > 0, we obtain (37.28) from (37.31). Consequently, q ( f )> 0. Lyapunov also noted that such a transformation in the case of p ( t ) > 0 for every nonzero k leads to the case in which 9(t) > 0. However, it may happen that, f o r some k, one of the tests f o r boundedness of the general solution of Eq. (37.4) i s satisfied in accordance with Theorem 34.1 and yet none of these are satisfied for Eq. (37.1).* If one of these tests is satisfied for Eq. (37.1), then there exists a k such that this test will also be satisfied for (37.4) since (37.1) is obtained from Eq. (37.4) by taking the limit as k 2 + 00. As an example, Lyapunov considered the equation

and found sufficient conditions for boundedness of the general solution with the aid of such a transformation. He also considered the case u

Jpdf = 0,

(37.33)

0

.This is in accordance with Remark 35.1, where it was noted that among the bounded solutions of &. (37.1) there may be a periodic solution with period 20 or greater. If (37.1) has a periodic solution with period -or 20, then, as we have seen, none of the tests for boudedness of the general system is satlsfied. However, 4. (37.4), obtained from the transformapon (37.2), where iu is given by formula (37.30) may also fail to h e a periodic solution with period or 2 ~ . (u

208

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

assuming that

and that p ( t ) where n is an sarily change To see this, obtain

changes sign only at points of the form z + n y / 2 , integer and a a real constant. That ~ ( tmust ) necessign at the points a n J 2 is obvious from (37.34). when we substitute t - a + - n ~ / 2 + into (37.34), we

+

where B i s an arbitrarily small number. It follows from (37.34) that Eq. (37.33) is satisfied. On the basis of (37.33), the equation w=-p

(37.35)

which is obtained from (37.29) for k 0, admits periodic solutions that differ from each other by constant values. For w , Lyapunov took a periodic solution whose value is 1 at t = a. Assuming also (without loss of generality) that p > Oin the interval t = a , t = a +0/2, he showed that w attains its maximum at the point p defined by P

"Ttpdt a

+ w J p d f = 0.

(37.36)

LI

Assuming that P f(t-a)pdt

c 1,

(37.361)

. I

Lyapunov showed that the chosen function does not vanish and that q ( t ) >, 0 in Eq. (37.4). As an example, he studied the equation

8% + p s i n t . x = 0, p2 4 1 dt2

for which we can easily find

(37.3 7)

209

AN EQUATION O F THE FORM (33. I)

and 9 (in Eq. (37.4)) in the form q = pz(l + y . ~ i n f ) ~ s i n ~ t .

The period of q in Eq. (37.4) is found from the formula (37Jj1):

A necessary condition for the general solution to be bounded is* Lyapunov also noted that, in the case in which that 1. G 0.39

....

I' pdt 2.o we

may set

i, t

= p--ludt

,

.r = yeye-jc'dt. 7 = S e Z L * d t &, 0

where

v'

= p - B.

I W

5= -Jpdt. "0

Then, Eq. (37.1) is reduced to (37.4), in which

with period

By means of such a transformation, he found sufficient conditions for boundedness of the general solution of the equation

dP

+

(i2

L p sin t )n = 0.

for which we may take v' = psint, v (37.4)

=

-- IL

s

(37.38)

cost.. We then obtain in

*We obtain this result on the basis of the t e s t w pdt < 4for Eq. (37.4). However, as 0

Lyapunov noted, if we assume thatp >0, we can obtain p < 0 by using plukovskiy's m e t h e

21 0

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

with period

T

Here, the test (34.1), which in present notation is T \ ' q ( i ) d r ~

4

b

yields (37.39) where

If we set

we obtain

and (37.39)becomes

We shall not go into a detailed exposition of this equation of Lyapunov. In certain cases, he has been able to strengthen the methods described by choosing m in a different manner. We shall not go into this matter but refer the reader to Lyapunov's book 126, p. 3331.

38. Starzhinskiy's Transformation [69] We shall now show how Starzhinskiy proposed to study the case in which P ( I ) changes sign. He transformed an arbitrary system

,

STARZHINSKIY'S TRANSFORMATION

[69]

21 1

of two linear homogeneous differential equations

dX = XP(t), dt

+ m) = P ( t ) ,

P(t

(38.1)

where P ( t ) i s a second-order matrix, into Eq. (33.1) with P(t) > 0. He introduced the new unknown matrix

Y =Xexp-

2kx0

Jt,

J=

dY =YQ, dt Q=-

(--2k

2kr J + e x p w

x

[I ' 1 -1

(38.3)

J t ) Pexp

(%J t ) .

In expanded form, the system (38.3) is written (Q =

dyl dt = 911Y1 -k 91&2,

(38.2)

0 '

(38.4)

1

dya dt = 9%1!.hf9221/%

911

"l

912

qnz

1 ).

*

(38.5)

where cos2~-p11 - sin T -cosT -prl- sin 7 .cos 7 . plr + sin2T pzz 2k A 412 = - u ) cos2r-p,l +sin z .cos 'i.pI1%I=

+

1

+

- cos-:.sinr.p,,-- sin3T.pl2 991

=

2k x

0

'*

-

912

(38.6)

sin ~ a c oTs .pll - sin27 .psl + C O S ~ T.p12sinT.cosT.pse

= sins-cosr.p2, +'sin2r.pll

+ sin

j-.cos2:.pzz+

7.cos T -plz

and 7=-

2k x

t.

0

For sufficiently large integral k , the elements 912 and qZl (which are periodic with period w ) are arbitrarily largefor all t and

21 2

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

The functions QU and 4% are periodic with period o and they are uniformly bounded with respect to k. The unknown gl satisfies the equation

.. -(411 + + i1dq12)A! +

Y1

422

(411422

-q12421 - i l l + 411 ild412) K = 0.

(38 81

Here, we use the notation ctz/ dt = z. Furthermore, by making the substitution

s

+ + dt [: I Starzhinskiy shifted from (38.8) to the equation Y1

-+ i[-

412421

= 2 exp

+4ll422 -

(411

41d412)

422

1

411

-F 4ll41d412 - 4(411

+ + 412

(38.9)

9

ild412)2

+

or i+q(t)z=O.

(38.10)

If the first and second derivatives of the elements of the matrix P ( t ) a r e bounded, then the coefficient q ( t ) is periodic with period w and is positive for all values of t ; that is, Eq. (38.10) is an equation of the type (33.1) with p ( t ) >/ 0. For sufficiently large k, we can get q (t) arbitrarily large. Specifically, this function is of the form

(38.11)

Here, 41 (t. k) and qo (t, k) are periodic functions with period 14. ( t , k) I < M , = 0. 1

0,

and the constant M is independent of k. There a r e several remarks that may be made inconnection with this. Remark 38.1. Let us suppose that we have shown by use of one of Lyapunov's tests that the general solution of Eq. (38.10) is bounded. Then, a s we can see from (38.2) and (38.9), the system (38.1) has only one characteristic number and it coincides with the characteristic number of the function

TRANSFORMATION OF AN ARBITRARY SYSTEM OF TWO EQUATIONS

213

(38.12)

But since the integrand in this expression is periodic with period w, the characteristic number of the system (38.1) will be (38.13)

For the general solution of the system (38.1) to be bounded, it is necessary and sufficient that v = 0. If the general solution of Eq. (38.10) is unbounded, the question of the boundedness of the system (38.1) is also answered by the characteristic number (38.12). Let us find the characteristics numbers vl and v, of Eq. (38.10) and then find the characteristic numbers v1 + v and v, + v of (38.8). Remembering that the characteristic numbers of the system (38.1) do not depend on k., we can obtain them too from the numbers v , + v and v , + v . R e m a r k 38.2. Starzhinskiy’s transformation converts an arbitrary system (38.1) with period coefficients into a canonical system with periodic coefficients since the canonical system* x = y, y = -qx

(38.14)

corresponds to Eq. (38.10). Remark 38.3. If we take a large value of k i n Eq. (38.10), then inequalities (34.16). (34.17), or (34.18), which ensure a solution of the problem on the basis of Theorem 34.1, will contain a large number of t e r m and will therefore be difficult to verify. For this reason, we should choose for the number k in (38.6) the smallest of the numbers that ensure (in 38.6) that q(t) 2.0.

39. Transformation of an Arbitrary System of Two Equations into a Canonical System Consider a system of two linear homogeneous differential equations

dY

= YP dt YThe general form of a canonical system of two equations is as follows: x=-

or, in matrix form,

bx-cy,

i=m+by

(39.1)

2 14

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

with periodic second-order matrix f (t + W) = P (t). Let us make the substitution

x = YZ,

(39.2)

where 2 and Z-' are both continuous for It I<.. matrix X, we obtain the equation X

-

XQ, Q = Z-' (PZ

To determine the

+ i),

(39.3)

2 = ZQ- PZ.

(39.4)

Let us choose the matrix 2 in such a way that the system (39.3) will be a canonical system, that is, so that

(39.5) where a , 6 and c a r e arbitrary functions. We shqll, in addition, try to arrange for a,, 6 and c to be bounded and periodic with period w, Let us denote the elements of the matrix Z and zkl and those of the matrix P by PM.. Then, we can write Eq. (39.4) in expanded form as follows:

(39.6)

These equations need to be satisfied in an arbitrary manner provided only the matrices 2 and 2-' a r e continuous in the region [ f I< 03 and the functions a, 6, and c are bounded and periodic with period o. Let us show thatthesolutionof (39.6)is determined algebraically when we set 2zp a relationship between zll ,z,, z12 and zzl Suppose that we a r e given the relation

.

Then, from the system (39.6). we have

TRANSFORMATION OF AN ARBITRARY S YS TEM OF TWO EQUATIONS

On the other hand, from (39.7), we have 222

+ Vl,,

=

- 'pj,,

(211, 212, ZZl,

(ZIh 212, 5 1 ,

-'pjZI(211,

212. 221,

0 [ S l ( b + P11) + Zl&

+

ZZ,Pl,l

215

+

4 [zn (6 - P11) + 211a -P1a cp (211, z12, 221, 41 (39.12) 4 [ Z 2 l ( b Pzz) + ZllPZl + 'p (211. 212, ZZl, t ) CI cp&l, 21% 221, 0 .

+

+

+

By equating the right-hand members of Eqs. (39.8) and (39.12), we obtain the equation

Let us suppose that, from this,

Then, from (39.7), we obtain 222 = $z(Z1a,

221-

0.

(39.15)

If we substitute the values of zll from (39.14) into the right-hand members of (39.9), (39.10) and (39.11), we obtain zii = -$1 (212, zait t ) (6

+

PII)

- zig - Z N P I ~

(39.16) (39.17)

Similarly, from (39.14). we obtain

(39.19)

2 16

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

If we equate the right-hand members of Eqs. (39.16) and (39.19). we obtain the relation $2(Z12r 2219 f)= 0. (39.20)

Let us suppose that, for this, we have (39.2 1)

= 9 2 (212, t).

221

If we substitute this value of (39.17), we obtain ,712

291

into the right-hand member of

= 93 (Z18,

0.

(39.22)

On the basis of this same equation, we obtain from (39.18) 221

-

$4 (Z18,

0.

(39.23)

From (39.21), we obtain &I

=

(ZlZ,

f) i12

+

";,

(Zm

f)= +;z,2

(212, f) +3 (ZlZ,

0

+ e;,

(21%

0.

(39.24)

By equating the right-hand members of (39.23) and (39.24), we obtain

Let u s suppose that, for this, we obtain 212

=-'Pl(f).

(39.26)

Then, from (39.21), we obtain

Equation (39.20) is also satisfied on the basis of (39.21). With the values (39.26) and (39.27), the right-hand members of (39.16) and (39.19) coincide. Therefore, the values of zlldefined by (39.16) and (39.19) also coincide. We obtain the value of zllon the basis of (39.16), (39.27) and (39.14). Thus, Eq. (39.13) is also satisfied. But then, the right-hand members of (39.8) and (39.12) coincide. From this it follows that the values of z, defined by Eqs. (39.8) and (39.12) coincide. But we obtain the value of zZ2 from (39.7) (from which (39.12) follows). We have obtained explicitexpressions for q,, z,, z12and zZ1with the aid of algebraic operations* in terms of the undetermined *That is, from algebraic equations,

THE

CASE IN WHICH (39. 7 ) IS OF THE FORM

222

=0

21 7

functions a , 6 and c and their derivatives. Let us suppose that the values found for z l 1 , ~ , z l and Z z, satisfy the first three equations of the system (39.6). Then, the fourth equation of this system will be satisfied since &, as found from (39.7) satisfies (39.12) and also (39.8), from which the fourth equation in the system (39.6) follows for the values found for zu,zzz,zlzand221. Using the values found for zll,z,,zlz~ and h1 to satisfy the first three equations of the system (39.66 we obtain three equations from which we then find the functions a , b andc. Then Eqs. (39.10) and (39.11) andalso Eqs. (39.17)and (39.18) will be satisfied. We shall not stop to consider the various cases that can be encountered in this procedure. For example, it may happen that the relation (39.13) does not contain zll.. Then, we immediately have a relation of the type (39.20). Subsequent reasoning in this case differs only slightly from the case considered. Since the zk1 a r e found algebraically, they will be periodic functions when a , 6, c and pkl are periodic. We can choose a , 6 and c arbitrarily. Then, the zkl will be determined as the solutions of the system (39.6), that is, not algebraically. Let us now consider certain special cases of the function 'p in (39.7) that allow us to get explicit expressions for a, 6 and c easily.

40. The Case in which (39.7) is of the Form z g 2 = 0 Suppose that if in (39.71, z2z

= 0.

(40.1)

PZlZlZ

(40.3)

Consequently, 22,

-

'2

a

and (40.4)

2 18

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

(40.5)

On the basis of (40.5), we obtain from (40.3)

(40.7) Equating the right-hand members of Eqs. (40.6) and (40.7), we obtain

which, if we define

(40.9) becomes

zI1 = M z , ~ .

(40.10)

Then, on.the basis of (40.5),we obtain

ill= Mz12+ M [ z , (6 ~ -pll) + a M 4 . If we equate the right-hand members of this equation and Eq. (40.4) and divide by z12, we obtain & $IM2b 4- M2a=

ac 4- PiGai

(40.11)

a

On the basis of (40.10). Eqs. (40.5) and (40.6) take the forms 21.2

41

=-

(

= (6

pzlM

-P11f

+

~

+

a

d) 212. P2l

)ZIP

(40.12)

(40.13)

we must now satisfy Eqs. (40.11), (40.12) and (40.13). 'I'his can be done in several ways. In (40.11). let us set M-0, ac

+ p1.p21= 0.

(40.14)

THE CASE I N WHICH

I S OF THE FORM Z2*

(39. 7)

=0

219

From M=O ,we easily find t

a = Ap,, exp (26 4-px

pll)dt.

(40.15)

0

where A is an arbitrary constant. Then, from (40.12), we find I

~ 1= 2

B exp [ (b -pll) d t .

(40.16)

b

From (40.13), we find on the basis of (40.3) I

*?l=Ke xPI-~(b+p,)d t ],

(40.17)

0

where B and

K are constants. From (40.10), we have 211

= 0.

(40.18)

From (40.15) and the second of Eqs. (40.14), we obtain t

c = - 4~exp [ ---

A

f (20 + pZ -pl1)df1.

(40.19)

I3

We have obtained the matrix 2 of the transformation (39.2) and the elements a and c of the matrix (39.5) of the canonical system (39.3). The function b is still to be determined. For the matrix Q in Eq. (39.5) of the canonical system (39.3) to be periodic, we need only set (40.20)

On the basis of (40.3), we still have K = B / A . Now, the matrix Z and its inverse 2-l, which a r e determined by Eqs. (40.16), (40.17), (40.1) and (40.18), will be continuous in the interval It1 < 03. Example. Consider the noncanonical system

where p is a constant. On the basis of (40.16), (40.17), (40.18) and (40.1), we obtain by setting B = K = 1

220

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

and, on the basis of (40.15). (40.19) and (40.20).

Consequently, the system (39.3) is now put in the form

Then, on the basis of (39.2),

X

= Ce

11: bllt = Y Z .

where C is an arbitrary constant matrix. But, obviously,

Therefore

This yields

Suppose now that, in (40.9).

(40.21)

a = 1, b = 0 .

Then, :M

=:

;

. -I P ' ,

Pz.2-

1

pzl P21

. (40.22)

and, from (40.11). c = - pzlpIp

- M - M'.

(40.23)

THE TRANSFORMATION O F

n

LINEAR EQUATIONS

22 1

Equations (40.12) and (40.13) become (40.24)

121

=-

1

2(PI,+ P32 - P2l/P,l)

(40.25)

221.

Therefore,

5 ' = V'-/p2, ex p

[f$ + 1

.

(pI1

P??) d t ]

.

(40.27)

0

The quantity z,, is determined from formula (40.10) and z,, = 0. The matrices 2 and 2-I will be continuous if p1f 0. The canonical system (39.3) will be of the form (40.28)

where c is determined from formula (40.23). The coefficients of the system (40.28) will be periodic with period w if the coefficients of the system (39.1) a r e periodic with period 0 . The system (40.28) is equivalent to the equation

i+ cx = 0.

(40.29)

We saw above that this equation canbe reduced to the case in which by Lyapunov's or Starzhinskiy's method. It can also be done by the method shown in Sect. 39. c;; 0

41. T h e Transformation of n Linear Equations Into a Canonical System Suppose that the system (39.1) is an nth-order system. If the system (39.3) is canonical, then [34] (41.1)

where the h, a r e the elements of the matrix H.

222

LINEAR S Y S T E M S O F ORDINARY DIFFERENTIAL EQUATIONS

We leave the question open as to whether 2 can be found in closed form in such a way that the system (39.3) will be canonical with periodic bounded platrix Q. .We note that Arzhanykh's work [70] contains a method for transforming a given system (which may be nonlinear) into a canonical system.

42. Necessary and Sufficient Conditions for a Polynomial to Have Roots Located on the Unit Circle Consider a polynomial with real coefficients [go] P(x)=x"+U,x"-'+U#-"+..

+un-lX+Un.

(42.1)

We shall give a method for determining the number of roots of P ( x ) that are located on the unit circle. We introduce y defined by 2y=x+r', x=y+v-

(42.2)

x=y-vy2

--

1.

(42.3)

Let us substitute the first of these values of xinto (42.1). We obtain P ( x ) = Q (y) = Pf')(p)

+ vmP(51) (y).

(42.4)

Here, Pf')(y) and P(;-*)(y)are polynomials of degrees n and n-1, respectively. Since the coefficients of the polynomial (42.1) a r e real, to every root x1 = exp i cp there corresponds a root x2 = exp (-ii). To these two roots there corresponds a single real value y in Eq. (42.2). The absolute value of this y = coscp will be less than unity if y + k 1. Let us suppose that y + 5 1 and that its value is such that both values of x, given by Eqs. (42.3) will be roots of the polynomial (42.1). Then,

Consequently , P?) (y) = 0, P(;-')(y)

= 0.

(42.5)

I f x = + 1, theny= k 1 andP@)(+1)=0. Let us denote by N(y) the polynomial that is the greatest common divisor of the polynomials (42.5). We have proven

223

POLYNOMIAL WITH ROOTS ON UNIT CIRCLE

a root of the equation

Theorem 42.1. I f x = exp i pis

P(x) =0

(42.6)

w!th real + k r (for integral k), then the cowesponding value of y is a real root of Eq. (42.5) or of the equation (42.7)

[ P @ )(y)]'

+

p - 1 )

(y)]' = 0

(42.8)

with Iy 1 < 1. Conversely, every real root y of Eq. (42.7) o r (42.8) such that I y~ I yields two conjugate roots of Eq. (42.6) located on the unit circle. Remark 42.1. If Eq. (42.7) has a real root y such that1 y I 7 1, then the polynomial (42.1) has two roots (42.3): one with 1x1 < 1, the other with 1x1 7 1. If there is no polynomial N(y)(or if Eq. (42.8) has no real root), then there are no values of y that provide two roots (42.3) of the polynomial (42.1). The number of real roots Y of Eq. (42.7) (or (42.8)) such that I y I 4 1 is, as we know, easily found [351. Suppose that the degree of the polynomial (42.1) is even: n =2m. Then, there a r e no roots x = f 1 (their possibility is easily excluded) and the remaining roots a r e located on the unit circle in the form of pairs of complex conjugates: x, = C O S T isincp, x, = coscp - i sin 'p. Consequently, the polynomial (42.1) is a product of pairs of factors:

+

( x - XI)( x

-x*) = 2x

[

(x

+

1 -COS

rl)

2

.

From this it follows that Eq. (42.1) is of the form x"Q,(y)=O,

2y=x+x-',

x=y+vy2-1,

(42.9)

where Q,(y) is an mth-degree polynomial with real roots y. that are less than unity in absolute value and Eq. (42.1) is reciprocal. It is easy to find the polynomial Q,(y). This proves Theorem 42.2. For all roots of Eq, (42.1) oreven degree n = 2m

to be equal to unity in absolute value (but not 1). it is necessary and sufficient that (42.1) be reciprocal and that the roots of the equation Q, (y) = 0 be real with I y I < 1. Instead of (42.9), let us obtain an equation all the roots of which lie in the left half-plane and within the unit circle. With this in

224

LINEAR SYSTEMS

OF

ORDINARY D I F F E R E N T I A L EQUATIONS

mind, we rewrite (42.9) a s follows:

Q, (Y)

= Qi (Y')

+ Y Q (y2) ~

(42.10)

= 0.

If m = 2k, then Q1(z) and Qz(z)a r e polynomials of degree k and k - 1 respectively. If m = 2k + 1, then both a r e of degree k. From (42.10) we obtain (42.11) L (2) = Qi(- Z ) + zQ:(- Z ) =O, z = - .'y The roots of this mth-degree polynomial L ( z ) a r e real, negative, and of absolute value less than unity. Every root of Eq. (42.11) such that I z I < 1 provides (on the basis of the equation y i v / t a n d Eqs. (42.3)) two roots of Eq. (42.6) that are located on the unit circle. With the aid of methods that we a r e familiar with from algebra [35], we can easily find in this way the number of roots of the polynomial (42.1) that a r e located on the unit circle. For other methods of solving this problem, see [71]. If we show that the polynomial (42.1) does not have roots on the unit circle, then the number* of roots N that a r e located inside the unit circle** can, a s we know, be found from the formula

-

(42.12) This number is a nonnegative integer, and which integer it is, is easily found by an approximate calculation.

43. Investigation of the Roots of the Polynomial (42.1) as Functions of a Parameter Appearing in the Coefficients ak [901 Suppose that the coefficients of the polynomial P (x, E ) shown in (42.1) a r e functions of a parameter E that a r e analytic in a neighborhood of the point E = 0. Suppose that, for E = 0, we have m distinct pairs of complex conjugate roots situated on the unit circle: -xik)= e x p i (PR, x i k ) = e x p (-- i (Pk) (k

where Tk#

vx

(where

v

-

1.

. . ., m).

(43.1)

is an integer). Corresponding to these

*If the polynomial (42.1) is reciprocal, then, as we know, the number of roots within the circle is the same as the number outside if **For other methods, see the book by I. S. Arzhanykh [70].

ROOTS OF POLYNOMIAL AS FUNCTluNS OF A PARAMETER

225

pairs of roots there a r e m common distinct real roots y = y k , where I yk I < 1 (for k = 1, . . ., m) of the equations *

or L(y)= [P'")(y,O)]'f [P'"-')(y,0)J2=o.

(43.3)

The question arises a s to the distribution of the roots of the equation P ( x , E) = 0

for small values of

8.

(43.4)

Since the coefficients of the equations

P(n)(y, E ) = 0, P(n-1) (y, E ) = 0

(43.5)

or L ( y , 8 ) = [ P @ )( y , 4

2

+ [P(n-l)(y,

.)I2

=0

(43.6)

will in this case obviously be analytic also in a neighborhood of the point E =O (with radius of convergence a t least a s great a s that of the coefficients of the polynomial (43.4), Eq. (43.5) or (43.6) yields m distinct holomorphic roots

in a neighborhood of the point E = 0. For small values of c, these roots will be real and distinct and I y,(e)I < 1 if P ( x , E)isreciprocal. This i s obvious from (42.9). If follows from this that the corresponding pairs of roots of Eq. (43.4) will remain on the unit circle for small values of e. This proves Theorem 43.1. I f P ( x , 8 ) i s reciprocal and if P ( n , 0)has m distinct pairs of solutions (43.1), then f o r all sufficiently small real values of x , Eq. (43.4) also has m distinct conjugate pairs of solutions located on the unit circle. Suppose now that, for e = 0, we have a p-multiple real root yo of Eq. (43.6) such that I yo I < 1. Then, the following cases may arise: I. There exist p distinct real (for real E ) functions (43.8)

ZThese equations are the same as (42.5) except that the coefficients ah contain the parameter c.

226

LINEAR SYSTEMS O F ORDINARY D I F F E R E N T I A L EQUATIONS

that satisfy Eq. (43.6). Corresponding to these functions are p pairs of complex conjugate roots of Eq. (43.4). 11. There exist functions y = yv (e) (for v = 1, .. ., p.) satisfying Eq. (43.6) that possess the property that yv ( 8 ) -+ yo as E -+ 0 and that are real only when E > 0 or only when E < 0. Then, we have p conjugate pairs of roots of Eq. (43.4) located on the unit circle for small (in absolute value) positive or negative values of E,respectively. These roots y , ( ~ ) can be represented by series of positive powers of the quantities el/q, where q is a positive integer, that converge for small values of E . 111. There exist functions y = y,(e) (for v = 1, . . ., m < p) that are real for real values of E throughout some neighborhood of E = 0 and that can be represented by convergent series in positive powers of d l q , where q is a positive integer. The regions of convergence of these series have been determined. All these conclusions a r e obtained on the basis of [32]. Let us now look at the case in which x = 1 is a root of Eq. (43.4) for E = 0. In this case, a s can be seen from (42.4), we also have, for E = 0, P ( n ) (I, 0 ) = 0.

(43.9)

In accordance with [32, Sect. 61, we shall have a solution of the first of Eqs. (43.5) in one of the following forms: (43.10) where q is a positive integer, or (43.11)

or (43.12)

or (43.13) Here, the ak are real and the series converge for small values of

E.

ROOTS OF POLYNOMIAL AS FUNCTIONS

OF A PARAMETER

22 7

We may obtain one of the series (43.10), (43.11), (43.12), o r (43.14) o r several such series. These series are real either for Io I < E,, o r only for small positive e 7 0 o r only for e < 0..

But to answer the question as to whether the corresponding roots of Eq. (43.4) that assume the value x = 1 for e = 0 remain on the unit circle for small values of E , we need to consider the following fact: If the solution found for the first of Eqs. (43.5) is not a solution of the second of Eqs. (43.5), it does not generate solutions of Eq. (43.4). Let us suppose now that the solutionfoundsatisfies both of Eqs. ( 4 3 3 , that is, that it is a solution of Eq. (43.6). For it to generate roots of Eq. (43.4) that are located on the unit circle, it is necessary and sufficient that it be realandthat its absolute value be less than unity. Suppose, for example, that we have (43.10). Then, if is an odd number, y remains real for I E I < e,. But if q is an even number, then y will be real only when a > 0. Here, if we have a l < 0, i t is obvious that, for small E > O , the functiony will be real and 1 y I < 1. This gives us two conjugate roots of Eq. (43.4) that are located on the unit circle for small values of e. The number of such pairs of roots of Eq. (43.4) is equal to the number of real roots of (43.6) such that I y I < 1. If we have (43.11), we repeat the same line of reasoning for e < 0. We again follow this line of reasoning if we have (43.12). Here, the deciding factor for the existence of a real y such that I y I < 1 is the sign of a1 and the region e > 0 o r 5 < 0. If, for example, q is even and a1 > 0, then y, a s defined by the series (43.12), will not be real even when I yI < 1. To see this, if E > 0, then y = y(e) is real but lyl> 1 since eak > 0 , but if E < 0, theny = y(e) is not real. On the other hand, if y = y ( ~ is ) real and I y I > 1 and if y satisfies the equation* (not necessarily both of Eqs. (43.5)) lq

then, since x = y + v F l , the corresponding x will be greater than 1 in absolute value. If Eq. (43.4) has roots x = x ( e ) on the unit circle for small (in absolute value) positive or negative e and if x - 1 for e=O., then, in accordance with Theorem (42.1), there exists a real root of Eq. (43.6) such that Iyl < 1. We obtain this root in the form of the series shown above. We note also that when we examine Eq. (42.9), which corresponds to a polynomial of even degree n = 2m, we can also consider the case in which P ( x , e) is continuous with respect to 0.. We need only use the methods expounded in Sects. 2-5 of [32]. *"his equation is obtained by maldng the substitution x = y+/l/"--into

(43.4).

228

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

44. Questions Regarding the Stability and Boundedness of Solutions of Linear Systems of Differential Equations with Periodic Coefficients on the Basis of the Methods of Section 43 [93] Consider the canonical linear system of differential equations (44.1) k=O

where the series converges in the region I o I 4 r , where the P k (t) are continuous nth-order matrices that a r e periodic with period 2x, and where X is the integral matrix, We have seen (cf. Sect. 6) that the integral matrix X can be represented in the form

where the x&(t)are continuous matrix functions of t and the series converges in the region 181 < r. This integral matrix possesses the property

x (t + 2a,

E)

where

x (275 x (t, 8)

- x x,

E),

(44.3)

m

x (2n,

-

E)

( 2 4 Ek.

(44.4)

k=O

-

The question as to when the matrix (44.2) possesses the prop(( 0 11 as t 03 o r when it is bounded is solved as erty that X (f,E) follows: If all the characteristic numbers of the matrix (44.4) a r e less than unity in absolute value, then X(t, e ) + ~ O I ~as f + a,. However, if all the characteristic numbers of the matrix (44.4) are equal to unity in absolute value and if the elementary divisors a r e simple. then the matrix (44.2) does not possess the property that X (t, €1 I1 0 I1 a s f + 03 but it is bounded and oscillatory. Since we have assumed the system (44.1) to be canonical, it follows from a theorem of Lyapunov [26] that the characteristic equation +

P(x,E) = 0

(44.5)

of the matrix (44.4) is reciprocal. Therefore, whenever the matrix (44.4) has a characteristic number equal to p, there is also a

Q U E S T I O N S REGARDING T H E S T A B I L I T Y A N D B O U N D E D N E S S

229

characteristic number equal to p-I. From this it follows that the + as matrix (44.2) cannot possess the property that X ( ~ , E ) -(1011

t-03. Let us find the conditions under which the matrix (44.2) will be bounded (without approaching the zero matrix a s t + 00) for sufficiently small values of the parameter E . If the matrixX(2x.O) defined by the series (44.4) has acharacteristic number p such that l p I < 1, then there exists (in view of the fact that, even for a = 0, the system (44.1) is canonical) a characteristic number p such that I p I > 1. Then, it is obvious that, for all sufficiently small a, there are characteristic numbers p such that I p I 7 1. Therefore, the matrix (44.2) is unbounded. Let us suppose now that the absolute value of every characteristic number of the matrix X(2q0) is equal to unity, that is, that all the characteristic numbers lie on the unit circle. The characteristic numbers x of the matrix (44.4) a r e roots of the nth-degree polynomial (44.5), where the coefficients of all powers of x a r e holomorphic functions of E in a neighborhood of E = O . These series converge for I E I < r. For c = 0, (44.5) becomes the characteristic equation of the matrix X(2r, 0). Consequently, all roots of Eq. (44.5) lie on the unit circle for E =O. In Sect. 43 it was shown how one could prove Chat, for all sufficiently small values of t, all the roots of Eq. (44.5) are distinct (and.hence that all elementary divisors of the matrix (44.4) are simple), and that they a r e located on the unit circle. For example, this will be the case when all characteristic numbers of the matrix X(2q 0) a r e simple and located on the unit circle. It will also be the case whenever all roots of Eq. (43.6) are, for sufficiently small c (possibly only for E > 0 o r t < 0), real, simple, and less than unity in absolute value. (In this case, the matrixX(2r 0) may have nonsimple elementary divisors.) Let us suppose that the system (44.1) of even order n = 2m is not canonical. Then, for the integral matrix (44.2) to be bounded for small values of E without approaching the zero matrix a s t + 00 , it is necessary that the characteristic numbers of the matrix X (2x, 0) lie on the unit circle. If they are of the form (43.1) and are distinct, then, by Theorem 43.1, they are distinct and located on the unit circle for all sufficiently small a. For the characteristic numbers of the matrix X(2r, 0) to lie on the unit circle, it is necessary that the characteristic equation of the matrix X(2x, 0)be reciprocal (Theorem 42.2). If it is reciprocal and if Eq. (42.10) has rn (where n = 2m) distinct real roots y of absolute value less than 1 for (44.5), then all the characteristic numbers of the matrix X(2q a) lie on the unit circle for all sufficiently small e. We may even say that the characteristic numbers of the matrix X (2r, E) will in fact remain on the unit circle for all real I e I < ec, such that D ( E f ) 0 and Qm(+ 1, E ) f 0 (cf. Eq. (42.11)), where D ( E )is

230

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

the discriminant of Eq. (42.10). For such values of e, the roots of Eq. (42.10) will all remain distinct and less than unity in absolute value. These singular values so, which are allclose together, must also satisfy the equation A (E,,) = 0, where A (e) is the discriminant of the characteristic Eq. (44.5). This proves Theorem 44.1. For the integral matrix (44.2) of the system (44.1) (which i s noncanonical) to be bounded without approaching the zero matrix for small values of s , , i t is necessary that the characteristic equation of the matrix X(2x, O)be reciprocal. If this condition i s satisfied and Eq. (42.9), which i s set t@ f o r (44.5) (the characteristic equation of the matrix X(2x, E ) ) , has m distinct real roots y such that 1 yI’< 1, then the matrix . .(44.2) will be bounded without approaching the zero matrix as t --c CO. Remark 44.1. All other possible cases in which we have the matrix (44.2)a r e considered in Sect. 43.

45. A Sufficient Condition for the Integral Matrix of the Noncanonical System (44.1) to Possess the Property that x ( t , z)-llOll ast-+co In order for X ( t , ~)-+IlOll

as t - 7 0 0 ,

(45.1)

it i s necessary and sufficient (as was shown after (44.4)) that all the characteristic numbers x of the matrixX(2x. E) be of absolute value less than unity. Let u s note first of all that if the characteristic numbers x of the matrix X ( ~ P0) , a r e of absolute value l e s s than unity, they will remain so for all sufficiently small s. This is obvious. Consequently, we need to find those conditions under which the characteristic numbers x of the matrix X (2x, 0) will be of absolute value less than unity. We proceed as follows. Let us write the characteristic equation for the matrixX(2x. 0) in the form (-1)”D

(x(2x, 0) - X I ) = p ( X ) =0,

(45.2)

where P(n)is a polynomial of the form (42.1). Now, we can use the methods expounded in [71]to show whether all the roots x are lees than unity in absolute value. However, it will probably be more convenient here to use the method proposed by Arzhanykh [70]. Specifically, he obtains equations characterizing the conditions under which 1x1 <-1 for all roots of Eq. (45.2). This is equivalent to Hurwitz’ equations for the matrix lnX(2x, 0). which ensure that

ANOTHER METHOD OF SOLVING ARTEM'YEV'S PROBLEM

23 1

the real parts of the characteristic numbers of this matrix are negative. But it is more convenient since we do not need to find the matrix 1nX (2a. 0). These conditions obtained by Arzhanykh are both necessary and sufficient for 1x1 to be less than unity for all roots of Eq. (45.2). Altogether, for an nth-order matrixX(2q O), there a r e n of these conditions of Arzhanykh. We write them momentarily in the form A l > 0, Az>O, .... A,>O. (45.3) These inequalities will be satisfied for all sufficiently small real values of E . if they are satisfied for E = 0. Specifically, we shall have A1 ( E ) > 0, *-., A n ( E ) > 0. (45.4) if we have A1

(0) > 0,

..., An (0) > 0.

We may go further and say that Arzhanykh's inequalities a r e always satisfied for those values of E that are less in absolute value than cg, where E~ is the smallest of those real numbers E, for which we have* (by Sect. 43)

( p @ (y, )

EO)]*+

[p@-')(y,

=0

for real values of y such that I y ( < 1; hence, P ( n ) (+ 1, to) = 0.

(45.5) (45.6)

Thus, to determine E ~ we , have equations in the form of series that converge for those values of E for which the series in Eq. (44.1) converges. Remark 45.1. If it is shown on the basis of Theorem 42.1 that Eq. (45.2) does not have a root x lying on the unit circle, then the number N of roots x such that ] x I < 1 is easily found from formula (42.12) since we know that this is a nonnegative integer.

46. Another Method of Solving Artem'yev's Problem Consider again a system of the form (18.1) dX = X P ( t , dt

p1 )..., p v , 2).

(46.1)

+Since, for such cg. the equation (-1)"D(X(2;:,

has roots lying on the unit circle.

fo)-YI)

=o

232

L I N E A R S Y S T E M S OF ORDINARY D I F F E R E N T I A L E Q U A T I O N S

where the matrix P ( t , pl ,..., p v , E ) is of order n= 2m and is periodic with respect to f with period 2 r and where p l r ..., pv, and I are parameters. Now, we do not assume the system (46.1) to be canonical. Let us find conditions that the parameters pI, ..., p,,, and E must satisfy to ensure that the integral matrix of the system (46.1) will be bounded and oscillatory (without approaching the zero matrix a s t --+ a). In accordance with Theorem 42.2, we must require that the characteristic equation of the matrix X ( 2 n , p l r . . . , p v , E) be reciprocal. This gives us rn equations since these conditions for Eq. (42.1) a r e a,,= 1,

a,,+=a,

(k-0, 1,..., rn-

(46.2)

1).

Then, we require that all the roots of the equation (cf. (42.9))

be real and of absolute value less than 1. If the system (46.1) is canonical, the conditions (46.2) a r e automatically satisfied and only conditions (46.3) remain. Thus, if we a r e given the system (46.1), Artem’yev’s problem can be solved immediately on the basis of conditions (46.2) and (46.3) without .using the matrix W. This is much simpler and, in addition, we shall be dealing with series that converge for the same values of E a s do the series for the matrix P ( t , pl,..., p v , E ) in Eq. (46.1). A s we know, the series representing the matrix Wconverge in a smaller region. Formula II of Sect. 18 is solved in the same way on the basis of the methods expounded in Sect.44directlywiththe aid of the matrix X ( 2 x ; p l , ..., pvrE), The problems of Sect. 27 are also solved in this way. We might note that many of the problems that we have considered here are studied in greater detail in the works of Golokvoschus [50] in terms of the exponential matrix.

47. Supplement to the Theory of Implicit Functions as Studied in [32,73,97] We shall present some results from [32]. We know [72] that if an implicit function y a s defined by F ( x , y) = ax

+ by + ... = i,

(47.1)

...

, converges where the series F ( x , y) with real coefficients a, 6, in a neighborhood of the point x = y = 0, approaches 0 a s x - 0 , then, for 6 # 0,

S U P P L E M E N T TO T H E THEORY OF I M P L I C I T F U N C T I O N S m

y = x

QkXk,

al=--

k=l

a b

233

(47.2)

and this series converges in a neighborhood of the point x = 0.. Here, a,, a,, are real constants. If b = O but a + 0 , then

...

(47.3)

where b,,, is the coefficient of the smallest power of y in the series (47.1). From this we have the unique real function m

(47.4) k=l

defined throughout a neighborhood of the point x = 0 if m is odd. On the other hand, if m is even, then, for pm > 0, the series (47.4) provides two real functions (since T = nilrnhas two real values differing in sign) but only for small positive x . On the other hand, if pm < 0, then (47.4) provides two real functions for negative values of x that are small in absolute value, where a1 = - y x a n d T = ( - x ) * / m . Now, let us consider an equation of the form

where the coefficients a are real constants, where the series converges in a neighborhood of the coordinate origin, and where the terms of degree greater than 4 in x and y are omitted. Let us find conditions under which there is a continuous real solution y =y ( x ) of Eq. (47.5) possessing the property that Y(x)-+O

as ~ 4 0 .

(47.6)

We seek Y in the form y

=u.

(47.7)

Substituting this expression for ginto (47.5) and dividing by I , we obtain

234

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

We denote by a, and a, the roots of the equation al+a,a+a3a2=0.

(47.9)

Let us introduce a new unknown v defined by u =v+ak

(k = 1 or F= 2).

(47.10)

To determine u , we obtain the equation (a,

+ 2aha3)o+ (a4+ aaak+ .,a: + x + + a39+ (aa + 2a8ah+ 3a7a3 + 17%)

(47.1 1)

+(ag+a,ak+al~a2k+alla3k + a 1 , a : ) x Z + . . . =o.

Here, all the omitted terms a r e of degree exceeding 2 in x . If a, + a,, we have a,

+2ak a3 + 0.

(47.12)

In this case, we have from (47.11) v = mlk)x'

+ mS$'

x,+l-t-

...

( k =: 1,2),

(47.13)

where 1 is a positive integer,

(47.14) and the A, is the coefficient of the lowest power of x in (47.11). Consequently,

+ mik' + ml21

y = x (ak

XI

xl+l

+ ... ).

(47.15)

If a, and a, a r e real, we have two real functions y(x)that a p proach 0 a s x + 0. There are no other y (x). If a, and a, are complex, there are no real functions y(x) that approach 0 as x + 0. Suppose now that a, = a,= a, Then, the coefficient of the firstdegree term in v in Eq. (47.11) is equal to 0; that is, a2

+ 2a a3 = 0,

3 :

-

4ala3= 0.

(47.16)

S U P P L E M E N T T O T H E THEORY O F I M P L I C I T F U N C T I O N S

235

Under these conditions, the coefficient A of the first-degree term in .c in Eq. (47.11) is of the form A

= (8a;)-'

(8a,ai

- 4a,a,as 2 + 2aza6a, - 7 9 23) .

(47.17)

If g3 + 0 and A + 0, then we have from (47.11) x = k,vz 4-k3v"

+ ... ,

(47.18)

where 8aj

k2 =---

8a4ai - 4a2a, a:

3 -I2aiaea3- apa2

'

(47.19)

From (47.18). we obtain two real functions y that approach 0 a s x -0 if k , > O :

y =x

2 ( a

+

k=l

A&/')

, A,

= k;"'

.

(47.20)

'

Both these y (remember that x'i' has two values differing in sign) a r e defined in some interval of the form 0 4 x < h. If k2<0, then, forx>O,

has real values. There are no other functions y that approach 0 a s x.+o ;

Let us consider a few particular cases. If a, = a2 = 0,

a3 +O,

(47.21)

we have a = 0. Then, for (47.22)

a4ZO

we have (47.18). If a, = as = 0,

a2 + 0,

(47.2 3)

then the coefficient of the first-degree term in v in Eq. (47.11) is a, + 0 Therefore, we obtain (47.15) (ak= 0:).Buthere, we can obtain

.

236

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

other functions Y that approach 0 as x - 0 a s follows: If we make the substitution x = uy in (47.5) and divide by g,we obtain a2u

+ Y -+a,ug +

+ar

u3y

a,

a12Y2

+

+

a, U2Y -I- a l l Y 2 ~

+ alou2y2+ a, y2u3-+ a8u4g2+ ...

-1

(47.24)

0.

Since a 2 + 0 . , we obtain (47.25)

where A . is a coefficient of the term in (47.24) of lowest degree in + 0, then A = a, and m = 1. On the basis of (47.25), we have

y. If, in particular, a,

x m

x = uy -=

k , y’+l

(47.25 1)

v=m

From this we obtain

x W

y=

A&.

(47.26)

k=l

Here, ‘F

= XWd-1)

, A,

= k-l/w+1), rn

if

krn > 0.

(47.27) 8

,If m+ 1 is an odd number in these equations, Y i s real throughout an entire neighborhood of x = 0. On the other hand, if m 1 is an even number, then y will be real only for x > 0. The function T has two real values differing in sign. Therefore, we have two real y defined for small nonnegative values of x. If k, < 0 and m 1 is an odd number, then T and A, will again be given by Eqs. (47.27). We have one real y defined in a neighborhood of the point x = 0. On the other hand, if km < 0 and m 1 is even, then

+

+

+

= (-x)l/(m+l),

A1-- (- km )-V@+I)

.

(47.28)

We have two real y defined for nonpositive values of s that are small in absolute value. We have found all real values of y that approach 0 as x -, 0 and that satisfy conditions (47.23).

SUPPLEMENT TO THE THEORY OF IMPLICIT FUNCTIONS

237

Let us suppose now that a2 = a3 = 0, but a, f

0.

(47.29)

In this case, Eq. (47.9) has no roots. !Let us introduce an unknown u defined by x = uy.

(47.30) If we substitute this into (47.5), divide by y 2 , and take conditions (47.29) into account, we obtain

+ a4 yu3 + aau2y+ + a , y + ag y2u4+ +a, y2u3+ alo y2u2+ a11uy2+ a12y2 + ... = 0.

alu2

~,UY

(47.31)

Here, if a,

=+0,

(47.32)

we obtain from (47.31)

+

y = ku2 k y 3

+ ... ,

k2

2

a1 . --

(47.33)

a7

Let us suppose that k,> 0.. Then, from (47.33), we have m

(47.34) k-. I

From (47.34) and (47.30), we obtain (47.35) Thus, if (47.29) and (47.32) hold and if k , , 0, we have a unique real solution of Eq. (47.5) that approaches 0 a s x-• 0 and that is defined throughout a neighborhood of the point x = 0. Suppose now that k2

0.

(47.36)

Then, we have y in the form (47.37) which is defined for nonpositive x of small absolute value.

238

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

Let u s suppose now that (4 7.38)

a, =a2,

but that the quantity (47.17) is equal to 0: A = 0:

a, f

0.

(47.39)

Then, Eq. (47.11) is of the form (47.5) and we can repeat the preceding line of reasoning for it. Repeating this reasoning, we can either show that there are no real functions y that approach 0 a s x -+ 0 or we can find them in the form indicated above. But it may happen that we will again arrive at an equation of the form (47.5). However, it i.s easy to see that we cannot arrive in this way at an equation of the form Pm(x,y ) + P , + , ( x ,

y)+ ...= 0, m 9 3 ,

(47.40)

where the Pk ( x , y) are all homogeneous kth-degree polynomials, unless 1, = a, = as = 0. It may turn out that no matter how many times we repeat this procedure, we always obtain an equation of the form (47.5). Then, as was shown in [32], we have one y that approaches zero a s s - * 0 and it can be represented in the form of the convergent series (47.41) k= I

where all the ck a r e unique (or multiple) roots of equations of the form (47.9). If we can show that there a r e real functions y that approach zero a s x + 0 (we a r e thinking of the case in which a multiple root a of Eq. (47.9) f 0), then all these functions have the same principal part (infinitesimal of lowest order) y

= ax.

(47.42)

Let us now consider Eq. (47.40), where tii > 3. If we substitute Y = 2x (cf. (47.7)) into (47.40) and divide by xm, we obtain Pm(l,u ) + P , , + , ( l , u ) x + P m + s ( l , u ) x 2 + ... =o.

(47.43)

Let us denote by a the roots of the equation P,"(l, a ) = O .

(47.44)

239

SU PPLEMENT TO THE THEORY OF IMPLICIT FUNCTIONS

If we set u=u+a,

(4 7.45)

we obtain

...= 0.

P m ( l , u+a)'+Pm+.,(l, u+a)s:-

(47.46)

If we expand this in a series of powers of u , we obtain* Pm+l(l, a)x+Pm+,(l,

a ) ~ ~ + P m + s ( l , a)x'+

+ [ P h a ) + Pt;l+l(l:a)x+PCn+Z(l,

+- 2!1

[PT,(1, a)

+ Pi,+, (1.

a).x2+

a)x

...+

...I u t (47.47)

+

i-PL+z ( 1 , a ) x 2 + ...I 3; ...= 0. Let us suppose that a is a simple root of Eq. (47.44). Then, P i ( ] , a) + 0 and, from (47.47), we have

...

where Pn(1, a) is the first of the quantities P,"+ (1, a ) ,P,+2( 1, a ) , that is nonzero. On the basis of (47.7), (47.45) and (47.48), we find

+

y = ~ ( akrP

+

kn+1x"+I

+ ...).

(47.49)

If a is real, then the function (47.49) i s real. Conversely, if a is complex, then the function (47.49) will also be complex. If all roots a;, a2, ,amof Eq. (47.44) are real and simple, then we obtain nz real solutions y of Eq. (47.40) in the form (47.49) that approach zero as x 0, and there a r e no other such solutions. If all roots of Eq. (47.44) a r e complex simple roots a,, ,a,,,, then there is no function y that approaches 0 as x --c 0. Suppose now that a is a v-multiple root of Eq. (47.44). Then, we have

...

-.

P,

...

(1, a ) -L P;, (1, a ) = ... = P!?

(1, a )

1

0,

P2)(1, 4)+o.

(47.50)

Suppose also that (47.51) *In this equation, P i k ) ( I , a) denotes the kth derivative with respect to a.

240

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

Then, from (47.47). we have x = k, U'

+

k,+l~'+l

+ ...,

(47.52)

where

Taking into account the sign of k, and the evenness o r oddness of the number Y , we find real functions y that approach 0 as x -+0 by a line of reasoning analogous to that above if a is a real number. On the other hand, if a is complex, these formulas will not give us a real y corresponding to this root a that approaches 0 a s x approaches 0. Let us suppose now that, instead of (47.51). we have

P,+,(1, 4 3 0 .

(47.53)

Then, Eq. (47.47) takes the form

+ P ( x , v ) =0, where P ( x , 2 ) contains terms of degree exceeding 2 in x and v . Let u s note now that the polynomial L ( v ) in Eq. (47.47), which is made up of terms not containing x , is of degree no greater than m and no less than v since P!,?(1, v + a ) = O

for k > m , and Pk'(1, a ) # O .

If (47.54) does not have second-degree terms, that is, if we do

not have

P,n+*(l,a)=Pln+1(1,a ) = P h , a ) = 0 ) , then we have an equation of the form (47.5), which we examined above. Of course, it may happen that the set of terms of lower degree in Eq. (47.54) constitute a form of third or even higher degree. But since, in accordance with the remark made above, the polynomial L ( v ) is of degree not exceeding m , either we obtain an equation conta'ining linear terms (with the aid of not more than m successive transformations) or we run into the case in which these transformations, no matter how many times we perform them, lead

S U P P L E M E N T T O T H E THEORY OF I M P L I C I T F U N C T I O N S

24 1

to an equation with the lowest form of the same degree p. In the latter case, the equation corresponding to (47.44) will each time have only one p-multiple root. In that case, as was shown in [32], Eq. (47.40) is of the form [a9

+ 6,y + p ( x , Y)Y

Q, ( x ,

= 0.

where 61# 0, where P ( x , y) is a convergent series in positive powers of x and y that does not contain a free term o r a first-degree term in ?c o r y, and where O(x, y) is a convergent series possessing a free term. Thus, if a is a w-multiple root of Eq. (47.44)that is real, we have a p-multiple real holomorphic solution y of Eq. (47.40) in a neighborhood of the point x = 0 that approaches 0 asx -+ 0.. To get all real solutions y of Eq. (47.44) that approach 0 a s x - , 0, we need to reason along the same lines, making the substitution x = uy.. But this should be done only in the case in which the polynomial .Pm(x,y) does not contain the term 9 y" because, if 8 Z 0 in this case, the transformation .u = uy does not lead to a new solution. Here, we have given the results of [32], which contains detailed proofs. Reference [32] also examines systems of equations containing nonholomorphic functions. It is also shown there that analytic continuatiop of these implicit functions cannot lead to singular poinis x = x such that the function y = y(x) fails to have a limit a s x --., x and the infinite set of values of y ( x ) fall in a finite closed region D embedded in the domain of definition of the functioqF(x, y), In other words, there isnosequence of points (xk, y&)-, (x, y) D , where D is the domainof definitionof the function F ( x , y). It is also shown how one may determine the entire domain of definition of an implicit function and the radius of convergence of series representing an implicit function in a neighborhood of the point X =

0,

Example: Suppose that 6 2 0 in (47.1) and that we have the function (47.55)

defined by Eq. (47.1). The point analytic function only when C

t

F ( x , y)

=o,

can be a singular point of this +

t

F; ( x , y) = 0.

(47.56)

Thir condition is necessary but not sufficient. For the point defined by Eqs. (47.56) to be a singular point, it is sufficient, for

242

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

example, that, in addition, (47.57) On the other hand, if (47.56) is satisfied but F Z ,

y') = 0,

(47.58)

then the point cannot be a singular point. For example, if (47.56) and (47.58) a r e satisfied and the roots of the equation

a r e real and simple, then the point ;cannot be a singular point of the function (47.55). However, let,us suppose that (47.56) and (47.58) are satisfied. Then, the point x is a singular point. But here it q a y turn out that a real. function (47.55) exists even when 1x1 > x.. But forx 7 x (with x real), for example, this function cannot, by virtue of That was said above, be represented by a series in powers of ( x - X ) * / P , where p is an even positive integer. If in the domain of definition of the function (47.1) there are no real x that satisfy Eqs. (47.56), then the domain of definition of the function F (x, y)'contains no real singular points x. Then, the real function (47.1) will be defined for all real values of x in the domain of definition of the functionF ( x , y ) although the radius of converg; ence of the series (47.55) can be bounded by a singular value x which is complex. Example. Let y be defined by the equation

Here, Eqs. (47.56) have the solutions eyI = 0, xI = - 2 and neY*

For the series

+ n - 1 =0.

z

k=I

which is a solution of the equation @ ( x , y) = 0 , the points x1 and x2 a r e singular points since condition (47.57) is now satisfied (in

243

MI0 IMPLICIT FUNCTIONS

view of the fact that

(x' ') ax

=

- 2).

If n is positive and even,

then x, c 0 and lxll < Ix21. Therefore, the function y = q) ( x ) is real in the region - 2 2 x < a and the series written will converge in the region 1x1 < 2. On the other hand, if n is positive and odd, then lxzl 4 lxll and the series 'p ( x ) converges for 1x1 < 1 ~ ~ 1 ' .

48. T w o Implicit Functions Suppose that a function F(x,, holomovphic in aneighborhoodof the coordinate origin:

T h e o r e m ( W e i e r s t r a s s ) [32, 721.

...,x,,y)iis where

are constants and the series shown converges f o r (fork= 1, .... m). Here, r i s a p o s i t i v e number.Su+ pose alsothatthepower-seriesexpansionofthe function F ( 0 , ... , 0 , y) begins with the nth-degree term in y : F ( 0 , ..., 0, y) = Any" An+1y"+1+ ... (48.2) where A,, it; 0. Then, apl... Pm

Ixk(
+

F(x1,

..., xm , y)

(y"

+ alY"-' + ... + an-ly +

Un)Q(xi

.... xm , y), (48.3)

where a1,..., 4 are functions of xl,..., x, that are holomovphic at the point x1 = . ...' .x, = 0 and that ,vanish at that Point and where Q(x,, , s, , y.) i s holomovphic at the point xl = .X= x, = y = 0 but does not vanish at that point.

...

Suppose now that we are given the two equations

c P& OD

P ( x , y, 2 ) =

(x. y. z) = 0.

(48.4)

k=m

where P ( x , y, z) and Q ( x , y, z) a r e holomorphic in a neighborhood of the point x = y = z = U and the functions Pn(x, y, z) and Qk (x, y, z) a r e homogeneous polynomials of degree k. T h e o r e m 48.1 [73]. I f Eqs. (48.4) and (48.5) define functions

244

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

then these functions can be represented in the forms* (48.6) &=I

(48.7)

except in the case in which Eqs. (48.4) and (48.5) have a common root x = x (y, z ) for arbitrary y and z. Proof: Suppose that

+ Ap+pP+' + ..., Q ( x , 0, 0)= BqX4 + Bq+1X4+' + ...,

P (x, 0, 0)= AGP

(48.8) (48.9)

where A p f O andB,#O. (4 8.10) Then, on the basis of Weierstrass' theorem, Eqs. (48.4) and (48.5) can also be written

Q ( x . Y, z ) = [fl +%(!I,

+h(Y*

211 Y ( X ,

Y.

Z)M-'+...+

4 = 0.

(48.12)

where 'pk(y, z ) and $&(g, z) are functions that are holomorphic at the point y= z = 0 and that vanish at that point and where O(0, 0, 0) + 0 andW (0,0, 0) .f. 0. From this it follows that, to determine the functions

that satisfy Eqs. (48.4) and (48.5) in a neighborhood of the point x = y = z = 0, we may write

*In [74], see N. Chebotarev's survey of the methods of obtaining formal expansiona (48.6) and (48.7) Newton's method. Unfortunately, this work was not known to us at the time of writing7321. Therefore, it was not cited in that work. However, [74] does not change [32E

245

TWO IMPLICIT FUNCTIONS

L,(x) = f l + +,( z)xq-' !I,

+ ... +qJ,(Y,

z) = 0.

(48.15)

If we eliminate x from these equations, we find the' following necessary and sufficient condition for compatibility of Eqs. (48.14) and (48.15):

where @ is a known polynomial [35]incp,, ..., ' p p , q,, ..., qq, where Z(y, z) is a holomorphic function in a neighborhood of the point y = z = 0 , and where the Zk(y, z) are homogeneous polynomials of degree k. With regard to Eq. (48.16), we may encounter the following

cases:

Case I. @ ( y , , ..., cp,, ..:, qq)= 0. Then, Eqs. (48.14) and (48.15) and hence (48.4) and (48.5) will have common roots x=x,(y,

for arbitrary y and z . function

2)

(v=

1, 2, ...)

(48.1 7)

Consequently, we can take an arbitrary

and find x ( z ) in the form x = x, [y(z), z ]

(v =

1, 2, ...).

(48.19)

However, it may happen that x, (y, z ) -+, 0 a s y-Oand 2-0. Then, (48.18) and (48.19) will not be a solution of the desired form (*). R e m a i k : When Case I holds, Eqs. (48.14) and (48.15) can be written in the form

L, ( x ) L ( x ) = 0 , L, (x) L ( x ) = 0, where L,, LP and L a r e polynomials in x with coefficients depending on y and z . In this case, we shall also seek a solution of (48.14), (48.15) from the equations

or

L,(x) = 0,

L(x)=0

L ( x ) = 0,

L, ( x ) = 0

246

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

or L1 ( x )

L, ( x ) = 0,

= 0,

which returns us to equations of the form (48.14), (48.15). Case II. Equation (48.16) is not solvable in the form y = ap (z) 70 Then, Eqs. (48.4) and (48.5) do not have solutions of the as 2-0. required form p). Case 111. Equation (48.16) gives us the solution

as

y=y(z)--0

2-0.

(48.20)

This solution may not be unique but it can always be represented in the form (see Sect. 47)

2: a,z -

y=

k

(48.21)

k= I

From Eqs. (48.14) and (48.15), we see that we can obtain x = x ( z ) in the form

I

Here, the ak and pk are constants andv and p are positive integers. It may also happen that Eq. (48.16) does not contain y (Case 11) but has the solution z = 0. Then, (48.4) and (48.5) can have the solution z=O,

x=x(y)-0

as

y-0

(4 8.2 3)

and x(y)can be represented in the form (48.24)

where the ' in a r e constants and 6 is a positive integer. Suppose now that we have (48.8), (48.9), that is, that we have either P ( x . 0 , 0) = 0

or Q ( x , 0, 0) = 0.

Then, we shall not have (48.11) and (48.12) simultaneously. But, in this case, Eqs. (48.4) and (48.5) can be reduced to a form such

24 7

TWO IMPLlCtT FUNCTIONS

that (48.8) and (48.9) hold. To do this, we introduce new variables defined by

+ b p + clw

x

a,u

Y=

w + b*v + c,=J .

I

(4 8.25)

WJ

z=a.++bsV+

Let us substitute these values of x, y (48.5). Then, we obtain

and z into Eqs. (48.4) and

OD

P(x. y. z) =

ak,,ukv'w"

=

0.

(48.26)

k+l+v=m

These series converge in a neighborhood of the point u = z) = w = 0. It is easy to see that the coefficeints of urnand un in Eqs. (48.26) and (48.27) are, respectively, equal to P = P m (a1 v

9

(48.28)

US),

(4 8.29)

Q=Qn(alT

Let us choose constants a,, a, and a3 such that (48.30) PzOandQzO. We choose the remaining coefficients in the transformation (48.25) in such a way that the determinant composed of them will be nonzero. When conditions (48.30) hold, we have, on the basis of Weierstrass' theorem, from (48.26) and (48.27)

where the 'pR(v, w ) and qr(v, w) are holomorphic functions at the point v = w = 0 and vanish at that point and where @ (u,V , w) and Y(u, v, w ) are holomorphic functions at the point u = v w =0 but do not vanish at that point.

-

248

LINEAR S Y S T E M S OF ORDINARY DIFFERENTIAL EQUATIONS

If Cases I and I1 above do not hold, we obtain a solution of Eqs. (48.31), (48.32) in the form (48.33)

(48.34)

where the ak and 8k a r e constantsand v and p a r e positive integers. When we substitute these values of u and v into the last of Eqs. (48.25), we obtain (48.35)

where the 7a.are constants and 1 is a positive integer. On the basis of (48.25),we againobtain x and y in the form (48.6), (48.7). This completes the proof of Theorem 48.1.* In [75], a study is made of an implicit function of k variables.

49. The Construction of Functions (*) Defined by Equations (48.4) and (48.5) Suppose that we are given two equations of the form Az

+ By + M x + P ( x , y, z) = 0,

Cz+Dy+Nx+Q(x,

(49.1)

Y, z ) = O ,

(49.2)

where A, B , M, C, D , and N are constants and P ( x , y, z) and Q(x, y, z) are power series with constant coefficients not possessing free or first-degree terms in x , y , o r z. These series converge in a neighborhood of the coordinate origin. Let u s seek solutions y = y (2). x = x ( z ) of Eqs. (49.1) and (49.2) possessing the properties x(z)-O,

and y(z)-O

as 2-0,

.?he question of the existence of a solution y1 (2) equations

is investigated in an analogous manner.

+ 0,.,.y& (2)

(49.3) * 0 as 2

--f

0 of the

24 9

THE CONSTRUCTION O F FUNCTIONS

or, in general, such that x(zl), x(z,),

...

-+

0 and y(z,), y!tz), ... .+0 as zl,z,, ...

-

0. (49.4)

First of all, we note that if the rank of the matrix composed of the coefficients of the first-degree terms is equal to 2, we can immediately answer the question of the existence of such a solution, For BN - M D Z 0, such a solution exists, is unique, and is easily found. Suppose now that A D - B C Z O . Then, wz have a solution of the form '(49.5) k=m

k=n

where the ak and pk a r e constant coefficients. We see from this that the desired solutions exist and we can find all of them. We h o w sufficient conditions (depending on the signs of am and pn and the evenness o r oddness of the numbers m and n) under which these real solutions a r e defined for both z .> 0 and z < 0 Or in only one of these regions. Let us suppose now that the rankof the matrix of the coefficients of the first-degree terms is equal to unity, that is, that at least one of the coefficients A, B, M , C , Dand N is nonzero. For example, let us suppose that A # 0. Then, from Eq. (49.1), we find 2 = cp ( x ,

(49.6)

Y).

where q(x, y) is a series that converges in a neighborhood of the coordinate origin. lf we substitute this value of z into Eq. (49.2), we obtain an equation of the form F ( x , y) = 0.

Consequently, the question that we posed will be solved if we do not consider the case in which F (x, y) = (UX

+ by + cxa + dug I- y2+ ...)', 2

In this case, the process of determining the function y = y(x) is transcendental and difficult to find (see Sect. 6 [32]). Suppose now that A = B = M = C = D = N = 0. Then, Eqs. (49.1) and (49.2) can be written

250

LINEAR SYSTEMS OF ORDINARY D I F F E R E N T I A L EQUATIONS

'=

Qn

(

~

Y* 2 )

9

+

Qn+1

( x , Y, 2 )

+.

*.

= 0.

(49.8)

where Pk ( x , y, z ) and Qk ( x , y. 2) a r e homogeneous polynomials of degree k. To answer the-question posed, we introduce new functions u and v defined by* x=uz,

y=vz

(49.9)

and assume that the sequences

(49.10) remain bounded as zl, %.. -* 0. If we substitute the values of x and y given by (49.9) into (49.7) and (49.8) and divide by zm, we obtain

u1, u2

,...

a and vl, v2.... .* 6,

(49.12)

-+

where a and 6 a r e solutions of the equations

P,(u, 6, 1) = 0, Q,(u,b,

(49.13)

1) = 0.

In Eqs. (49.9), let us make the substitution where

T

lL=T+a.vEe+6, (49.14) and 0 a r e new unknowns possessing the properties that 7

(ZJ,7(z2),... -' 0,

ep,), qZ2) ,,... + o

(49.15)

a s zl, z2,... 0. If we substitute the values ofuandcr given by (49.14) into (49.11) we find -+

P,,,

(7

+

U,

0

+ b, I ) + Pm+l (z i

0,

Q

+ 6. 1) +

+ P r n + , ( z + u , 6 + b , l ) ? + ... = O , *We first used this substitution in [32].

2

(49.16 )

(OZ'6P)

(6 'c'6P) n=a

0=1

m

I SZ

S N 0 1 1 3 N n r l JO N 0 1 1 3 n M I S N 0 3 3 H I

252

LINEAR S Y S T E M S O F ORDINARY D I F F E R E N T I A L EQUATIONS

Here p and 4 a r e positive integers. Ifa a n d b are realnumbers, the solution (49.21) is real. At the beginning of -this section, we saw how the question may be answered as to the existence of the required solution if at least one of the quantities

,

-,

-,.

db

aa

I

is nonzero. Specifically, in this case, the matter reduces to the

case of a single equation in two variables. Let us suppose that all successive transformations of the form (49.9) and (49.14) give us Eqs. (49.7) and (49.8) with rn 7 1. Then, we cannot find the desired solution.* Of course, it may happen that Eqs. (49.13) are incompatible,** which will prove that a solution possessing properties (49.4) and (49.10) does not exist. We may seek a solution u in the form x=uz,

2-vy,

(49.23)

where

~ ( z l ) u, (zz),... and v ( Z I ) ~ V ( Z Z ) ~ . . .

(49.24)

a r e bounded. Here, we have

where the sequence ~ ( z , ) ,p(zz),... is bounded. Thus, we have the case

where the sequences P @I), P

and 4 ( Z d . 4 (ZZ),...

(ZZ),.~..

(49.26)

are bounded. We may then proceed just as in the case of (49.9). T h e o r e m 49.1. I f the system (49.71, (49.8) has a solution x = x (4, Y = Y (z),

*Cf.p. 40 of [32]. **Or have solution b = q~(a).

(49.27)

T H E CONSTRUCTION O F FUNCTIONS

25 3

such that x + 0 and y -+ 0 as z 0 , then, f o r such a solution one of the following three cases holds: -+

1. x = u ( z ) z , y = v ( z ) z

It. x = u ( y ) y , z = v ( y ) y 111. z = u ( x ) x , y = v ( x ) x

...

I. '

(49.28)

..

where the sequences ulr k , and v,, v,, , corresponding to 21, 22, are bounded. Proof: If v-03 in (49.9) a s 2-0, then v ( z ) - 0 in (49.23). If the sequence u(z,), u(z,), ... in (49.23) is bounded, we have (49.25). On the other hand, if u(z)-. 00 in (49.23) as z + 0, consider

...

*

(49.29)

x = P ( Z ) y.

If the sequence p (y,), p (y3),... is bounded, we have (49.25) and (49.26). On the other hand, if p(z)--, 00 in (49.29) as y 0, we set -+

y = Pl(x)x9

where p , ( x ) -0,as x -+0. Since z = v ( y ) y, where v

-+

(49.30)

0 as y --f 0, we have, on the basis of

(49.231, z =4 ( x ) x ,

where q=vp,

-c

Oas

x

-+

(49.31)

0; that is, we again have

where 4 ( x ) + 0 and p1 ( x ) +. 0 a s x Theorem 49.1.

+

0. This completes the proof of

*Of course, all three of these cases may hold simultaneously.

Appendix TABLE OF LAPPO-DANILEVSKIY'S COEFFICIENTS *

-1 0 -2

p3

0

-1

-2

0

-2

-1

0 -1 -2 0 -1

-1 -2* -1 0 -1

-1 -1

'

-2 -2

P1

P2

'

0 0 0 0 0 -1 -1 -1 -1

-1 -1

-1

-2

-1

-2 -2 -2

0 -1

-2

-2

-1 0

-2

(1) aP1P2P3

11

P3

P4

-2

1 -

0

1 0 71

pz

-2

0

-1 0

P1

-2

0

-2

0 -1 -2

P3

1 0

a(1) P 1P2

p2

PI

PZ

1 0 -1 0 -1 0 1

PS

0

0

-1

-2

-2

0

0

-2

-1

-2

0

0

-2

-2

-1

0

-1

0

-2

-2

2 1

0

-1

0 -1

-2

-1

0

-1

-1 -1

-1 -2

-2 -1 0

-1 -1

-2 -2

-2 -2 -2 -2 -2

0 0 0 -1 -1

0 -1 0

-1 0 -2

-1 0 -1

-1 0 -1 0

-2

-2

0

0

5/;

-2

-1 -1

-2 2

0

-2

-2

0 1 0 -1 1

-_2 (1)

P5

-1 -1 -1

2 1

-

P2

1

0 -1 0 -1 -2 0 1 0 -1 0

-1 -1

-1

-2

-2

-1

-2 -2

-0 -1

-2 0 0 -1

-2

-2 -1 -0 -2 -1 -2

-1

-2 0

a P IPZ .P5

..

1 0 -1 0 -1

-2 0 1

l T h i s table was prepared ~y the author's c o gues at the Leningrad di.-sion of the mathematics institute of the Academy of Sciences of the USSR, 0. A.Fedorova, and I. M. Varentsova, under the direction of Candidate of the Physical-Mathematical Sciences K. E. Chernin.

254

APPENDIX

PI

-

_

P3

P4

P5

-1

-1

0

0

- 1

P:

- _

- 0 0 0 0

I Lap _ -

-1

-1 -1

-1

-2

0

-1

-1

-2

-1

0

-4

-2

-1

(

0

-2

-2

-1

(

-1

-2

-1

(

-1 -1

-1 -1 -1 -1

c -1

-1

-1

-1 -1 -1 -1 -1 -1 -1 -1 -1 -

-1 -1 -1 -1 -1 -2 -2 -2 -2

P1

(

D2

-2 0 0

0

2 0 0

-2 -1

3 0

-1

-2

-1

-1

-1

0

- 1

-2 -2

-1

0 -2

0 3 0

-1

0

0 -1

2 0

-1

0 0 0 -1

-

4

- 3 0

-1 -2

-2 -2

(

-

0 2 5 _

P3

-

P4

- - - -

P6

-

-2

0

0

-1

-2

-2

1 -

0

0

-1

-2

-1

0

-2

1 -

8

4

-2

0

0

-2

-1

-2

0

0

-2

-1

-1

0

0

0

-2

-2

-2

- -1

0

0

-2

-2

-1

2 0

0

0

-2

-2

0

- _1

0

1

0

-1

-2

1

0

-2

-2

-2

0 0

0

0

4 1 -

2

2 1 4 ,1

0

1

0

-1

2 0

0

1

-1

0

-2

1 -

0 0 0

1

-1 -1 -1

-1

-2

1

-1

-1

0

-

1

1 -2 - -

-2 -

P3

-

-

-

-

-1

-2

-1

-1

0

- 3

-1

-2

-2

0

0

- -5

-2

a a

0 0

-1

-2 -1

a

-1 -1

-2 0

4 2 0

-2

- 1

-1

-1 0 -1

0

-2 -2 -2 -2

-2 -2

-2 -2

2

- 1

- inut - -

a a a a

-1

-1

-2

Pi

-2

-1

-2

PI

0

0

0

6 (CC

- 1

0

0

tici

-2 -2

1 12 1 -

0

---

-1 -1 -1 -1

$1) PlPZ..-Prj

255

c I

P4

PS

(1) PlPZ...PS

1

-2

-2 0

-2

-1

0

0

0 -2

0 0

-1

-1

0

-2 0

0

- 1

-1

0

-1

0

1

-1

-1

-1 -1

0 - 2 - 1

-2

-1

-2

0

0

-2

-2

0

0

-1

2 0

-2

-2

0

-1

0

1 -

-2

-2

-1

0

0

1 -

-

-

P1

'2

P3

P4

P6

(1) %lP2-.*P6

-1 -1

-1

-2 -2

-1

0

0 0 0 0 0 0 0 0 0

-

-

- - -1 -1 -1 -1 -1 -1 -1

-1 -2

-

0

0 -2

0

-1

-2

1

-2

-1

-1

0

-2

-1

0

- 1

-2 -2

-2 -2

-1

-1

0 2 - -5 4 - 3 0 2 0 0 3 0

-2

- 1

0

0

0

-2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

-2

0 0 0 0 0

-1 -1

-2

-

-1

-1

-2

-2

-1

-2 0 0

-2

-1

-1

-2 -2

-1 -1

-2

-2 -2 -2 -2

-1

0

-1

0

-1

0

- 1

-2 -2 0 0 0

-1

0 3 0 0 2 0

- - -1 -

-2

-1

-2

-1 -1

-2 -2 -2

-2 -2

1

- 2 0

-1

-2

-2 -2

4

-2 -2

0

-2 -2 -2 -2

2

0 -2 -1

0 -1 -

25 6

LINEAR S Y S T E M S OF ORDINARY D l F F

Table of

-

P5

0

P4 P3 -1 -2 , -2

0

-2

' -2

-2

0

-1

0

0

-1

-2

-1

0

0

-2

-1 -2

PI

P2

-1

-1

0

0

-2

-1

0

-1

0

-2

-1

-1

-1

-1

-2

-2 -2

-2 -2

0 -1 -2 -1 -2 0 -1 -2 0 -1

0

0

-2

0

-1

-1

0 0 0 0 -1 -1 -1

-1

-2 0

-1

0 0 0 0 0

-1 -1 -1 -1 -1

-1

0

-2

-1

0 0 0 0 0 0

-2

-1 -1 -1 -1 -1

-2

-1

-1

-1 -1 '-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

-1

0 0 -1 -1

-1 -1

-1

-2

-2

-2 -2

0 0 -1 -1

0 -1 0 -1 -2 0

-1

-1

-2

-2

0

-1 -1

-2

0

-1

-2 -2

0

0 0 -1 -1 -1 -2

-2 0

-1 0

0

, -1

-1 -1 -1

-2 -2

-2 -2

-1

-1 -1 -1 -1 -1 -1

-2 -2

-1

-2

-1

-2 -2 -2

0 0 0

-2

0

-2

0 -1 -1

-2

~

I ~

-2 0 0 -1 -1 -1

-2 -2 -2

-2 0

-1

-2 -1

-2 0 -1

-1

-2 0

-1

-2 0

RENTIAL EQUATIONS

APPENDIX

-

fficients (contin

-

p1

P:

P3

-

-

-

-2

-i

0

p4

ps

0

-1

(1) ?lp2...ps

p6

5

p6

--

-1

0

I

-1

-2

-1

0

Q

-2

Q

1 -

L

O

-2

-'

0

-1

0

-1

0

I

0

-2

-i

0

-1

-1

0

0

-1

0

-2

0

0

- -1

I

-2

-

L

I

P6

2

- -21 4

2

P1

72

33

'4

-

-

-

-

-

-

-

0

0

a

-1

-2

-2

-2

P5

P7

-f-

&I) P lPZ.-.P? 1 36 1 -

P1

-

D;

P3

- -

PI

P5

-

P6

p7

L -

0

C

-2

-2

C

-2

-1

0

0

C

-2

-2

-1

0

-2

-1

0

C

-2

-1

-1

-1

2 0

0

C

-2

-2

-2

c

- _1

C

0 -1

-1

0 1

-2

-. -.

0

0

C

-1

-1

-2

-2

0

0

C

4

-2

-1

-2

0

0

C

-1

-2

-2

-1

0

0

0

Q

-1

C

-2

-2

-2

1 -

0

C

-2 -2

-i

-2 -2

0

-1

0

C

-2

-2

0

-1

0

-1

-1

-2 -2

0

-1

0

-1

-2

-1 -2

0

-1

0

-1

-2

-2

0

1

0

2

0

.2 -1

0

1

0

2

-1

-1 -2

0

0

-1

-1

-1

-2

-2

0

0

-1

.I

-2

-1

-2

0

0

-1

-1

-2

-2

-1

0

0

-1

.2

0

-2

-2

0

0

.1

2

-1

-1

-2

0

0

.1

2

-1

-2

-1

0

0

1

2

-2

0

-2

0

0

1

2

-2

-1

-1

0

0

1

2

-2

-2

0

0

0

.2

0

-1

-2

-2

0

0

2

0

-2

-1

-2

0

0

2

0

-2

-2

-1

0

0

2

1

0

-2

-2

0

0

2

1

-1

-1

-2

0

0

2

1

-1

-2

-1

0

0

2

1

-2

0

-2

0

0

2

1

-2

-1

-1

0

0

2

1

-2

-2

0

0 -

-

2

2

--

0

-

-

0

-1

-2

-

24 1 12

24 1 16

-1

8 0 -1 8 1 4 0 - -1 4 0 - -1 4 -1 8 -1 4 0 -1 4 -1 2 0 1 2 0 - -1 2 - 1

-_

-1

c

-1

0

1

0

2

-1

-2 -1

0

1

0

2

-2

0 -2

0

1

0

2

-2

.1 -1

0

1

0

2

-2

-2

0

1

-1

0

-1

.2 -2

0

1

-1

0

-2

.1 -2

0

1

-1

0

-2

.2 -1

0

1

-1

1

0

.2 -2

0

1 1 1 1

0 0

0 0 0

0 0 0 0

1 1 1 1 1 1

-1 -1 .1 -1 -1 -1 -1 -1 -1

-1

1

1 1 1 1

2 2 2 2 2

-1 -1

-2 -2 -2 0 0

-1 -1

-1

- - - - -

.1 -2 0 .1

0

-2 -1

-2 -1

-2 0 -1 -2

-2 -1 0 -2 .1 -1 .2 0

-

2

1 12 1 8 1 4 0 1 4 1 2 0 1 - _ 2 0 1 2 1 4 1 2 0 -1 2 1 0 - 1 0 - 1 - 2 0 1 0 - 1

-_

258

L I N E A R S Y S T E M S O F ORDINARY D I F F E R E N T I A L E Q U A T I O N S

tblt -

fl -

milevski

P2

P3

P4

Ps

- -

-

-

- -

-

p1

P6

P7

0 0

-1 -1

-1 -1

-2 -2

-2 -2

0

-1

-1

0

0

-1

-2

0

0

-2

-2

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1

-2 -2 -2

0 0

-2

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

-1 -1 -1 -1 -1 -1

-2

0

-1

-2

0

-1

-2 -2 -2 -2

-2 -2

-2 -2 -2

-2 -2 -2

-2

0 0 -1 -1

-1 -2 0 -1 -2 -1 -2 0 -1 -2 0 -1 0 -1 -2 0 -1

-1 -2 -1 0 -1 0 -2 -1 0 -1 0

-2

-2

0

0

0

0 -1 -1 -1 -1 -1 -1 -1 -2 -2

-2 -2

-1 -1 -2 -2 -2 0 0 -1 -1 -1 -2 -2 0

(1) PIP2

-

- 3

-1 -2 -1 0

0

0

-1

-2

-2

-2

0

0

-2

-1

-2

0 2

0 - 3 - 5-

-

4 _3

0

-2

0

0

-2

-2

-1

0

-2

0

-1

0

-2

-2

0 0 0 0 0 0 0 0 0 0 0 0

-2 -2 -2

0 0 0 0 0 0 0 0 0

-1 -1 -1 -1 -1 -2

-1 -1 -2 -2

-1 -2 0 -1 -2 -1

-2 -1 -2 -1 0

-2

-2

-1

0

0 -1 -2 0 -1

-2 -1 0 -1 0

- 2 0 0 0 1 17 8 6

0 0 0 0 0 0 0

-2 -2 -2 -2 -2

-2 -2 -2

-2 -2 -2 -2 -2

-2

0

-2 -2 -2

0

-2

0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1

-2

-2 -2

0 0 -1 -1 -1 -2

-2

-2

-2 -2 -2

0 0 0 0 0 0 -1 -1 -1

- - - -

0 -1 -1 -2 -2

-2 0 0 -1

-

-2 -1 -2 0 -1 -2 -1 -2 -1

-

-2 -2 -1

0

-2

- 3

-1 0

0 - 1 - 4 0 1

-2 -1 -2

-

P3

-2

P4

Ps

Pt

P7

- - -

-

-1 -1 -1

-1 0 -1 0

0 0 0 0 0 0 0

-2

0

-2

-1

0

-2

0 0

-2

-2 -2

0 0

-2

-2 -2

0

0 0

-2

-2

0

-2

-2

0 0 0 0 0

-2

0

-2 -2 -2 -2 -2 -2

-2 -2 -2 -2

-1 -1 -1 -1 -1

0 0 -1 -1 -1 -2 -2 0 0 0 -1 -1

0

-2

-2

-1

-2

0

0

0

-2

-2

-2

0

0

-1

0

-2

-2

-2

0

-1

0

0

-2

-2

-2

-1

0

0

_7 4 _3

-1

0

0

0

-2

-2

-2

--5

- 7_

3 0 - 1 0 - 1 0 3 0

P2

-1 -1 -1 -1 -1 -1 -1 -1 -1

2 0 0

-2

0

C C

4

-2

-2

PI

- - -

0 2 _5

4 - -7 4 0 - _7 4 - 4 0 3 0 1 5

0

...P:

C ffi in d) - !Ill - (cc -

-2 -2 -2 -2 -2

-2 -2

-2

-2

-2 -2 -2 -2 -2 -2 -2

0 0 0 0

-1

-1

-1

-2

-2 -2 0

C -1 C

0

-1 -1

-1 -2 C -1

-2

0

0

-1

-1 0 -1 -2 0 -1

-2 -1 -2 -1 0 -1 0 -2 -1 0 -1 0

0

-2 0 -1 -2 0

-2 -1 0

-1 0

-1

0

0

-1

-1

-2

-2

-I -1

0 0 0

-1 -1

-2 -2 0

-1 -2 -2

-2

-1

0 0 0

-2

-1 -2

0 - 1

0 4 1 0 3 0 - 6 17 8 -1 0 0 0 0 0 - 5 - 1 0 - 3

-

0

4

7 4

0

4

18

16 - 1 0 - 1 - _9

-1

0

0

-2

-1

-1

-2

-1

0

0

-2

-1

-2

-1

0

-1

0

0

-2

-2

0

-2

-7

-1

0

0

-2

-2

-1

-1

0

-1

0

0

-2

-2

-2

0

_3

-1

0

-1

0

-1

-2

-2

4

4

-1

0

-1

0

-2

-1

-2

-1

0

-1

0

-2

-2

-1

4 - 1 9 4 0

-1

0

-1

-1

0

-2

-2

_ 9_

-1

0

-1

-1

-1

-1

-2

- 5

- -

- -

-

- -

-_ 4

259

APPENDIX Table of Lappo-Dtdlevskiy's Coefficients (continued)

P:

- - P:

P.

- - - -1 -1

(

-1

(

-1 -1

P

-2

-

-

-1 -2

-1

0

-2

-1

-1 C

(

C

-2

-2

(

-1

-1

-2

(

-1 -2 -2 -2

-1

-2

- 5

-1

-1 C -2 -1

-1

-1

-1

-1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1

0

0 - 1 - 7 0 2 0 0 0 1 1 0 1 0 - 3 - 7 8 23 16 31 8

-1 -1 -1

-1 -1 -1 -2 -2 0 0 0 -1 -1

-2 0 -1 -2 -1 -2 0 -1 -2 0 -1 0 -1 -2 0 -1

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

C

-1

C

-2

-1 -1 -1 -1

C C C

C C C -1 -1 -1 -1 -1 -1 -1 -2 -2 -2 -2 -2

-

-

-1

-

-4 -1

C

a

(

-, c

-1

C

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1

a a a 0 0 0 0 0 0

-2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2

-1

0

-2

-2

-2

-1

-1

0

0

.1

-2

-2

-1

-1

0

0

.2

-1

-2

-1

-1

0

0

.2

-2

-1

0

-1

-1

0

-1

0

-2

-2

31 -

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

-1 .1 .1 1 1

0 0 0 0 0 0

1

0 0 0

.1 .1 ,2 2 2 0 0 1 1 1 2 2

-1 -2 0 -1 -2 -1 -2 0 .1 .2 0 .1

-2 -1 -2 -1

1

-1 -1 -1 .1 .1 .2 .2 .2 2 2 2 2

-

1

1 1 1 1

0 0

0

I

-1 -1 -1 -2

-1 -1 -1 -1 -1

0

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

-1

c

C

-

P'

-

-

P!

-2

-I

a

-

D

c

(

-1

-

PI

-

(

-1

-

P:

(

-1 -1 -1 -1 -1 -1 -1

C

-

- -

-1 -2 -1

-

-1 -1

c c

-

P: 0 4 0 2 7 0 - 3 0 1 0 - 1 27 8 9 0

(

(

Pf

- - -1

-1 -1

(

P!

- - -

a a

-2 -1

0 -1 0 -2 -1 0

-1 0 0

0

-2 -1 -2 -1 0 -1 0

- - -

8 10 0 - 6 0 - 2 - 9 0

3 0 - 1 0 3

-

-

-

-1 -1

-1

-1 -1

-1

-1

-2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2

-1

-1 -1 -1 -1

-1

-

,

I

-

-

I I

I

I

I

I (

-1

-1 -1

-.-.

c c (

C C C

a

(

-1 (

(

-

(

-1

c

c

-1

-1

-1 C

-2

C

C

-2 -1

a -1 -1 -1 -2 -2

-2

0

-2 -1

1 1 1 1 1

0 0 -1 -1

-1

0 -1

0

0 -1 0

-1

-1

-2

1

-2

0

-1

-1

-2

2

0

.1

-1

.1

-2

2

0

0

-1

.1

-2

2

.1

0

-1

.2

0

0

0

2

-1 -1 -1 -1 -1 -1 -1

2 .2 2 2 2 2 2 2 2

0

0 0

1 1 2 2 2 0 0 1 2

2 1 2 1 0 2 1 2 1

-1 -1

-

-

0

0 0

0 0 0 0

0 -

0 0 D 1

1 1 1

-

-

-

...

(1)

'PIP2 P7 49 16 -10 0 4 0 0 5 0 - 1 0 - 1 0 5 0

0 4 0 -10 49 16 3 0 - 1 0 3 0 - 9 - 2

--

0

- 6 0 10 31 8 0 31 8 23 16

- 2 8 - 3 0 1 0

1 1 0 0 0

260

LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Table of Lapp-Danilevskiy's Coefficients (continued)

P1

p2

- -

-

P1

P4

P!

PI

-

-

-

-

-1 -1 -1 -1 -1 -1 -1 -1 -1

-2

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

-2

-1

-2 -2 -2 -2

-1 -1 -1

-1

-

-

-i

(

-2

(

-2

(

-1 -1 -1

-2 -2 -2 -2 -2

(

-i

(

(

-2

(

-1

(

-2

(

-1

( (

-2 -2

(

-2 0 0 0

-1 C

0

(

-1

(

-2 C -I -2

-2 -1 -2 -1 0 -1 0 -2 -1

-

-1

-1 -1

-1

-1 -1 -1 -1 -1

(

-2

-1 -1 -1 -1 -1

-2

-1

-1

-1

-2

-1

-2

-1

-2

-1

-2

-1

-2

-1

-2

-2 -2 -2

-2 -2

-1 -1

0 -1 0 -2 -1 0 -1 0

0 0 0 0

-2

P7

-' -'

(

-1 (

C

a -1

(1)

'P lP2.. .P7 2 0 - 7 - 1 0 - 5 0 9 27 8 - 1 0 1 0 - 3 0

-

P:

P4

- -2 -2 -2 -2 -2 -2

-2

C

-2

-2

C C

-2

C

7

-2

2

-2 -2 -2

a a

-1 -2 -1 0

-2 0 0 0 -1 -1

-1

-1 0 -1 0

0 -1 -2 0 -1

-1 0 -1 0

1

-2

-2

0

0

2 2 2

0 0 0 0 0

0

-2 -1 -2 -1 0 -1 0 -2 -1 0 -1 0

0

0

0

4 0 - _9 4 - 1 3 4

-1

-2

-2

0

0

0

-2

-1

-2

-2

0

0

-1

-1

0

-1

-2

-2

0

0

-2

0

7 4

;

-2

1 0

-2 -2

0

-2 -2

0

-2

0

-1

0

-1

0

0

-1

-1

0

-9

-1 -1 -1

-2 -2 -2

-2 -2

0

-2

0

0 -1

-2

-1 -1

0

0 0 -1

0

- 1 0 - 1

-1

-2

-2

-1

-1

0

0

1 -_

16

-2 -2 -2

-1

-2

-2

-2

0

0

0

5 -_

-2

-2

0

0

0

-1

-2

2

18 3 4 7 4 0 7 4 4

0

-2

-2

-2

-

0 -

0

-1

- -

-1 -

-1

-

-2

-

0

0 0 0 0 0

-2 -2 -2 0

0 -1

-1 -1

-2

-1

-2 -1 -2 0 -1 -2 0

-2 -1

-2

-2

I

-2

-1

0

-2

-2

0

0 0 0 0

-2 i 0 -2 0 -2 j 0

-2

0

0

-2 -2 -2

-1

4

0 0

-2 -2 -2

-1

-2

-1 -1

-2 0

0

-1

-2

-1

-1

-2

-2

C 0

0

.2

-1

-2 -2

-1

0

-1

-2

0

0

-2 -2

0

2 2

0 0 -1 .1 -1 -1 -1

.2 0 0 0 .1 .1

-1 -2 0 -1 -2 0 -1 0 -1 -2 0 -1

2

-1

2

0

0

0

2

0 0

2

0 0 0 0

2 2 2

-2 -2

0

0

-2

0

-2

-2

2

2

0 -1 .1 .1

.2

I

-2

j

0

0 0

-2

-1

a

0

-1

-1

-1

9

-2

c

-1 -1 -1 -1 -1 -1 -1 -1 - 1 -1 -1 -1 - 1 .1 - 1 - 1 -1 .1 - 1 .1 -2 .1 - 2 .1 -2 .1 -2 1 -2

- _9

.2

C

-2 -2

-2

- 5

-1

-2

-2

0

.2

C 0

C

-2

0

-1 0 -1 0

-1

0

-1

0

- 5

-1

a

-2 0 -1 -2 -1 -2 0 -1 -2 0 -1

-1

0

0

-2

( (

0 0 0 0 0

...P7

0 - 3 0 - 1

_ I

C C

(1)

aP1 PZ

-1 -2 -1 0

-1 -,

C

P7 -

-1 -1

(

P6

-

C C C

(

-2

P

-

C

-1 0

-2

- -

- - -

a

4

-1

-

9

-2

0

-1

-

P1

-2

0 -1 -2 0 -1

C C

-

0 0 0 - 1 17

-_ 8

- 6 0 3 0 1 4 0 - 1 0

1 0 - 4 - 1 0

- 3 0 6 17 8 1 0 0 0

- 2 0 5 1 0 3 0

- 4

4z4

2

.2

0

0

-1

0

2

.2

0

-1

0

- -7

2

.2

1

0

0

-3

-

-

-

- -

4

4

26 1

APPENDIX

- -

-

-

Table of Lapp-Danilevsl - -

P1 P1

'3

P4

P5

P6

-

-2

-1

0

0

a

-2

-2

-2 -2 -2 -2 -2

-I

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -2 -2

-1 -1 -2

-1 -2 0 -1 -2 -1 -2 0 -1 -2 0 -1 0 -1 -2 0 -1

-2 -1 -2 -1 0 -1 -2 -1 0 -1 0 -2 -1 0 -1 0

-1 -1 -1

-2 -2 -2 -2 -2

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

-2 -2 -2 -2 -2 -2 -2 -2 -2 -2

-2 -2

-2

-2 -2

a 0

0 0

-2 -2

-1

0

-2

-2

0

0

-1 0 -1 0 -1 0 -1 0 .1 0 -1 0 .1 0 .1 -1 . I -1 .1 -1 .1 -1 .1 -1

0 0 -1 -1 -1 -2

-2 -2

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -2 0 -1 -2 0 -1 0 -1 -2 0 -1

-2 -1 -2 -1 0 -1 0 -1 -1 0 -1 0

-2

-1

.1

-1 -2

0

0

-2

-1

.1

-2

0

0

-1

-2

-1

.1

-2

0

-1

0

-2

-1

.1

-2

-1

0

0

-2

-1

.2

0

0

0

-1

-2

-1

.2

0

0

-1

-2

-2

-1

.2

0

0

-2

0

-2

-1

.2

0

-1

0

-1

-2

-1

.2

0

-1

-1

0

-2

-1

.2

0

-2

0

0

-2

-1

.2

-1

0

0

-1

-2 -2 -2

-2

-2

- -

-

'3

-2

-1

-2

-1

0

-1

0

1 -

-2

-1

-2

-1 -1

0

0

1 -

-2

-1

-2

-2

0

0

0

1 -

-2 -2

-2

0 0

0 0

-1

-2

0 0

-2 -1

-2

-2

0

0

-1

0

-2

-2

-2

0

0

-1

-1

-2

-2

0

0

-1

-2

-1 0

-2 -2

-2 -2

0 0

0 0

0 -1

-1 0

0 0 2 0

-2

-2

0

-2 -2 -1 0

0

-2

12 1 0 1 2 0 1 2 0 - 1 - -1

-2

0

-1

0

-1

-2

-2 -2

0

-1

0

-2

-1 0

- _1

- 3

-2

-2

0

-1 -1

0

-5

-2

-2

0

-1 -1

-1

-2

-2

0

-1 -2

0

0

-2

-2

0

-2

0

0

-1

-2

-2

0

-2

0

-1

0

-2

-2

0

-2

-1

0

0

1 -

-2

-2

.1

0

0

0

-2

- _1

-2 -2

-1

0

0 1

-2 -2

-1

'0

0 0

-1 -2

-1 0

- -1

-2

-2

-1

0

-1

0

-1

1 -

-2

-2

-1

0

-1

-1

0

-2

-2

-1

0

-2

0

0

-2

-2

-1 -1

0

0

-1

-2

-2

-1 -1

0

-1

0

-2

-2

.1

-1

0

0

-2

-2

-1 -2

0

0

0

-2

-2

-2

0

0

0

-1

-2

-2

-2

0

0

-1

0

-2

-2

.2

0

-1

0

0

-2

-2

.2

-1

0

0

0

P1

0 0

-

-2

-2 0 0

0 -1 -1

- _5

4

- 3 0 2 0 0

-2

-1 -1 -1 -2 -2 0 0 0 -1 -1

- - - -

- - - - - -

-

P7

- - - - -

-

'a Coefficienta (continued)

3 0 - 1 0 - 1 0

3

-

-

4 2 0 1 0 1 0 1 1 0 1

2 0 1 2 1 4 - _1 2 0 - _1 2 0 1 2 1 4 0

92

P4

P5

P6

- - - -

-1

-2

P7 -

-1 0

4

8

-_

2

0

2 0 1 2 1 4 0 1 4 8

4

0

4 0 1 . 4 1 8 0 1 8 1 16 1 24 0 1 12 1 24 1 36

-

-

Bibliography 1. Lappo-Danilevskiy, I. A., Primeneniye funktsiy ot matrits k teorii lineynykh sistem obyknovennykhdifferentsial’nykh uravneniy (Application of functions of matrices to the theory of linear systems of ordinary differential equations), GITTL, 1957. 2. Smirnov, V. I.. A Course of Higher Mathematics, Vol. 111, Reading, Massachusetts, Addison-Wesley, 1964. 3. Gantmakher, F. R., The Theory of Matrices, New York, Chelsea, 1959. 4. Erugin, N. P., Metod Lappo-Danilevskogo v teorii lineynykh differentsial’nykh uravneniy. (Lappo- Danilevskiy’s method in the theory of linear differential equations), Leningrad University Press, 1956. 5. Erugin, N. P., “Opokazatel’noy podstanovke sistemy lineynykh differentsial’nykh uravneniy” (problema Puankare) (On the exponential substitution of a system of linear differential equations (the Poincard problem)), Matem. sb., 3 (45), No. 3, 1938. 6. Goursat, E., Cours d’Analyse mathe’matique, 7th ed., vol. 11, Paris, Gauthier-Villars. 7. Hukuhara, M., Tokyo Daigaku ruga Kubu Ku&, S. Fac. Sci. Univ. Tokyo. Sec. 1, 7, No. 1, 1954. 8. Artem’yev, N. A., “Issledovaniye osushchestvimosti periodicheskikh dvizheniy” (An investigation of the attainability of periodic motions), Zzv. Akad. nuuk, USSR, Seriya matem., 3, No. 2, 1941. 9. Artem’yev, N. A., “Metod opredeleniya kharakteristicheskikh pokazateley i prilozheniye ego k dvum zadacham nebesnoy mekhaniki” (A method of determining characteristic exponents and its application to two problems in celestial mechanics), Zzv. Akad, nuuk, USSR, seriya matem., g, No. 2, 1944. 10. Shtokalo, I. Z., “Kriteriy ustoychivosti i neustoychivosti” (Stability and instability tests), Matem. sb., 19 (61), 263-286, 1946. 11. Erugin, N. P., “Nekotoryye obshchiye problemy kachestvennoy i analiticheskoy teorii differentsial’nykh uravneniy” (Some general problems in the qualitative and analytic theory of differential equations), Priklad. mat. mekh, 19, No. 2, 1955. 12. Yakubovich, V. A., “Metod malogo parametra dlya kanonicheskikh sistem s periodicheskimi koeffitsiyentami” (The small262

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264

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D I F F E R E N T I A L EQUATIONS

26. Lyapunov, A. M., “Obshchaya zadacha obustoychivosti dvizheniya” (A general problem on the stability of motion), Sobmniye sochineniy (Collected works), Vol. 11, Akad. nauk press, USSR, 1956. 27. Donskaya, L. I., “Postroyeniye resheniya lineynoy sistemy v okrestnosti regulyarnoy osoboy tochki v osobykh sluchayakh” (The construction of a solution of a linear system in a neighborhood of a regular singular point in special cases), Vestnik, Leningrad State University, No. 6, 1952. 28. Donskaya, L. I., “0 strukture resheniy sistemy lineynykh differentsial’nykh uravneniy v okrestnosti regulyarnoy osoboy tochki” (On the structure of solutions of a system of linear differential equations in a neighborhood of regular singular points), Vestnik, Leningrad State University, No. 8, 1954. 29. Erouguine, N., “Sur la substitution exposante pour quelques sist&mes irr&guliGres,” Matem. sb. m. 42:6, 745-753, 1935. 30. Erugin, No P., “Ob asimptoticheskoy ustoychivosti resheniya nekotoroy sistemy differentsial’nykh uravneniy” (Onthe a s y m p totic stability of a solution of a certain system of differential equations), Priklad. mat. rnekh,XIJ, No. 2, 1948. 31. Erugin, N. Po,Priklad. mat. mekh, XXIII, no. 5, 1959. 32. Erugin, N. P.,“Nefavnyye funktsiiSr(Implicit functions), Leningrad University press, 1956. 33. Breus, K. A., “0 privodimosti kanonicheskoy sistemy differentsial’nykh uravneniy s periodicheskimi koeffitsiyentami” (The reducibility of a canonical system of differential equations with periodic coefficients), Doklady Akud. nuuk, USSR, 123, No. 1, 1958. 34. Krein, M. G., “Osnovnyye polozheniya teorii A-zon ustoychivosti kanonicheskoy sistemy lineynykh differentsial’nykh uravneniy s periodicheskimi koeffitsiyentami” (Fundamentals of the theory of )i-zones of stability of a canonical system of linear differential equations with periodic coefficients), Sb. “Pamyuti A . A . Andronova,” Izd. Akad. Nauk, USSR, 1955. 35. Kurosh, A. G., Kurs vysshey algebry (A course inhigher algebra), 1946. 36. Chetayev, N. G., Priklad. mat. mekh., IX,no. 2, 1945. 37. Chetayev, N. G., Ustoychivost’ dvizheniya (Stability of motion), OGIZ, 1946. 38. Shtokalo, I. Z., “Lineynyye differentsial’nyye uravneniya s peremennymi koeffitsiyentami” (Linear differential equations with variable coefficients), Zzd. Akad. nuuk, Ukr. SSR, Kiev, 1960. 39. Levitan, B. M., Pochti-periodicheskiye jknktsii (Almostwriodic functions). GITTL. 1953. 40. Gel’man, A. Ye.,Doklady Akad. nuuk, USSR, 116,no. 4, 1957.

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41. Shtelik, V. G., “K voprosu o resheniyakh lineynoy sistemy differential’nykh uravneniy s pochti-periodicheskimi koeffitsiyentami” (On the question of solutions of a linear system of differential equations with almost-periodic coefficients), LRr. matem. z h u m l , lo,No. 3, 1958. 42. Ivanov, V. N., Doklady Akud. nauk, 109, No. 5, 902-905, 1956. 43., Khalanay, A., Doklady Akud. nauk, USSR, 88, No. 3, 419-422, 1953. 44. gandor, gt., Bul. St. Akud. R. P. R., sec. de #t. mat. gi fiz., XII, No. 2, 339, 1955. 45. Lyashchenko, N. Ya., Doklady Akud. nauk, SSSR, 111, No. 2, 295-298, 1956. 46. Bogolyubov, N. N., and Yu. A. Mitropol’skiy, Asymptotic Methods in the Theory of Nonlinear Oscillations, 2nd ed., Delhi, Hindustan Publishing Corporation, 1961. 47. Vinogradov, I. M., “Obshchiye teoremy o verkhney granitse modulya trigonometricheskoy summy” (General theorems on an upper bound of the absolute value of a trigonometric sum). Zzv. Akud. mu&,USSR, seriya matem., l5,No. 2, 1951. 48. Vinogradov, I. M., “Osobyye sluchai otsenok trigonometricheskikh summ” (Special cases of bounds for trigonometric sums), Zzv. Akud. nauk, USSR, seriya matem., 20, No. 3, 1956. 49. Yakubovich, V. A., Vestnik, Leningrad University, seriya matem., mekhan. i. astr., l3,No. 3, 1958. 50. Golokvoschus, P. B., “Neobkhodimyye i dostatochnyye usloviya periodichnosti fundamental’noy sistemy resheniy nekotorykh lineynykh sistem differentsial’nykh uravneniy’ ’ (Necessary and sufficient conditions for periodicity of a fundamental system of solutions of certain linear systems of differential equations), Doklady Akad. nauk, Belorussian SSR, 3,No. 7, 1959. 51. Golokvoschus, P. B., “Otyskaniye kharakteristicheskikh pokazateley sistem dvukh lineynykh odnorodnykh differentsia1’nykh uravneniy s periodicheskimi koeffitsiyentami, soderzhashchimi malyy parametr” (The finding of characteristic exponents of systems of two linear homogeneous differential equations with periodic coefficients containing a small parameter), Doklady Akud. nauk, Belorussian SSR, 3 No. 9, 1959. 52. Golokvoschus, P. B., “Otyskaniye kharakteristicheskikh pokazateley sistemy dvukh differentsial’nykh uravneniy 8 periodicheskimi koeffitsiyentami, analiticheskimi otnositel’no malogo parametra” (The finding of characteristic exponents of a system of two differential equations with periodic coefficients that are analytic with respect to a small parameter), Doklady Akud. nauk, Belorussian SSR, 5,No. 6, 1960. 53. Golokvoschus, P. B., “Zamechaniya ob ogranichennykh i periodicheskikh resheniyakh sistemy dvukh lineynykh

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54.

55.

56. 57. 58. 59. 60. 61. 62. 63.

64.

65.

LINEAR SYSTEMS

OF ORDINARY DIFFERENTIAL EQUATIONS

differential’nykh uravneniy s periodicheskimi koeffitsiyentaqi, integriruyemoy v zamknutoy forme” (Remarks on bounded and periodic solutions of a system of two linear differential equations with periodic coefficients that is solvable in closed form), Zzv. vysshikh uchebnykh zavedeniy, Matem. No. 3, 1960. Golokvoschus, P. B., “Ob ogranichennykh i periodicheskikh resheniyakh nekotoroy sistemy dvukh lineynykh differentsia1’nykh uravneniy s periodicheskimi koeffitsiyentami” (Onbounded and periodic solutions of a certain system of two linear differential equations with periodic coefficients), Ucheniyye zapiski Vil‘nyusskogo gos. universiteta im, V. Kapsukasa, 2, seriya matem., i fiz. nauk, 1960. Golokvoschus, P. B., “Dostatochnyye usloviya ogranichennosti vsekh resheniy sistemy dvukh uravneniy, soderzhashchikh malyy parametr” (Sufficient conditions for boundedness of all solutions of a system of two equations that contain a small parameter), Ibid. Gremyachenskiy, A. P., “Obobshcheniye odnoy teoremy Lyapunova” (A generalization of a theorem of Lyapunov), Priklad, no. 5, 1948. mat. mekh.,= Lyapunov, A. M., “Sur une s&ie dans la theoris, line’aire du second ordre a coefficients periodiques,” Zapiski Akademii naukpo fiz.-matem, otd. seriya, 8, l3, No. 2, 1902. Erwin, N. P., “Obobshcheniye odnoy teoremy Lyapunova” (A generalization of a theorem of Lyapunov), Priklad. mat. mekh. XI1 No. 5, 1948. g r o z o v , V. V., “ 0 kommutativnykh matritsakh” (Oncommutative matrices), Uchenyye zapiski, Kiev State University, 112, book9, 1952. Erugin, N. P., “ 0 periodicheskikh resheniyakh differentsia1’nykh uravneniy” (On periodic solutions of differential equations), Priklad. mat, mekh..2Q, No. l, 1956. Vinogradov, I. M., Elements of Number Theory, New York, Dover, 1954. Zhukovskiy, N. Ye., Matem. sb.,=, 3, 1892. Burdina, V. I., “Ob ogranichennykh resheniyakh sistemy dvukh lineynykh differentsial’nykh uravneniy” (Onbounded solutions of a system of two linear differential equations), Dokludy Akad. nauk, USSR, XCIII, No. 4, 1953. Kusarova, R. S., “Ob ogranichennosti resheniy lineynogo differential’nogo uravneniya s periodicheskimi koeffitsiyentami” (On bounded solutions of a linear differential equation with periodic coefficients), Priklad. mat. mekh. l4, No. 3, 1950. Gusarov, L. A., “Ob ogranichennosti resheniy lineynogo uravneniya vtorogo poryadka” (On boundedness of solutions of

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a linear second-order equation), Doklady Akad. nauk, USSR, 17, No. 2, 1949. 66. Krzhinskiy, V. M., “Obzor rabot ob usloviyakh ustoychivosti trivial’nogo resheniya sistemy lineynykh differentsial’nykh uravneniy s periodicheskimi koeffitsiyentami” (A survey of articles on the conditions for stability of a trivial solution of a system of linear differential equations with periodic coefficients), Priklad. mat. mekh. l8, No. 4, 1954. 67. Yakubovich, V. A., “Voprosy ustoychivosti resheniya dvukh lineynykh differentsial’nykh uravnenly kanonicheskogo vida s periodicheskimi koeffitsiyentami” (Questions on the stability of a solution of two linear differential equations in canonical form with periodic coefficients), Matem. sb., 37 (79), No. 1, 1955. 68. Starzhinskiy, V. M., “Ob ustoychivosti periodicheskikh dvizheniy” (On the stability of periodic motions), (I)Zzu. Yasskogo politekhn. in-ta., Novaya seriya (New series), IV (VIII), No. 3-4, 1958 (11), V (IX),NO. 1-2, 1959. 69. Starzhinskiy, V. M., “ Zamechaniye k issledovaniyu ustoychivosti periodicheskikh dvizheniy” (A remark on the investigation of stability of periodic motions), Priklad. mat. mekh., l9, No. 1, 1955. 70. Arzhanykh, I. S., “Vsesoyuznaya mezhvuzovskaya konferentsiya po teorii i metodam rascheta nelineynykh elektricheskikh teepey” (All-union inter-university conference on the theory of methods of solving non-linear electric circuits),Sb. dokladou, No. 7 , Tashkent, 1960. 71. Krein, M., and M. Neimark, Metod simmetricheskikh i ermitovykh form v teorii otdeleniya komey algebraicheskikh urawneniy (The method of symmetric and Hermitian forms in the theory of the spacing of roots of algebraic equations), Kharkov, 1936. 72. Markushevich, A. I., The Theory of Analytic Functions, Delhi, Hindustan Publishing Corporation, 1963. 73. Erugin, N., “K teorii neyavnykh funktsiy” (On the theory of implicit functions), Doklady Akad. nauk,Belorussian SSR,No. 1, 1963. 74. Newton. Isaac. 75. Zanbek; O., Math. Annalen, 137, 167-208, 1959, 137, 477, 1959. 76. Feshchenko, S. F., and N. I. Shkil’, Asimptoticheskoye resheniye sistemy lineynykh differentsial’nykh uravneniy s malym parametrom pri proizvodnykh (Asymptotic solution of a system of linear differential equations with a small parameter in the coefficients of the derivatives), Ukr. matem. zhumal,XJ, No. 4, 1960.

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77. Feshchenko, S. F., and L. D. Nikolenko, Dopovidi, Akad. nauk,

ukr. SSR, No. 8, 1961. Ukr. matem. zhurnal,XIII, No. 3, 1961. Matematika za 40 let (Mathematics in the past forty years), Moscow, 1957. Bogolyubov, N. N. and Yu. A. Mitropol’skiy,AsymptoticMethods in the Theory ofNonlinear Oscillations, 2nd ed., Delhi, Hindustan Publishing Corporation, 1961. Bogolyubovr N. N., “0 nekotorykh statisticheskikh metodakh v matematicheskoy fizike” (On certain statistical methods in mathematical physics), Zzd. Akud. nauk, USSR, 1945. Malkin, I. G., Teoriya ustoychivosti dvizheniya (Theory of stability of motion), GITTL, 1952. Lyapunov, A. M., Zzbrannyye trudy (Collected works), Zzd. Akad. nauk, USSR, 1948. Yakubovich, V. A., “Rasprostraneniye metoda Lyapunova opredeleniya ogranichennosti resheniy uraveniy p (f) y - 0 na sluchay znakoperemennoy funktsii p ( t + m) = p (f)” (An extension of Lyapunov’s method of determining the boundedness of solutions of equations of the form y p ( t ) y - 0 to the case of a sign-alternating function p ( t + w) = p ( t ) ) , Priklad. mat. mekh., No. 6, 1954. Demidovich, B. N., “0 nekotorykh svoystvakh kharakteristicheskikh pokazateley sistemy obyknovennykh lineynykh differentsial’nykh uravneniy s periodicheskimi koeffitsiyentami, ” (Certain properties of characteristic exponents of a system of ordinary linear differential equations with periodic coefficients) No. Uchenyye zapiski, Moscow State University, Matem., 163, 1952. Makarov, S. M., “Issledovaniye kharakteristicheskogo uravneniya lineynoy sistemy dvukh uravneniy pervogo poryadka s periodicheskimi koeffitsiyentami” (An investigation of the characteristic equation of a linear system of two first-order equations with periodic coefficients), Pm’klad. mat. mekh. 25, No. 3, 1951. Goursat, E., Cours d’Analyse mathematique, 7th ed., Vol. 11, Paris, Gauthier-Villars. Erugin, N. P., Doklady Akad. nauk, Belorussian SSR, 1, No. 2, 1963. Massera, T., Boletin de la Facultad de Ingenieria, IJ, No. 1, Mago, 1950. Kurtsvetl’, Ya., and 0. Voyevoda, Chekhoslovutskiy matem. z h u m l , 3, 5(80), 1955. Erugin, N. P., Vestsi Akad. nauk Belorussian SSR, ser. fiz.tekhn. navuk, No. 1, 1962. Breus, K. A., “Wo asymptotychnyi razvyazok lyneinykh differentsyal’nykh rivnyan’ z periodychnymy koefitsientamy”

78. Feshchenko, S. F.,

79. 80. 81. 82. 83.

+

+

u,

84.

m,

85.

86. 87.

88. 89. 90. 91.

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92. 93. 94. 95. 96. 97.

269

(Asymptotic solutions of linear differential equations with periodic coefficients), Doklady Akad. nauk, Ukr. SSR, No. 5, 1955. Tikhonov, A. N., Uspekhi matem: nauk, 7, 1 (471, 1952. Erugin, N. P., Vestsi Akad. nauk, Belorussian SSSR, ser.fiz.tekhn. navuk. No. 1, 1962. Erugin, N. P., Doklady Akad. nauk Belorussian SSR, 6, No. 7, 1962. Bogdanov, Yu. S., Doklady Akad. nauk, Belorussian SSSR, 1, No. 3, 1963. Erugin, N. P., Vestsi Akad. nauk, Belorussian SSR, ser.fiz.tekhn. navuk. No. 1, 1963. Lefschetz, S., Differential Equations: Geometric Theory, 2nd ed., New York, Interscience, 1963. ADDITIONAL REFERENCES

Cesari, L. “Sulla stabilitadelle soluzioni dei sistemi di equazioni differenziali liueari a coefficienti periodici.’’ Mem. Acad. Italia (6)11(1941), 633-695. Hale, J. K. “Evaluations concerning products of exponential and periodic functions.” Riv. Mat. Univ. P a m 5(1954), 63-81, “On boundedness of the solution of linear differential systems with periodic coefficients.” Riv. Mat. Univ. P a m 5(1954), 137-167, “On a class of linear differential equations with periodic coefficients,” Illinois J. Math., 1(1957), 98-104, “Linear systems of first and second order differential equations with periodic coefficients.”Illinois J. Math.,2(1958), 586-591). Gambill, R. A. ‘‘Stability criteria for linear differential systems with periodic coefficients” Riv. Mat. Univ. P a m 5(1954), 169-181, “Criteria for parametric instability for linear differential systems with periodic coefficients.” Riu. Mat. Uniu. P a m 6(1955), 37-43, ‘‘A fundamental system of real solutions for linear differential systems with periodic coefficients.” Riu. Mat. Uniu. P a m 7 (1956),311-319. Cesari, L. and Bailey, H. R. “Boundedness of solutions of linear differential systems with periodic coefflcients.” Archive Rat. Mech. Anal., 1(1958), 246-271. Cesari, L. Asymptotic Behavior and Stability Problems in Ordinary Difleerential Equations, 1959, Springer-Verlag, BerlinGottingen- Heidelberg.

Index kitexmating series, 176 p a t o . V., 38 Analytic function, 1 Arbitrary system, conversion of, 213 Artem’yev, N. A., 23, 76 Artem’yev’s problem, 106 Arzhanykh, I. S., 222

Jacobi’s formula, 67 Jordan cell, 78 form, canonical, 111 matrix, 3 Kroneckerdelta, 66, 147

Bogdanov, Yu. S., 28 Bogolyubov, N. N., 129 Bound, 82 Bounded functions, 103

Lagrange’s formula, 2 limiting value of, 82 Lagrange’s polynomial, 4, 60 Lappo-Danilevskiy, I. A., 3, 55 coefficients, table of, 254 recursion formulas, 144 theorem, 63 Laurent series, 50 Limitmatrix, 8 Linear equations, transformation of, 221 Linear systems, construction of solutions of, 44 Lyapunov’s equation, 169 Lyapunov’s theorem, 47, 103

Canonical systems, 101 aebotarev, N., 38, 244 Chernin, K E., 254 hmidovich, 6. P., 132 Determination of coefficients by Stokalo‘s method, 116 in series expansion, 75 Differential transformations, 52 hnskaya, L I., 52

Mathieu’s equation, 161 Matrixcalculus, 1 Metacanonical integral matrix, 66, 147 Morozov, V. V., 38

Entirefunction, 1 Exponential matrix, expansion of, 68 Exponential transformation, 51 Fedorw, F. I., 39 Fedorova, 0. A, 254 Functions of several matrices, 33-36 of a single matrix, 1-22 Funemental integral maPlx, 137

Negative characteristic numbers, 72 Nonsymmerric functions, 2 Oscillatory functions, 103 Ostrogradskiy- Jacobi theorem, 111 Periodic matrices, 64, 107 Periodic solutions, finding of, 175 Poincad-Lappo-Danilevskiy problem, solution of, 49 Prime elementary divisors, 90 Principal value, 20

Gantmakher, F. R , 2, 53, 117 Gel’man, A. E, 128 Golokvoschus, P. B., 232 Gremyachenskiy, A. P., 169 Holomorphic functions, 23, 101 HurwitZ determinants, 125 inequalities, 107

Realfuncdons, 60 Reducible systems, theory of, 109 Regular singular point, 51 Riemann plane, 7 sheet, 9

Integral transformation, 50 Invariants of a matrix, 2 2 70

INDEX

Salakhova, 1. M., 41 Scalar coefficients, 1 Shtelik, V. G., 129 Shtokalo, I. Z., 23 Shtokalo's method, 112 Shtokalo's theorem, 124 Smirnov, V. I., 176 Starzhinskiy, V. M, 210 Starzhinskiy's transformation, 210 Taylor series, 1

Unit circle, 222 Varenrsova, I. M., 254 Vector-valued solution, 130 Vinogradov, 1. M., 132 Weierstrass theorem, 243 Zhukovskiy, N. E., 202

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Linear Differential Equations With Periodic Coefficients

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Linear Differential Equations With Periodic Coefficients, volume 1

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