LINEAR
SYSTEMS
OF
ORDINARY
DIFFERENTIAL
EQUATIONS N.
A.
Izobov
UDC 517.941.92
The p r e s e n t s u r v e y deals chiefly with the Lyapunov t h e o r y of c h a r a c t e r i s t i c exponents of the g e n e r a l f o r m of linear differential s y s t e m s , x=A(t)x
with bounded p i e c e w i s e continuous coefficients and p a r t i c u l a r l y with its m o s t r e c e n t advances. A sufficiently c o m p l e t e bibliography on such other independent b r a n c h e s of the t h e o r y of linear s y s t e m s as " L i n e a r S y s t e m s with P e r i o d i c Coefficients" (already s u r v e y e d in p a r t [469] in the s e r i e s G e n e r a l Mechanics) and the "Analytical T h e o r y of L i n e a r S y s t e m s " can be found in Sec. 15. The bibliography p r e s e n t e d in our s u r v e y b a s i c a l l y e n c o m p a s s e s all works on l i n e a r s y s t e m s a b s t r a c t e d in R e f e r a t i v n y i Zhurnal (Matematika). 1.
DEFINITION
AND
BASIC
PROPERTIES
OF
DIFFERENT
EXPONENTS
The Lyapunov c h a r a c t e r i s t i c exponent ([349], p. 27) of the solution x (t) # 0 of a linear s y s t e m is the number
-- ~
11 )It
A linear s y s t e m cannot have m o r e than n nonzero solutions with p a i r w i s e different exponents (Lyapunov [349], p. 34). A fundamental s y s t e m of solutions of a linear s y s t e m is called n o r m a l (Lyapunov [349], p. 34) if the sum of the exponents of its solutions is a m i n i m u m in the set of all fundamental s y s t e m s and is said to be b i n o r m a l (R. l~. Vinograd [126]) if in addition X-1 (t) is n o r m a l r e l a t i v e to the adjoint s y s t e m ~ = - A T ( t ) y . The exponents ~I - ~2 - 9 - ~n of a n o r m a l o r d e r e d (i.e., a r r a n g e d in n o n d e c r e a s i n g o r d e r of the exponents of the solution) s y s t e m will be called the exponents of the l i n e a r s y s t e m while the e x t r e m e exponents, Xi and 7~n, will be called its lowest and highest exponents. The solutions a r e distributed with r e s p e c t to t h e i r exponents in the following way (A. M. Lyapunov [349], p.34, and Yu. S. Bogdanov [48]): the e n t i r e s p a c e of initial values is divided into m l i n e a r s p a c e s L k embedded in each other, such that e v e r y nonzero solution outside it h a s an exponent l e s s than ~k (~'l < 9 . . < ~'m). Stability of the Exponents: The exponents of a linear s y s t e m a r e said to be stable if for any e > 0, a 6 > 0 can be found such that e v e r y exponent Xy of any p e r t u r b e d s y s t e m
~=A(Oy+Q(t)y, IIO(t)ll--<~, s a t i s f i e s the inequality
t>0,
rain Ikr-- kil ~< .~. i
T h i s concept a p p e a r e d in P e r r o n [808] w h e r e it was e s t a b l i s h e d for the f i r s t t i m e that exponents can be unstable. T r a n s l a t e d f r o m Itogi Nauki i Tekhniki ( M a t e m a t i c h e s k i i Analiz), Vol. 12, P a r t 1, pp. 71-146, 1974,
9 Plenum P~+blishing Corporation, 22 7 West 17th Street, New York, N. ~: 10011. No part o f this publication may be reproduced, stored ht a retrieval system, or transmitted, in an), form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of tile publisher. A copy o f this article is aPailable from the publisher for $15.00.
46
The lower exponent ( P e r r o n [808]) is given by
~_[xl ~ li__~m+ in Ilx (011. Unlike the c h a r a c t e r i s t i c exponents of a linear s y s t e m , t h e r e m a y e x i s t m o r e than n different lower e x p o nents ( P e r r o n [808]). M o r e o v e r , a s y s t e m has been c o n s t r u c t e d whose set of lower exponents was the segment [0, 1] (N. A. Izobov [251]). In addition, the lower exponents a l s o obey a p r o p e r t y s i m i l a r to that obeyed by the c h a r a c t e r i s t i c exponents (N. A. Izobov [248]). T h i s p r o p e r t y m a y be stated a s follows: A l m o s t nil solutions beginning in a k - d i m e n s i o n a l (k = 2, . . . . n) s u b s p a c e of the s p a c e of initial data have a 2 o w e r exponent which is the g r e a t e s t exponent for the given solutions. The exact exponent o c c u r s in those solutions in which the c h a r a c t e r i s t i c and lower exponents coincide. t
The g r e a t e s t lower bound of the c o n s t a n t s R satisfying
llX(t, s)fI..
all s --- t of the n o r m
$
X(t, s) of the Cauchy m a t r i x is called the singular exponent ~20 (K. P. P e r s i d s k i i [434]) or index (P. Bohl [651]; cf. a l s o Yu. L. Daletskii and M. G. K r e i n [181], p. 171). The concept of a s i n g u l a r exponent is due to K. P. P e r s i d s k i i . A. D. Myshkis and I. M. Rabinovich [413] have noted in t h e i r r e c e n t study of the work of P. Bohl that Bohl [651], which w a s not well-known at the t i m e , a l s o contains this concept. Likewise the t
g r e a t e s t lower bound of the upper integral m e a n s
/~
~
1 ~ R (~) dz of bounded m e a s u r a b l e functions
R (t) in this l i m i t will be called the c e n t r a l exponent fit. ]~. Vinograd [123-124]). A p r o b a b l e exponent (V. M. Millionshchikov [381, 383]) of a l i n e a r s y s t e m is a n u m b e r ~ if t h e r e e x i s t s an i n v a r i a n t m e a s u r e (V. V. N e m y t s k i i and V. V. Stepanov [416], p. 514) on a dynamic s y s t e m DA of d i s p l a c e m e n t s (bounded and u n i f o r m l y continuous on a line) of the m a t r i x A (t) defined on a s p a c e R A of m a t r i c e s of the f o r m ([416], pp. 533-535) A(t)~limA(t~q-t) ( s e c t i o n - u n i f o r m limit), such that for n e a r l y e v e r y ~ (t) E R A the s y s t e m ~ = ~ (t)x p o s s e s s e s X of its exponents. The set of p r o b a b l e exponents of a l i n e a r s y s t e m will be called its probable s p e c t r u m , denoted Ap (A). Auxiliary exponents (V. M. Millionshchikov [398]) of a l i n e a r s y s t e m a r e those n u m b e r s ..- : - -
|
,~.=lim hm --~ ~]
lnd~,[X((i+l)T, iT)], k = l ,
n,
w h e r e d i [X] >- . . . ~ d n [X] a r e positive s q u a r e r o o t s of the eigenvalues of the m a t r i x X* X, and the set of a u x i l i a r y exponents will be called its a u x i l i a r y s p e c t r u m (v i coincides with the c e n t r a l exponent ~2). The Lyapunov n o r m X (Yu. S. Bogdanbv [ 51]) is defined on an a r b i t r a r y linear s e t A, h a s its own v a l ues X(a) of an e l e m e n t of a given o r d e r e d s e t A , and is such that: l) X(ca) - A(a), a E A and c is a r e a l n u m b e r , and 2) k (a + a ' ) ~ m a x {~.(a), ~.(a')}, w h e r e a, a v • A. The c h a r a c t e r i s t i c d e g r e e (B. P. Demidovs v Ix], such that * [xl ~- lira
[187]) of a solution x (t) with exponent X is that n u m b e r
* in Ilx (t)e-~qE.
We r e f e r the r e a d e r also to Refs. 57, 103, 122, 125, 219, 247, 389, 399, 400, 436, 456, 465, 690, 791, and 827. 2.
REDUCIBLE
AND ALMOST
REDUCIBLE
SYSTEMS
The Lyapunov t r a n s f o r m a t i o n s ([349], p. 42) keep the: a s y m p t o t i c c h a r a c t e r i s t i c s of s y s t e m s invariant and a r e therefDre used for c l a s s i f y i n g these c h a r a c t e r i s t i c s . The s i m p l e s t and m o s t i m p o r t a n t c l a s s of s y s t e m s singled out by Lyapunov include the following. 1_~, Reducible S y s t e m s ([349], p. 43). A l i n e a r s y s t e m is called r e d u c i b l e if t h e r e exists a Lyapunov t r a n s f o r m a t i o n that c o n v e r t s it into a s y s t e m with constant coefficients.
47
Lyapunov ([349], p. 195) indicated one i m p o r t a n t c l a s s of reducible s y s t e m s , n a m e l y s y s t e m s with periodic coefficient m a t r i c e s . F u r t h e r investigations of reducible s y s t e m s a r e a s s o c i a t e d with the n a m e N. P. Erugin and deal with the g e n e r a l theory he c o n s t r u c t e d of such s y s t e m s in the monograph "Reducible S y s t e m s . ~ In this monograph N. P. Erugin p r o p o s e d two methods for investigating r e d u c i b i l i t y p r o b l e m s . The f i r s t method is that of s u c c e s s i v e a p p r o x i m a t i o n s (fromwhich, incidentally, follow well-known a s y m p totic decompositions and which p e r m i t e s t i m a t e s to be obtained of the r e m a i n d e r t e r m in a s y m p t o t i c f o r m u las for the c a s e 6f an i r r e g u l a r singular.point, something that had not p r e v i o u s l y been done) (cf. V. V. Khoroshilov [556], L. I. Donskaya [192, 193], and I. N. Zboichik [232]). The second method c o n s i s t s in r e duction to s y s t e m s of a special type suitable for investigations (cf. A. E. G e l ' m a n [143; 144], L. Ya. A d r i a n o va [3], and L. V. T r i g u b o v i c h [525]). Let us d i s c u s s c e r t a i n c o n c r e t e r e s u l t s of N. P. E r u g i n ' s monograph. Reduciblity C r i t e r i o n ([199], p. 9). A l i n e a r s y s t e m is reducible if and only i f its fundamental s y s t e m of solutions can be r e p r e s e n t e d in the f o r m X(t) = S(t) e x p A t with Lyapunov m a t r i x S (t). R e a l n e s s of the T r a n s f o r m a t i o n ([199], p. 17; [213], p. 119). A r e d u c i b l e s y s t e m of a r e a l Lyapunov t r a n s f o r m a t i o n can be c o n v e r t e d into a s y s t e m with constant r e a l canonical coefficients m a t r i c e s . N e c e s s a r y Reducibility Condition ([199], p. 17). The integral of the t r a c e of the coefficient m a t r i x of t
a r e d u c i b l e s y s t e m can be r e p r e s e n t e d in the f o r m f SPA (Q d~=at-b~
(t)
with bounded function ~0(t).
0
Reducibility of an A n t i s y m m e t r i c System ([199], p. 20). A l i n e a r s y s t e m with an anti,symmetric m a t r i x (ajk = - akj for all j and k) is reducible, I. A. L a p p o - D a n i l e v s k i i [3051 C a s e and Reducibility ([1991, p. 24). If the m a t r i x A(t) c o m m u t e s with its integral ~ A (~)dr
for e v e r y s --- t and if we have the r e p r e s e n t a t i o n
s
fA (~)de =A + B(t)
with bounded
0
m a t r i x B (t) the linear s y s t e m can be r e d u c e d to the s y s t e m ~r = Ay. T r i a n g u l a r S y s t e m s ([199], p. 36). A t w o - d i m e n s i o n a l upper t r i a n g u l a r linear s y s t e m is r e d u c i b l e to a diagonal f o r m if and only if its diagonal coefficients have the r e p r e s e n t a t i o n aii (t) = a i + 9 i (t) with bounded t
integrals
I ~'(:)d:
t
and, if at = a2 = a, t h e integral
](t)=fax~(~)ex p I(y2--*t)d~d*
0
0
sional upper t r i a n g u l a r linear s y s t e m is r e d u c i b l e to the s y s t e m
(~
is bounded. A t w o - d i m e n -
0
y = Oa
y. if the function Cvl (t) + C2t,
w h e r e C 2 ~ 0, is bounded. In p a r t i c u l a r ([199], pp. 28-32) the reducibility p r o b l e m for the t w o - d i m e n s i o n a l s y s t e m (1) with upper t r i a n g u l a r m a t r i x
A(O= ~ A~t-.,.
(*)
nt~O
has been exhaustively investigated. The nonanalytic c a s e for a m a t r i x A (t) as well a s the n - d i m e n s i o n a l c a s e have a l s o b e e n examined. R i c c a t i Equation and Reducibility. N. P. Erugin ([199], p. 68) e s t a b l i s h e d a r e l a t i o n between r e d u c i b i l ity of a t w o - d i m e n s i o n a l s y s t e m to t r i a n g u l a r f o r m and the existence of a bounded solution or a solution that tends to +oo a s t --~ + ~ for the equation u' = - al2 u2 + ( a 2 2 - - all) U + a 2 1 . New reducibility t e s t s w e r e obtained using it, for example, a reducibility c r i t e r i o n for a s y s t e m with a nontriangular m a t r i x of the f o r m (*) as well as ([199], p. 82) a linear s y s t e m i n w h i c h aj2, a~2--aH>~>O, a~t>O, akk~ak+~k(t), at-~a~, t >0,
48
is reducible to the s y s t e m ~ --- diag[a I, a2]y if the i n t e g r a l s
t
t
l?k(~.)d~, and
~a~x(Qd~ a r e
0
-0
bounded.
R e m a r k . In r e d u c i n g a linear s y s t e m to t r i a n g u l a r f o r m it is n e c e s s a r y to find a given bounded solution. But s o m e t i m e s such a r e a l bounded solution does not exist while a complex bounded solution does exist, which also leads to a solution of the p r o b l e m (see e x a m p l e [199], p. 80). P e r t u r b a t i o n s that P r e s e r v e the Reducibility P r o p e r t y . N. P. Erugin ([199], p. 63) set forth the p r o b l e m a s t o how s m a l l Q (t) m u s t be such that the p e r t u r b e d s y s t e m ~r = [A (t) + Q (t)]y r e m a i n s r e d u c i b l e to the s a m e s y s t e m a s initially. Its solution was given in the f o r m of u n i m p r o v a b l e (differential) e s t i m a t e s or an a l g o r i t h m for obtaining t h e m by m e a n s of the f i r s t method. N. P. E r u g i n ' s m o n o g r a p h [213] in which his r e s u l t s w e r e set forth and studies of A. M. Lyapunov, I. A. L a p p o - D a n i l e v s k i i , N. N. Bogulyubov, and L Z. Shtokalo on linear s y s t e m s w e r e brought t o g e t h e r , also c o n s t i t u t e s an i m p o r t a n t contribution to the l i t e r a t u r e on t h e s e p r o b l e m s . T h i s book h a s e x e r t e d a significant influence on the f u r t h e r development of different lines of r e s e a r c h into l i n e a r s y s t e m s . We r e f e r the r e a d e r also to Secs. 6 and 12 and to Refs. 24, 27, 54, 103, 137, !41, 207, 203, 209, 211, 218-220, 228, 271, 319, 332, 333, 361,362. 370, 373, 442, 463, 600, 605, 757, and 826. 2 ~ A l m o s t Reducible S y s t e m s (B. F. Bylov [85, 88] and Lillo [766]). A linear s y s t e m is said to be almosY-reducible to the s y s t e m
:~=B (t) y.
(2.1)
if for any 6 > 0 t h e r e e x i s t s a Lyapunov t r a n s f o r m a t i o n y = Ss(t)x that t r a n s f o r m s it into the s y s t e m ~ = [B (t) + Q6(t)]z if the inequality IIQ6 (0tl <- 6 holds, and is said to be a l m o s t r e d u c i b l e if the m a t r i x B (t) t u r n s out to be constant. The concept of a l m o s t r e d u c i b i l i t y was introduced by B. F. Bylov in 1954 inhis C a n d i d a t e ' s D i s s e r t a t i o n [85] (published in 1962 [88]) in which n e c e s s a r y and sufficient a l m o s t reducibility conditions e x p r e s s e d in t e r m s of e s t i m a t e s of the G r a m i a n w e r e a l s o given. The f i r s t journal publication on a l m o s t reducible s y s t e m s , called a p p r o x i m a t i v e l y s i m i l a r , was that of Lillo [776], which also derived the f i r s t i m p o r t a n t r e s u l t s on the a l m o s t reducibility of linear s y s t e m s with a l m o s t periodic coefficients ([766], Sec. 6). Concept of A l m o s t Reducibility Is Reflexive and T r a n s i t i v e (B. F. B$1.ov [85] and Lillo [766]). Bylov [99] also proved that a l m o s t r e d u c i b i l i t y is s y m m e t r i c in a c l a s s of s y s t e m s that a r e a l m o s t r e d u c i b l e to the P e r r o n - s e p a r a b l e diagonal s y s t e m s [811] (cf. a l s o Sec. 7). B. M. Millionshchikov [399] e s t a b l i s h e d a f t e r o v e r c o m i n g c e r t a i n t h e o r e t i c a l difficulties the n o n s y m m e t r i c i t y of the a l m o s t r e d u c i b i l i t y r e l a t i o n s . T h e r e e x i s t s a m - d i m e n s i o n a l linear s y s t e m that is not a l m o s t r e d u c i b l e to the s y s t e m ~r = Oy, but such that the l a t t e r s y s t e m is a l m o s t r e d u c i b l e to a linear s y s t e m . This r e s u l t is b a s e d on the MiUionshchikov t h e o r e m [394] on the instability of singular exponents (for d i s cussion, see Sec. 9). S. A. Grishin [167] h a s since proved Millionshchikov's h y p o t h e s i s t o the effect that t h e r e e x i s t s a s y s t e m (2.1) for e v e r y linear s y s t e m with integral s e p a r a b i l i t y of solutions (see Sec. 7) that lacks t h e s e p r o p e r t i e s and to which the linear s y s t e m is a l m o s t r e d u c i b l e but that (2.1) is not a l m o s t r e d u c i b l e to a linear system. In conclusion let us note two r e s u l t s of B. F. Bylov [89, 95]. An a l m o s t r e d u c i b i l i t y c r i t e r i o n for a l i n e a r s y s t e m with different exponents is the e x i s t e n c e of the bound -~-~--hm x~)l/ for the solutions ~t--iT in ~x!~]. t I of a n o r m a l s y s t e m of uniform exponents (in the c a s e of a s y s t e m of uniform exponents (in the c a s e of a s y s t e m with identical exponents the e x i s t e n c e of uniform exponents in all solutions is a l r e a d y insufficient for its a l m o s t r e d u c i b i l i t y [90]) and an a l m o s t r e d u c i b i l i t y c r i t e r i o n for a linear s y s t e m to the P e r r o n s e p a r a b l e diagonal s y s t e m [811]. We r e f e r the r e a d e r a l s o to Secs. 4, 6, 7, and 12 and to Refs. 103,399, 400, and 791. 3.
REGULAR
SYSTEMS
R e g u l a r s y s t e m s a r e the next type in the Lyapunov c l a s s i f i c a t i o n of types of s y s t e m s . They include r e d u c i b l e and a l m o s t r e d u c i b l e s y s t e m s and play an i m p o r t a n t r o l e in linear a p p r o x i m a t i o n stability theory (cf. Sec. 13). The s y s t e m {2.1} is said to be r e g u l a r (A. M. Lyapunov [349], p. 38) if
49
n
t
and is i r r e g u l a r in the opposite case. The following a r e among the chief r e g u l a r i t y c r i t e r i a for a linear s y s t e m . Lyapunov C r i t e r i o n ([349], p. 39). A triangular s y s t e m is r e g u l a r ff and only if t h e r e exists explicit integral means for its diagonal coefficients. P e r r o n C r i t e r i o n [808]. A linear s y s t e m is r e g u l a r if and only if the exponents of the adjoint s y s t e m (cf. See. 1) a r e equal in absolute value and opposite in sign to the exponents of the initial system. Other c r i t e r i a will also be discussed below. T h e s e include: The C r i t e r i o n of V. P. Basov [26]~ D. M. Grobman [168], and Yu. S. Bo~danov [49]. A linear s y s t e m is r e g u l a r if and only if it is g e n e r a l i z e d - r e d u c i b l e , i . e . , by means of a generalized Lyapunov t r a n s f o r m a t i o n y = S (t)x, where S (t) is piecewise differentiable and (oJ[S (t)] = 0~[S-t(t)] = 0) is t r a n s f o r m e d into the s y s t e m y = diag [X1 . . . . Xn]y.
Vinograd DIS] C r i t e r i o n . A linear s y s t e m is r e g u l a r if and only if t h e r e exist explicit exponents for the solutions of a given n o r m a l s y s t e m X = [xi (t) . . . . . x n (t)] and explicit zero exponents for the lines of the angles between the solution x k (t) and the linear space of the ( k - i ) p r e c e d i n g solutions x 1. . . . . Xk_ 1. Regularity t e s t s for a linear s y s t e m were constructed by A. S. Galiullin [140]. The instability of the exponents of r e g u l a r s y s t e m s , both of the lowest and the highest exponents, was d i s c o v e r e d by R. I~. Vinograd [116, 117, 120]; they a r e stable for reducible and a l m o s t reducible s y s t e m s . Approximating Sequences. K. P. P e r s i d s k i i [436] has put f o r w a r d the hypothesis that the exponents Ai of the r e g u l a r s y s t e m (1) can be r e p r e s e n t e d by the equalities ~'i= lira ~z(:) (such sequences {~-k} a r e called ~---~oo
approximating) in t e r m s Of the exponents Xi (~') of the system y = A (t)y with m a t r i x A v (t) = A (t) for t E [0, ~'), and even by a periodically continued equality. R. l~. Vinograd [119] proved that this hypothesis, which is true for the c a s e of the t r i a n g u l a r m a t r i x A (t), is false under such a general formulation (for c e r t a i n sequences {Tk)). Yu. S. Bogdanov [55] has since given an effectively p e r f o r m a b l e t r a n s f o r m a t i o n (turning) of a tWo-dimensional s y s t e m into a s y s t e m , such that P e r s i d s k i i ' s a s s e r t i o n h o l d s , and has also p r e s e n t e d an example of a s y s t e m for which no sequence which tends to +~ is approximating. Generalizations of Regular Systems. The concept of a weskly i r r e g u l a r system, namely s y s t e m s in which the r e a u i r e m e n t that explicit exponents exist for solutions of a n o r m a l s y s t e m is r e p l a c e d by the weak r e q u i r e m e n t that the function ( l / t ) In [Ixi (t) I[ s lowly v a r i e s and that the second r e q u i r e m e n t is e n t i r e l y p r e s e r v e d for r e g u l a r s y s t e m s (cf. B. l~. Vinograd c r i t e r i o n ) , has since been introduced (N. A. Izobov [250]). B. F. Bylov [102] singled out a c l a s s of linear s y s t e m s , the asymptotic of whose solutions is d e t e r m i n e d t
relative to functions 0f-the f o r m exp I p (~) d~ . In this c a s e well-known Lyapunov, P e r r o n , and Vinograd IJ
r e g u l a r i t y tests a r e generalized using a given set of functions Pi' P~'" " "' Pq in place of exponents of the s y s t e m A l < A s < . . . < Aq. B. P. Demidovich [187] distinguished a class of r e g u l a r s y s t e m s , called completely r e g u l a r s y s t e m s (cf. Sec. 13). We r e f e r the r e a d e r also to Secs. 4-6, 12, and 13 and to Refs. 50, 52, 88, 115, 121, 189, 219, 240, 302, 361, 362, 442, 463, and 627. 4.
ABSOLUTELY
REGULAR
SYSTEMS
V. M. Millionshchikov introduced the following c l a s s of linear s y s t e m s . Absolutely Regular Systems [386]. A linear s y s t e m is said to be absolutely r e g u l a r if t h e r e exists a fundamental s y s t e m of solutions xt (t) . . . . . Sn(t) p o s s e s s i n g the p r o p e r t i e s !
1) lim--' In ll x~ (t) t[= x, i ~ l . . . . . n; It I-'~r
50
t
2) for e v e r y i = 1 , . . . ,
n and for e v e r y ~ > 0 a T can be found, such that the s e t of all h, f o r which we
h a v e for at l e a s t one solution x(t) such that lira + lnllx(t)ll=k ~, for s o m e v , 1TI
T,
Ft I--,c~
h a s r e l a t i v e m e a s u r e on the line l e s s than e. The concept of absolutely r e g u l a r s y s t e m s , which is i n t e r m e d i a t e between a l m o s t r e d u c i b l e and r e g u l a r s y s t e m s , h a s a deep meaning. A l m o s t e v e r y l i n e a r s y s t e m is a b s o l u t e l y r e g u l a r . That i s , w e have the following a s s e r t i o n . * Millionshchikov T h e o r e m [393]. Suppose we a r e given a l i n e a r s y s t e m with m a t r i x A (t) u n i f o r m l y continuous on a line. A l m o s t e v e r y ~(t) in a dynamic s y s t e m DA of d i s p l a c e m e n t s of the m a t r i x A (t) (in t h e s e n s e of any i n v a r i a n t m e a s u r e on D A) is such that the s y s t e m ~ = ~(t} x is a b s o l u t e l y r e g u l a r . Suppose X E An(A). Then a l m o s t e v e r y m a t r i x ~(t) f r o m the s p a c e R A (in the s e n s e of a given inv a r i a n t m e a s u r e on~) A) is such that the s y s t e m ~ = ~ ( t ) x is absolutely r e g u l a r , X being one of its c h a r a c t e r i s t i c exponents. The introduction of absolutely r e g u l a r s y s t e m s w a s p r e c e d e d by the definition of a s o m e w h a t wider c l a s s , s o - c a l l e d s t a t i s t i c a l l y r e g u l a r s y s t e m s [381]. The e x c e l l e n t p r o p e r t i e s of t h e s e s y s t e m s is shown to be the c a s e by a second a s s e r t i o n . Millionshchikov T h e o r e m [381]. The highest exponent of an absolutely r e g u l a r s y s t e m is u p p e r - s t a b l e while the lowest exponent is l o w e r h a l f - s t a b l e . /
The l a t t e r r e s u l t also e s t a b l i s h e s that the a r g u m e n t s of R. E. Vinggrad [116, 120] in his construction of a r e g u l a r s y s t e m with unstable exponents, such that the highest exponent is discontinuous above and the lowest exponent discontinuous below, is inapplicable to absolutely r e g u l a r s y s t e m s . M o r e o v e r the exponents" of an absolutely r e g u l a r s y s t e m a r e in g e n e r a l unstable. V. M. Millionshchikov [379] c o n s t r u c t e d an exa m p l e of an a b s o l u t e l y r e g u l a r s y s t e m for which s y s t e m s as close to it as d e s i r e d and which p o s s e s s exponents equal to the half s u m of the exponents of the initial s y s t e m could be found. In p r o v i n g this r e s u l t a t h e o r e t i c a l l y new method for the t h e o r y of linear s y s t e m s , called at the suggestion of R. E. Vinograd the Millionshchikov turning method, was used for the f i r s t t i m e . It.should be noted that this n a m e though a l r e a d y given is somewhat provisional. It is b a s e d on the type of perturbation, w h e r e a s the t r u e e s s e n c e of the method c o n s i s t s in a s c h e m e by which the p e r t u r b e d solution w a n d e r s c o r r e s p o n d i n g l y f r o m the s e l e c t e d solutions of the u n p e r t u r b e d s y s t e m . F o r e x a m p l e , a d i s c u s s i o n is available [395] in which turnings w e r e lacking, y e t which n e v e r t h e l e s s p r o c e e d e d on the b a s i s of a method of turning (cf. Secs. 6-9 and 13 for its use). Statistically A l m o s t Reducible S y s t e m s (V. M. Millionshchikov [38t]). A l i n e a r s y s t e m with exponent Xi is said to be s t a t i s t i c a l l y a l m o s t r e d u c i b l e if for e v e r y 6 > 0 there, e x i s t s a Lyapunov t r a n s f o r m a t i o n y = S6(t) x that r e d u c e s it to the f o r m = diag [xi . . . . . x,,] y ~- B6 (t) y ~u Q6 (t) y, w h e r e Q5 (t) is a t r i a n g u l a r m a t r i x with z e r o diagonal, IlQ~ (t)I[ <- 6, and the diagonal m a t r i x B~(t) is such that: 1) llBs(t)ll ~ C, w h e r e C is independent of t and 6; S
2) .~m ~ ' Ill 8~(~)H d~ ~<~. t..~,,.oo
0
V. M. Millionshchikov [381] p r o v e d that e v e r y absolutely r e g u l a r s y s t e m is s t a t i s t i c a l l y a l m o s t r e d u c i b l e (cf. a l s o Secs. 6, 7, and 11). * F o r notation see Sec. 1.
51
5.
IRREGULARITY
COEFFICIENTS
Different i r r e g u l a r i t y coefficients c h a r a c t e r i z e to v a r y i n g degrees the r e s p o n s e of a s y s t e m to p e r turbations that a r e either linear (cf. below) or a r e on a higher degree of s m a l l n e s s (cf. Sec. 13). The Lyapunov i r r e g u l a r i t y coefficient ([349], p. 51) of a linear s y s t e m is the variable e L defined in Sec. 3. The P e r r o n i r r e g u l a r i t y coefficieI{t [808] of a linear s y s t e m is the e x p r e s s i o n ~p-------max0.t-}-~}, w h e r e k 1 -< . . . <- Xn and/~l >- 9 9 9 > ~n a r e the exponents of the linear s y s t e m and of its adjoint s y s t e m y = - A T (t)y, r e s p e c t i v e l y . The D. M. Grobman i r r e g u l a r i t y coefficient [168] of a linear s y s t e m is the e x p r e s s i o n % = ifif max {~z+ ~}, u
X
i
where ki and 5 i a r e the exponents r e s p e c t i v e l y of the i-th column of the fundamental m ~ r i x X (t) and the i-th row of its inverse m a t r i x X -f (t). The g r e a t e s t lower bound exists for the m a t r i x X(t), whose columns f o r m a n o r m a l s y s t e m of solutions [168]. The asymptotic number of V. M. Millionshchikov [376] is the g r e a t e s t lower bound a M of constants e > 0 such that, for a given fundamental s y s t e m of solutions xi it), . . . . Xn(t) of a linear system, given C and t o and for all t >- t 0, we have
i e _ ~ ' , ~ (-~ )Iix('~ll ~ a ~ ~ l ~ < C ~IIx(t3[l ,
i = 1 . . . . . n,
for all solutions x (t) of the linear s y s t e m (the lower limit of integration is t o if the r i g h t side of the inequality tends to +~ as t--* § ~ or is +~ in the opposite case). Here we use the notation vi(t) -=-Ic o s e c cq it)I, where ~i it) is the angle between x i (t) and the hyperplane spanned by the v e c t o r s x k it), k ~ i. T h e s e i r r e g u l a r i t y coefficients a r e nonnegative. equalities (B. F. Bylov e t a l . [103], p. 284)
The f i r s t three of them satisfy the composite in-
0 ~
.
T h e r e f o r e a linear s y s t e m is r e g u l a r if and only if at least one of the i r r e g u l a r i t y coefficients eL, a p , or a G is z e r o . On the other hand the a s y m p t o t i c number a M may vanish also for i r r e g u l a r s y s t e m s . It is zero for e v e r y s y s t e m with integral separability. The exponents of the s y s t e m ~ = [A (t) + Q (t)]y with p e r t u r b a t i o n s of Q (t) ~-oo
SoilQ(~)l[e**a~ <
oo,
(5.1)
coincide with the exponents of the s y s t e m ~ = A (t) x for a > a G (D. M. Grobman [168]) and so also for a > e L (Yu. S. Bogdanov [49, 52]). Yu. S. Bogdanov [49, 52] obtained computational f o r m u l a s for calculating the exponents of a linear s y s t e m based on his own r e s u l t and on e s t i m a t e s of e L using coefficients. The highest exponents coincide in the initial and perturbed s y s t e m s when a > e p for the P e r r o n i r r e g u l a r i t y coefficient in the two-dimensional c a s e iN. A. Izobov [249]). A c o r r e s p o n d e n c e has been established for the c a s e of perturbations (5.1) with a > a M between the solutions of the initial and perturbed s y s t e m s , such that we have the r e p r e s e n t a t i o n iV. M. Millionshchikov [376])
.y(O=x(O+~(t), [l~(t)ll~to.
(5.2)
Unlike the i r r e g u l a r i t y coefficients, Millionshchikov's asymptotic number p r e c i s e l y c h a r a c t e r i z e s the r e sponse of a s y s t e m to perturbations in the sense that for e v e r y solution x it) of a linear s y s t e m , a m a t r i x Q (t) can be found that satisfies (5.1) with e < a M such that t h e r e ,exist n o solutions that can be r e p r e s e n t e d in the f o r m (5.2) in the perturbed s y s t e m .
52
6.
LINEAR
SYSTEMS
WITH
ALMOST
PERIODIC
COEFFICIENTS
In this section we shall provisionally divide works on linear (homogeneous) s y s t e m s with quasiperiodic and uniformly a l m o s t periodic coefficients into two trends, namely the construction of coefficient reducibility t e s t s and the investigation of a l m o s t periodic s y s t e m s of general f o r m . 1". The r e s u l t s of N. P. Erugin and I. Z. Shtokalo belong among the f i r s t r e s u l t s on uniformly almost periodic s y s t e m s . N. P. Erugin has derived the f i r s t c o n c r e t e r e s u l t s on q u a s i - p e r i o d i c s y s t e m s , in addition to general methods [199] for the study of reducibility (reduction to s y s t e m s of special f o r m , p r e s e r v a t i o n of the r e d u c ibility p r o p e r t y for variable coefficients, and others) that have been effectively applied to the t h e o r y of q u a s i - p e r i o d i c s y s t e m s by many authors (A. E. GePman, L. Ya. Adrianova, and I. N. Blinov). We should note above all his derivation of a reducibility test f o r a t r i a n g u l a r quasi-periodic s y s t e m ([199], p. 40), which established for the f i r s t time a t h e o r e t i c a l r e l a t i o n between reducibility and the a r i t h m e t i c nature of f r e q u e n c i e s , which many p a p e r s have since dealt with. M o r e o v e r , N. P. Erugin developed [206, 207] the I. Z. Shtokalo method (cf. below) and also set forth the existence problem for i r r e g u l a r uniformly a l m o s t periodic and q u a s i - p e r i o d i c s y s t e m s (cf. Sec. 3 ~ which h a s stimulated the development of the t h e o r y of quasi-periodic systems. I. 7. Shtokalo [597] was the f i r s t to give for the quasi-periodic s y s t e m ~ = [A + eQ (t)] x with m a t r i x A admitting of z e r o or purely imaginary c h a r a c t e r i s t i c n u m b e r s an effective method for constructing a s y m p totic stability and instability t e s t s for small e > 0. A. E. GelTman [143, 144] using the Erugin method for reducing a s y s t e m to a special f o r m (cL Sec. 2) and his own m a j o r i z i n g s e r i e s for two-dimensional quasi-periodic s y s t e m s and subsequently L. Ya. Adrianova [3] for n-dimensional q u a s i - p e r i o d i c s y s t e m s , constructed reducibility t e s t s by noting their relation to the a r i t h m e t i c nature of f r e q u e n c i e s and instability r e l a t i v e to the l a t t e r . We should also note that in these works the set of reducible s y s t e m s has a nonempty open hull in the space of all q u a s i - p e r i o d i c s y s t e m s = A (r x, where r = {~t . . . . . r with analytic m a t r i c e s A (~') 2It-periodic with r e s p e c t to ~" for a l m o s t e v e r y frequency basis. I. N. Blinov [40] subsequently provided a negative answer to a problem posed by V. I . A r n o P d [19] on the reducibility of a q u a s i - p e r i o d i c s y s t e m with an analytic m a t r i x and strongly incommensurable frequencies. We may also note the r e s u l t of Blinov [39], stating that the fundamental s y s t e m of solutions of a unif o r m l y a l m o s t periodic s y s t e m analytically dependent on a p a r a m e t e r X (the d e g e n e r a t e s y s t e m turns into a constant diagonal wtih different coefficients) can be r e p r e s e n t e d in the f o r m f
X (t, k) = Z (t,),) exp l A (:,),) d:, 0
where Z and the diagonal A a r e uniformly a l m o s t periodic in t and analytic in X. A. M. Samoilenko [473] proved that for the s y s t e m $r = Ax + Q (tat) x with 2 r - p e r i o d i c m a t r i x Q (~) analytic for IIm T[ < p and r e a l for r e a l T, having a strongly incommensurable f r e q u e n c y
I ( k , ~ ) l > a ( L k l i + . . . + l k m l ) -~, a, d > 0 , where k is any integral-valued vector, the inner Lebesgue m e a s u r e of the set M (Q) of constant m a t r i c e s A f r o m the unit sphere $1, such that the s y s t e m is reducible, satisfies the condition rues M (Q)-* mes S 1 when IIQtt -~ 0. Other proofs of the Adrianova-Samoilenko t h e o r e m can be obtained by means of the Lille t h e o r e m (cf. 2~ below). Yu. A. Mitropol'skii and A. M. Samoilenko's joint paper [405], which preceded [473], constructed reducibility t e s t s for such quasi-periodic s y s t e m s by t r a n s f o r m a t i o n with a quasi-periodic maVrix, while Yu. A. Mit-ropol'skii and E. A. Belan [401] used an a c c e l e r a t e d convergence method to c o n s t r u c t a uniform13, a l m o s t periodic t r a n s f o r m a t i o n of an a l m o s t diagonal (with s e p a r a b l e diagonal) uniformly a l m o s t periodic s y s t e m to a diagonal uniformly a l m o s t periodic f o r m whose e x i s t e n c e had been proved by B. F, Bylov. (cf. 2~
53
V. Kh. Kharasakhal and his colleagues studied the quasi-periodic s y s t e m ~ = A (t) x with an m - d i m e n sional basis by means of the s y s t e m of linear equations and partial derivatives with coefficients of different periods, Ox/Ou=F(ul . . . . . urn)x, F ( t . . . . . t)==A(t).
An analog of the Floquet t h e o r e m was successfully proved for the l a t t e r s y s t e m under additional a s s u m p tions on the p r o p e r t i e s of the initial s y s t e m , which p e r m i t t e d reducibility and r e g u l a r i t y t e s t s for quasiperiodic s y s t e m s to be derived. We r e f e r the r e a d e r also to Refs. 295, 296, 467, 528, 544-551, 593, 594, and 601. The f i r s t set also includes V. G. Sprindzhuk [487] in which r e p r e s e n t a b i l i t y conditions for an integral of a function f i x , r . . . . . Wmx) one-periodic in each independent variable in the f o r m at + O (1) w e r e indicated and as a c o r o l l a r y reducibility t e s t s w e r e obtained for two-dimensional quasi--periodic s y s t e m s . T h e s e p a p e r s (cf. also K. G. Valeev [110]) also contain stability t e s t s for quasi-periodic and uniformly a l m o s t periodic s y s t e m s . 2 ~ a) General Study of Uniformly Almost P e r i o d i c S y s t e m s : Lillo [766], in which the a l m o s t p e r i o d i c ity of the P e r r o n t r a n s f o r m a t i o n m a t r i x for a uniformly a l m o s t periodic system with separable solutions (cf. Sec. 7) to triangular f o r m was proved by means of a set of displacements (cf. Sec. 1), was one of the f i r s t works along these lines. In his p r e c e d i n g work [764], it was proved that the set of uniformly a l m o s t periodic s y s t e m s with separable solutions is open in the space of all uniformly a l m o s t periodic s y s t e m s . B. F. Bylov [91] constructed a reducibility test for the uniformly a l m o s t periodic s y s t e m ~ = A (t) x using a l m o s t periodic Lyapunov t r a n s f o r m a t i o n s , to a diagonal s y s t e m with uniformly a l m o s t periodic coefficients (and consequently (B. F. Bylov [88]), a l m o s t reducibility) in the f o r m of c e r t a i n constraints on e v e r y s y s t e m ~ = ~(t) y. In p a r t i c u l a r such a reduction is possible if a uniformly a l m o s t periodic s y s t e m p o s s e s s e s a s y s t e m of separable solutions, which is a strengthening of the above r e s u l t of LiUo. V. M. Millionshchikov [380] examined r e c u r s i o n m a t r i c e s A (t) (bourided and uniformly continuous on a line, such that e v e r y path of the s y s t e m DA is e v e r y w h e r e dense in RA; for notation see Sec. 1) and for these m a t r i c e s proved the following a s s e r t i o n : 1. A linear s y s t e m with a r e c u r s i o n coefficient m a t r i x can be r e d u c e d by a P e r r o n t r a n s f o r m a t i o n with a r e c u r s i o n m a t r i x to triangular f o r m also with a r e c u r s i o n matrix. 2. If the s y s t e m ~ = A (t)x with r e c u r s i o n m a t r i x is not a l m o s t r e d u c i b l e , an~(t) E R A can be found, such that ~ = ~(t) x is i r r e g u l a r . V. M. Millionshchikov [381, 397] also proved that for e v e r y uniformly a l m o s t periodic m a t r i x ACt) and for e v e r y e > 0, t h e r e exists a uniformly a l m o s t periodic (with the same frequency moduli) m a t r i x B(t), such that lIB ( t ) - A (t)[] ~ s and such that the s y s t e m ~ = B (t)x is statistically a l m o s t reducible (cf. Sec. 3). b) Exponents of Uniformly Alrnost P e r i o d i c Systems. This section and the preceding one join together in the-following a s s e r t i o n , which is fundamental in the general t h e o r y of uniformly almost periodic s y s t e m s . Millionshchikov T h e o r e m [380, 382, 393]. A uniformly a l m o s t periodic s y s t e m is almost reducible if and only if its exponents a r e stable. A r e f i n e m e n t of this t h e o r e m can be found in V. L. Novikov [420]. We note other r e s u l t s of the t h e o r y of exponents of uniformly a l m o s t periodic s y s t e m s . Bylov T h e o r e m s ([87] and [103], p. 190). 1. The upper (lower) singular and central exponents (cf. Sec. 1) of uniformly a l m o s t periodic s y s t e m s coincide. 2. The highest (lowest) exponents of a uniformly a l m o s t periodic s y s t e m is upper (lower) stable if and only if it coincides with the upper (lower) singular exponent. Millionshchikov's T h e o r e m s [381, 393,398]. 1. The highest (lowest) exponent for an absolutely r e g u l a r uniformly a l m o s t periodic s y s t e m coincides with the upper (lower) singular exponent ~o(~o).
54
2. The exponents of a uniformly a l m o s t periodic s y s t e m a r e stable if and only if the exponents of e v e r y s y s t e m ~/= :~(t) y, where ~(t) E R A , a r e stable. 3. F o r a l m o s t e v e r y ~(t) E R A (in the sense of any invariant m e a s u r e on DA), the highest (lowest) exponent of the s y s t e m ~ = ~(t) y is equal to its upper (respectively, lower) singular exponent. 4. We have for a two-dimensional uniformly a l m o s t periodic s y s t e m , a) its probable s p e c t r u m Ap(A) Consists of two n u m b e r s $t0 and COo;b) its P e r r o n t r a n s f o r m a t i o n can be r e d u c e d to triangular f o r m for int e g r a l mean diagonal coefficients Pi (t) under the equalities p~ = Pi and P2 = P2 (and the s y s t e m is almost reducible) or Pl = P2 and Pl = P2. 5. The probable exponents for e v e r y uniformly a l m o s t periodic s y s t e m coincide with the auxiliary exponents (cf. Sec. 1). 3 ~ Examples. N. P. E r u g i n ' s ([199], p. 88; [2i3], p. 137) c l a s s i c a l problem is considered one of the most important theoretical p r o b l e m s in the t h e o r y of linear s y s t e m s with a l m o s t periodic coefficients, This problem c o n s i s t s in determining the existence of i r r e g u l a r s y s t e m s with a l m o s t periodic and with quasiperiodic coefficients. V. M. Millionshchikov ([387]; cf. also Differents.Uravnen., 10, No. 3, 569 (1974)) constructed an e x a m ple of a not a l m o s t reducible s y s t e m with a l m o s t periodic coefficients (the question as to the existence of such s y s t e m s also r e m a i n s open); he also derived in this paper (proceeding on the basis of his r e s u l t s [380] p r e s e n t e d above) the existence of i r r e g u l a r s y s t e m s with almost periodic coefficients. Subsequently V. M. Millionshchikov [397] proved (using a proof principle identical to that of [387]) the existence as well of i r r e g u l a r s y s t e m s with quasi-periodic coefficients. V. L. Novikov, complicating the construction of Millionshchikov's example [387], constructed examples that proved: 1) T h e r e exists a uniformly a l m o s t periodic s y s t e m with highest exponent less than the upper singular exponent [420]; 2) T h e r e exists a not almost reducible uniformly a l m o s t periodic s y s t e m ~ = A (t)x, such that the highest exponent in e v e r y s y s t e m ~r = ~ ( t ) y , where ~(t) E RA, is equal to the upper singular exponent [421]. We r e f e r the r e a d e r also to Refs, 33, 41-47, 74, 83, 108, 114, 142, 180, 184, 188,222, 225-227, 241-245, 286,325, 352, 353,367, 378, 384, 386, 389, 390,399, 400, 403, 404, 425, 595, 632, 683, 672, 707, 708, 710, 762, 781, 786, 787, and 846. 7,
STABILITY
OF EXPONENTS
AND INTEGRAL
SEPARABILITY
OF SOLUTIONS P e r r o n [811], in which was proved (cf. also V. V. Nemytskii and V. V. Stepanov [416], p, 193) that the exponents of the perturbed ~ = [P (t) + Q (t)]y and initial diagonal of ~ = P (t)x of s y s t e m s when Q (t) "" 0 coincide as well as the separability condition Pi+t ( t ) - P i ( t ) >- a > 0, t >- 0 on the diagonal, was the f i r s t work on s y s t e m s with integral separability of solutions. 1
B. F. Bylov [85] weakened the separability condition to the r e q u i r e m e n t that i [P~+l(=)--Pl(~)]d:>~ s
> a (t -- s)-- d, a, d > 0, t > s, be i n t e g r a l - s e p a r a b l e and proved the stability of the exponents (of. Sec. 1) of a diagonal s y s t e m , under the assumption that it was r e g u l a r . The l a t t e r r e q u i r e m e n t was dispensed with by R. E. Vinograd [127]. Lillo [764] defined for a r b i t r a r y linear s y s t e m s the concept of separability of some of its solutions x 1 (t) . . . . . x n (t), such that the set of inequalities 1 gt(t--a)--~ fl--sl / ]] xi (t) [J / p ~.~(t-*)+v.lt-,f ~-e
-~ ~
with given r e a l n u m b e r s Xi and p =
~,~
, i=1
.....
n,
!
~ min[kj-),~l=#O , holds for all s and t.
Finally, B. F. Bylov [92] provided the following definition. 9Definition. A linear s y s t e m is said to be a s y s t e m with integral separability if it has solutions xl (t), . . . . xn(t), such that for all t > - s,
55
l[xt+, (t) li. If xi (t) II IIx~+,(s) II" Ii xt (s)It > dea(t-*)'
i = 1, 9
n - - 1,
(7,1)
with given c o n s t a n t s a, d > 0, and proved the reducibility of such s y s t e m s to P e r r o n - s e p a r a b l e diagonal s y s t e m s , m o r e o v e r e s t a b l i s h i n g the stability of the exponents of a linear s y s t e m with integral s e p a r a b i l i t y . The definition of integral s e p a r a b i l i t y is c l o s e l y r e l a t e d to the concept of exponential dichotomy; an i m p o r t a n t use and the h i s t o r y of the l a t t e r concept can be found in D. V. Anosov [16]. The theory of s y s t e m s with exponential dichotomy has been set forth in M a s s e r a and Shaeffer [364], Coppel [680], and Yu. L. Daletskii and M. G. Krein [181]. S
Subsequently B. F. Bylov and R. E. Vinograd {[96, 97]; [103], p. 207) proved a g e n e r a l t h e o r e m for the g e o m e t r i c location and e s t i m a t e of the growth of solutions of p e r t u r b e d s y s t e m s with s m a l l p e r t u r b a t i o n s , a s s u m i n g that the initial s y s t e m is b l o c k - t r i a n g u l a r with blocks of dimension n k, and such that we have the following integral s e p a r a b i l i t y conditions holding on the diagonal: t
t
t
t
J" r~d, 4 J Pi .(') d , 4 j' R~d,, ien~; ~ (rt+, -- Ok) d , > a (t -- s) -- d.
$
$
,$
$
The a s s e r t i o n that the set of s y s t e m s with integral s e p a r a b i l i t y is open in the m e t r i c s p a c e M n of n - d i m e n sional linear s y s t e m s with distance p[A it), B it) = sup ][ A ( t ) - B (t)[I i s implicit in this t h e o r e m and can be found in Bylov [98]. V. M. Millionshchikov [385] e s t a b l i s h e d that the integral s e p a r a b i l i t y condition (7.1) of a linear s y s t e m is equivalent to the p r o p e r t y that for e v e r y s > 0, a 6 > 0 can be found, such that if p[A (t), B (t)] <- 6, t h e r e e x i s t s a solution x (t) of a linear s y s t e m for e v e r y solution y (t) of the s y s t e m ~r = B (t, y), such that < {x(t), y(t)}~<~ for e v e r y t > - 0. We now p r e s e n t the following fundamental r e s u l t s in the stability theory of exponents and theory of s y s t e m s with integral s e p a r a b i l i t y obtained by V. M. Millionshchikov by m e a n s of his method of turnings. THEOREM 1 [396]. A set of linear s y s t e m s with integral s e p a r a b i l i t y coincidesxwith an open hull* of the set of s y s t e m s with stable exponents. THEOREM 2 [396]. T h e s e t of linear s y s t e m s with integral s e p a r a b i l i t y c o i n c i d e s w i t h the open hull of the s e t of linear s y s t e m s r e d u c e d by Lyapunov t r a n s f o r m a t i o n s to diagonal f o r m . THEOREM 3 [395]. The c l o s u r e of the set of linear s y s t e m s with integral s e p a r a b i l i t y Coincides with the e n t i r e spaceT M n. The use of the method of turnings h a s a l s o r e s u l t e d in the following a s s e r t i o n . Stability C r i t e r i o n for Exponents (V. M. Millionshchikov [396] and B. F. Bylov and N. A. Izobov [106]). The exponents of a linear s y s t e m a r e stable if and only if the l i n e a r s y s t e m can be r e d u c e d by a given Lyapunov t r a n s f o r m a t i o n to the b l o c k - t r i a n g u l a r f o r m hk = Pk (t) u k (k = 1 . . . . , m; u k is a v e c t o r of d i m e n sion nk, ~n i = n), where: 1. The blocks a r e i n t e g r a l - s e p a r a b l e , i.e., t h e r e e x i s t constants a and d > O, such that [IV;L 1 (t, , ) p ' > c U a " - ' [ [ V ~
(t, ")If, t > * .
k = 1. . . . . m -
1,
w h e r e U k It, T) is the Cauchy m a t r i x of the s y s t e m hk = P k it) u k. 2. The upper ~ k and lower c0k c e n t r a l exponents for e v e r y block coincide. [
The sufficiency of this c r i t e r i o n had been p r e v i o u s l y proved in I3. F. Bylov, R. E. Vinograd, et al. ([103], p. 208) in T h e o r e m 15.2.1, which is a g e n e r a l i z a t i o n of the P e r r o n t h e o r e m given above. To conclude this section we p r e s e n t a stability coefficient t e s t for exponents (N. A. Izobov [258]). The exponents of a t w o - d i m e n s i o n a l l i n e a r s y s t e m a r e stable if the g r e a t e s t and l e a s t c h a r a c t e r i s t i c n u m b e r s of the coefficient m a t r i x a r e s e p a r a b l e and if the c h a r a c t e r i s t i c v e c t o r s c o r r e s p o n d i n g to t h e m a r e c o m p l e t e l y s e p a r a b l e . T h e s i m p l e s t e x a m p l e of such a s y s t e m is one with positive (negative) nonzero coefficientm not *The union of all oPen s e t s contained in this set. *Cf. Diba [190] for n-th 9 linear equations.
56
on the diagonals. It is wellknown (K. P. P e r s i d s k i i [436]) that only i n t e g r a l s e p a r a b i l i t y of the c h a r a c t e r i s tic n u m b e r s is sufficient for the stability of the exponents of a l i n e a r s y s t e m with P e r s i d s k i i weakly v a r y ing functional coefficients. We r e f e r the r e a d e r also to Secs. 4, 6, and 8-11 and to Refs. 50, 85, 86, 100, 101, 105, 129, 149, 167, 190, 358, 377, 384, 386, 389, 393, 398-400, 504, 505, 516-518, 667, 682, 683, 742, 744, 751, 760, 761, 763, 765, 766, 783, 858, 859, and 878. 8.
UNIFORMLY
COARSE
SEQUENCES
OF
PERIODIC
SYSTEMS
The following r e s u l t s of V. A. P l i s s a r e r e l a t e d to the p r e c e d i n g section. Definition [444, 449]. A sequence of n - d i m e n s i o n a l s y s t e m s
~=Am(t)x, m =
1, 2. . . . .
(8.1)
with p i e c e w i s e continuous u n i f o r m l y bounded w i n - p e r i o d i c m a t r i c e s , where com --- oo when t >- 0, is said to be u n i f o r m l y c o a r s e if for s o m e e > 0 e v e r y s y s t e m y = [A m (t) + Qm (t)]y, w h e r e lIQm (t)l] <-- e, of the f o r m (8.1) lacks z e r o c h a r a c t e r i s t i c exponents. Uniform C o a r s e n e s s C r i t e r i o n (Pliss [444, 449]). 1 ~ A sequence of t w o - d i m e n s i o n a l s y s t e m s (8.1) with z e r o - s e p a r a b l e exponents of different signs is u n i f o r m l y c o a r s e if and only if e v e r y s y s t e m (8.1) is a s y s t e m with integral s e p a r a b i l i t y (cf. Sec. 8) with identical c o n s t a n t s a and d. 2". A sequence of stable s y s t e m s (8.1) is uniformly c o a r s e if and only if t h e r e exist constants ~ > 0 and T O> 0, such that for e v e r y T >- T O and m >- m (T),
where X m (t, r) is the Cauchy m a t r i x of the s y s t e m (8.1). T h e s e r e s u l t s w e r e used by P l i s s to p r o v e S m a l e ' s hypothesis [850] on the finiteness of the n u m b e r of stable periodic motions in the " c o a r s e c a s e " under the a s s u m p t i o n that the periodic s y s t e m itself and all s y s t e m s close to it lack periodic solutions with z e r o exponents [449], to analyze the n e c e s s i t y of the S m a l e Robbin c o a r s e n e s s conditions [484], and to study the r e l a t i v e position of the s e p a r a t r i c e s of t~vo-dimensional periodic s y s t e m s (cf. a l s o [443, 445-448]). 9.
CENTRAL
EXPONENTS
AND THEIR
REACHABILITY /
The singular (~0 and r 0) exponents [434, 651] and c e n t r a l (~ and w) exponents introduced by R. E. Vinograd [123] (cf. Sec. 1) of a linear s y s t e m connected by the r e l a t i o n s h i p s ~20 >- ~2 >_ oJ >- ~00 and which can be calculated (the f i r s t two) using the f o r m u l a s (B. F. Bylov, R. ~. Vinograd, et al. [103], pp. 116-117) ~0 = llm ~ sup lnll X((k + 1) T, kT)[I, T.-,~o I
k--1
~O----lim T~
~ - ~ lnl[X((i + 1) T, ir)ll, i~l
e s t i m a t e the j u m p s of its exponents for s m a l l p e r t u r b a t i o n s .
N a m e l y we have the following a s s e r t i o n .
Vinograd T h e o r e m ([123]; [103], pp. 164, 167). For e v e r y e > 0, a 6 > 0 can be found, such that the c h a r a c t e r i s t i c exponents of any p e r t u r b e d s y s t e m ~ = A (t)y + f ( t , y), where lit(t, y)II - 6 [lyJl, belong to the s e g m e n t [ : ~ - e , ~2 + e ]. t
R. E. Vinograd ([103], p. 164) also p r o p o s e d that these e s t i m a t e s cannot be improved and proved this is so for the c a s e of diagonal s y s t e m s . V. M. Millionshchikov [392] p r o v e d this a s s e r t i o n in the g e n e r a l c a s e using his method of turnings (cf. Sec. 4). We will now t r e a t in somewhat m o r e detail than in Secs. 4, 6, and 7 the application of the l a t t e r method. .Millionshchikov T h e o r e m [392]. The c e n t r a l exponents of a linear s y s t e m a r e r e a c h a b l e , i.e., for e v e r y s > 0, t h e r e exist p i e c e w i s e continuous m a t r i c e s B~ (t) and C~ (t), where lIBe (t)!l <- s [Ic~(t)II -< ~, such that the highest exponent of the s y s t e m
57
= [A (t) + B~ (t)] y
(9.1)
is at l e a s t f l - e , the lowest exponent of the s y s t e m ~r = [A (t) + C e (t)]y being l e s s than co + e. M e t h o d o f P r o o f of R e a c h a b i l i t y of ~ (V. M. Millionshchikov [392]). The set of s ~ 1 7 6 x (t) of a l i n e a r s y s t e m on the s e g m e n t [7, ~- + T] will be divided into two c l a s s e s : the slow solutions, c h a r a c t e r i z e d by 1
IIx (= + T) II< T IIX (-. + T, ~) !111x(~)llsin ~, and the r a p i d solutions if the inequality sign is r e v e r s e d . Our f u r t h e r d i s c u s s i o n s a r e b a s e d on the following two l e m m a s of V. M. Millionshchikov, which we p r e s e n t without proof. LEMMA 1. A r a p i d solution is contained in the angular c-neighborhood <{a, x (~)}~<s of a slow solution x (t) on [~', ~"+ T]. LEMMA 2. F o r e v e r y solution x (t) of a l i n e a r s y s t e m and for e v e r y v e c t o r a coinciding in n o r m with the v e c t o r x (~') and f o r m i n g an angle 6 < 7r/2 with it, t h e r e e x i s t s a turning m a t r i x U6(t), such that the v e c t o r
y(t)=Us(t)x(t), y ( ~ - - l ) = x ( z - - 1 ) , y ( z ) = a ,
(9.2)
on the i n t e r v a l ( 7 - 1 , ~') is a solution of the s y s t e m
:) = [A (t) + Q6 (01 v, IlQs(011~<(2xt + 1) ~, .a~ > IIA (t)ll. We s e l e c t a sufficiently l a r g e T > 1. We s e t Be (t) =- 0 on the i n t e r v a l [0, T] ( o n ( T - l , T), Be(t) m a y possibly vary) and w e t a k e as the solution y (t) of the s y s t e m (9.1) whose exponent we wish to be at l e a s t ~ - e , a r a p i d solution of the l i n e a r s y s t e m on [0, T]. If x it, y (T)) is a r a p i d solution on IT, 2T] of the l i n e a r s y s t e m , B e (t) -- 0 a l s o when t E [T, 2T]. But if it is a slow solution, then, i n a c c o r d a n c e with L e m m a 1, a v e c t o r a T can be found i n a n e - n e i g h b o r h o o d of the v e c t o r y iT), which we denote as b e f o r e by y (T), such that the solution x (t, y (T)) will turn out to be a r a p i d solutior/on [T, 2T]. We continuously connect the two values y ( T - 1) and y (T) of the solution y (t) of the s y s t e m (9.1) by m e a n s of the turning (9.2) indicated in L e m m a 2, which is not d e r i v e d f r o m the c l a s s of r e a c h a b l e p e r t u r b a t i o n s for 5 = e/(2M + 1). Thus in this c a s e B(t) = Q (t) when t E [ T - 1, T) and Be (t) - 0 when t E [T, 2T). By extending this c o n s t r u c t i o n to the entire h a l f : axis we will have a p e r t u r b e d s y s t e m (9.1) with solution y (t) whose n o r m s a t i s f i e s the r e q u i r e d bound k--1
[ly(kT)ll>~llY(0)tt H llX((Z+ 0T, ir)l[, ~ = ~ - sin~. i~l
The f i r s t p a r t of the t h e o r e m is proved. The r e a s o n why j u m p s a p p e a r in the exponents of linear s y s t e m s with s m a l l p e r t u r b a t i o n s was a l s o indicated in Yu. S. Bogdanov [60]. I
R. E. Vinograd ([103], p. 180), for diagonal s y s t e m s of even o r d e r , and T. E. Nuzhdova [423], for twodimensional s y s t e m s of g e n e r a l f o r m , each proved the s i m u l t a n e o u s r e a c h a b i l i t y of the c e n t r a l exponents, i.e., that t h e r e e x i s t s a m a t r i x Qe (t), such that both a s s e r t i o n s of the Millionshchikov t h e o r e m a r e t r u e . The C e n t r a l and Singular Exponents a r e Unstable iV. M. Millionshchikov [394]). T h e r e e x i s t s a l i n e a r s y s t e m of any o r d e r with nonzero singular exponents, such that for e v e r y natural m t h e r e exists a s y s t e m ~r = [A (t) + Qm (t)]y with z e r o singular exponents in which supllQm (t)ll --~ 0 as m--* o% However, t h e r e e x i s t v a r i o u s s e m i s t a b l e exponents (t3. F. Bylov, R. ~.. Vinograd, et al. [103] (p. 166). An r e a c h a b l e bound f o r the highest exponent of a s y s t e m with p e r t u r b a t i o n s IIQ (t)lI <-- C exp (-r has b e e n c o n s t r u c t e d (N. A. Izobov [254]) using the Millionshchikov method of turnings. We r e f e r the r e a d e r also to Secs. 10 and 13 and to Refs. 124, 128, 129, 131, 389, 399, 400, 422, 745, 763, and 765. I0.
LIMITS
ON
EXPONENTS
OF
LINEAR
SYSTEMS
The Lyapunov limits ([349], p. 31) obtained in proving the exponent finiteness theorem were the first limits derived on the exponents X i of systems of general form ~ = A (t)x iouly these systems will be con-
58
sidered here). The limits of Yu. S. Bogdanov [49, 52] t
,_~.
tl
2t } ~ la,,+a,,l l ,]~1
are directly related to them. We may also consider the bound (Wazewski [882],Wintner [885],Butlewski [662], and Kitamura [747]; cf. also Cesari [562] and B. F. Bylov et al. [103] (p. 128))
in which the least and greatest characteristic numbers of a matrix are denoted by ~ and A, respectively. S. M. Lozinskii [326] (cf. also [103], p. 128) subsequently, by means of the logarithmic norms I
l (t)=--lira t [lIE + hA (t)[[- 1], L (t)-:-l~_~o -~ [lIE + hA (t)H- 1] he introduced, obtained the limits l -< ki -< ~ and its particular cases
I~
L
kr
)
l,~lz
]
t
~,
kr
~
ir
J
R. E. Vinograd [103] (p. 127) (cf. also Eltermann [699]) derived Lozinskii's general bound in a different form and also for discontinuous A (t). He also obtained bounds on the exponents of triangular systems ([103], p. 138). V. M. Alekseev [10] using his freezing method for ann-dimensional system with piecewise differentiable matrix A (t) ( 6~ sup 1]A'(t)tt) obtained the bound X <- p + c61/2n on the highest exponent X in t e r m s of the least upper bound p of the greatest r e a l part X(t) of the characteristic numbers of the matrix A (t) and the p ar ameter 6, which, in V. M. Alekseev and R. E. Vinograd [13], was refined to l ~< p + cB'/n+1.
(10.1)
Examples of systems with highest exponents ~, > p + c061/n were presented there as well as in [103] (p. 138). Thereachability of the bound (10.1) in the two-dimensional* case was proved in N. A. Izobov [256]. Here, as well as in [255], refinements of it were obtained for n = 2, for example,an integral variant <- I + c6~/3 and bound A <- p + c61/2 of nearly identical characteristic numbers of A (t)o N. G. Chetaev ([564],p. 194), A. D. Gorbunov [163, 164], B. S. Razumikhin [458, 459], K. A. Karacharov and A. G. Pilyutik [265], and others have studied bounds on the exponents of linear systems by means of Lyapunov functions. All of these bounds are also bounds on the central and singular exponents of linear systems.
Bounds on exponents are in turn used for establishing asymptotic stability. Direct determination of stability by means of the second Lyapunov method and also by means of different varieties of the congruence method (cf. also Secs. 2, 5, 9, 11, and 12) can be found in Refs. 5, 11, 12, 34, 35, 81, 113, 130, 136, 139, t46, 162, 165, 183, 185, 188, 191, 195, 199,219, 221, 223, 224, 230, 259, 261, 262-264, 266,272, 273, 276278, 301, 314, 315,320, 327-330,335, 337, 339, 342-344, 346, 348-350, 355, 358, 365, 376, 406-408, 411, 417, 430, 437-439, 442, 451, 453, 462, 464, 474, 478, 483, 486, 506, 515, 526, 527, 529, 534, 542~ 543, 565, 566, 597, 604, 635, 637, 640, 644, 649, 654, 658, 669,670, 686, 693, 694, 698, 718, 721, 722, 763, 769, 776, 779, 782, 783, 810, 817, 819, 821, 829-831, 834, 838, 839, 842, 844, 849, 851, 852, 856, 857, 860, 874, and 886-889. *Its reachability for any n >- 2 was f i r s t proved, as far as we know, by Yu. I. Elefteriadi (Differents. Uravnen., 1~0,No. 8 (1974)).
59
11.
METRIC
THEORY
OF
LINEAR
SYSTEMS
In studying nonautonomous s y s ~ m s including linear s y s t e m s , V. M. Millionshchikov [375] introduced the concept of a g e n e r a l i z e d solution x (t) a s the uniform bound on e v e r y s e g m e n t of a sequence of d i s p l a c e m e n t s x k (t k + t) of its o r d i n a r y solutions. F o r the c a s e of a/bounded m a t r i x A (t) u n i f o r m l y continuous on a line, e v e r y g e n e r a l i z e d s o l u t i o n ~ (t) of the s y s t e m $r = A (t) x is the o r d i n a r y solution of s o m e s y s t e m ~ = A (t)x w h e r e A (t)---- lim A (t~ + t) and the bound is s e g m e n t - u n i f o r m [381]. We have for g e n e r a l i z e d solutions the following a s s e r t i o n . Millionshchikov T h e o r e m [381]. The m a x i m u m
~r.~l-Unh t J-
~
t-T
In I]~(t)ll ilZ('oll
and m i n i m u m ~[~] exponents of e v e r y g e n e r a l i z e d solution of a linear s y s t e m belong to its probable s p e c t r u m (cf. Sec. 1). The singular exponents also belong to it. Following V. M. Millionshchikov [380, 383] (cf. [399] for a s u r v e y of the h i s t o r y of this problem), we introduce the dynamic s y s t e m DA of d i s p l a c e m e n t s of the m a t r i x A (t) defined on the set of m a t r i c e s ~, (t) and its s p a c e R A. As a l r e a d y noted (Sec. 4), a l m o s t e v e r y s y s t e m k = ~ (t} x is absolutely r e g u l a r . we also have the following a s s e r t i o n .
F o r such s y s t e m s
Millionshchikov T h e o r e m [389, 398]. The c h a r a c t e r i s t i c exponents of e v e r y absolutely r e g u l a r s y s t e m a r e a l m o s t c e r t a i n l y stable, i.e., the c h a r a c t e r i s t i c exponents o f t h e s y s t e m ~ = A (t)y + a2C (t, w ) y (elements of a m a t r i x defining C (t, ca) in a given b a s i s a r e independent nonzero white noise) approach the c h a r a c t e r i s t i c exponents of a linear s y s t e m with p r o b a b i l i t y one as a --- 0. V. M. Millionshchik0v also derived the r e s u l t [389] that for a l m o s t e v e r y (in t h e sense of any inv a r i a n t m e a s u r e on D A) ~ (t) ERA, the upper (lower) c e n t r a l exponent of this s y s t e m k = A (t) x coincides with its g r e a t e s t ( r e s p e c t i v e l y , least) exponent. Definition~ [386]. The p r o b a b l e s p e c t r u m Ap (A) of a linear s y s t e m is said to be stable if for a l m o s t e v e r y A (t) 6 R A (in the s e n s e of any invariant m e a s u r e on DA), the c h a r a c t e r i s t i c exponents of the s y s t e m = ~ (t) x a r e stable. V. M. Millionshchikov [386, 389, 393] obtained for a s y s t e m with a r e c u r s i o n coefficient m a t r i x (cf. Sec. 6) a stability c r i t e r i o n on the probable s p e c t r u m , which a s s e r t s that it can be r e d u c e d by a Lyapunov t r a n s f o r m a t i o n to a t r i a n g u l a r s y s t e m , such that for any pair of diagonal coefficients Pkk (t) one of the following t h r e e r e l a t i o n s h i p s is t r u e :
x-p.-pi / < 0, _~p.-pii > 0, x-p~r.pjj=x~._pi~ = 0
Millionshchikov [386, 389] also in the c a s e of a s t r i c t l y ergodic ([416], Chap. 6) dynamic s y s t e m D A proved that the probable s p e c t r u m of a linear s y s t e m is stable if and only it it is a l m o s t r e d u c i b l e (cf. also Secs. 4 and 6). 12.
ASYMPTOTICALLY
EQUIVALENT
SYSTEMS
Two linear s y s t e m s a r e said to be a s y m p t o t i c a l l y equivalent if/u. S. Bogdanov [56, 62, 66] or k i n e m a t ically s i m i l a r ( C e s a r i [562], p. 93) if t h e r e e x i s t s a Lyapunov t r a n s f o r m a t i o n that c o n v e r t s one into the other. T h i s concept lies at the b a s i s of the Lyapunov t h e o r y of linear differential s y s t e m s . An i m p o r t a n t r e s u l t in the g e n e r a l t h e o r y of a s y m p t o t i c a l l y equivalent s y s t e m s is the following a s s e r t i o n . P e r r o n Triangulation T h e o r e m [809]. F o r e v e r y l i n e a r s y s t e m t h e r e e x i s t s a unitary Lyapunov t r a n s f o r m a t i o n that c o n v e r t s it into a linear s y s t e m with a t r i a n g u l a r bounded and piecewise continuous coefficient m a t r i x .
60
New proofs of this t h e o r e m have been given by P. A. Kuz'min [297], R. E. Vinograd [118]~ and Dilib e r t o [695]. Hardly e v e r y linear s y s t e m can be r e d u c e d by a Lyapunov t r a n s f o r m a t i o n to diagonal or to even blockt r i a n g u l a r f o r m . The c o r r e s p o n d i n g c r i t e r i a of this reducibility in t e r m s of the Gramian were given by B. F. Bylov [92, 94]. In p a r t i c u l a r to r e d u c e a linear s y s t e m to diagonal f o r m it is n e c e s s a r y and sufficient that det X(t)[ > 7 H [[xi(t)[I, where T > 0 for the fundamental s y s t e m of solutions X = [xl . . . . .
Xn]. However,
it is t r u e that e v e r y linear s y s t e m with "slowly varying" coefficients is a l m o s t reducible to diagonal form, namely, we have the following important a s s e r t i o n . P e r s i d s k i i T h e o r e m [436]. For e v e r y s > 0, a Lyapunov t r a n s f o r m a t i o n y = Ss (t) x can be found that t r a n s f o r m s a l i n e a r s y s t e m with a w e a k l y v a r y i n g m a t r i x A (t) and having c h a r a c t e r i s t i c numbers Xi (t) into the almost diagonal s y s t e m y = diag [kt(t) . . . . .
~At)l y + Q (t) y
(12~
with components satisfying the inequality [!Q (t)][ <- ~. The reduction of a linear s y s t e m to L-diagonal f o r m (the m a t r i x Q(t) in Eq. (12.1) is summable on the half-axis) has been studied by I. M. Rapoport [460], C e s a r i [562] (p. 68), N. Ya. Lyashchenko (3501, Conti [674], N. I. Shkil' [578], and others. The P e r s i d s k i i t h e o r e m admits of a generalization (N. A. Izobov [257]) for two-dimensional linear s y s t e m s with piecewise differentiable A (t)(5 =- supItA'(t)ll). Namely, t h e r e exists a Lyapunov t r a n s f o r m a tion that converts a linear s y s t e m to the f o r m (12.1) with components of Q (t) that satis~- the inequality IIQ (t)ll -- q61/3. The p r e c i s i o n of this bound was proved in [256]. L i n e a r System Asymptotically Equivalent to a System with P i e c e w i s e Constant Coefficients (D. M. Grobman [168], Yu. S. Bogdanov [149]). We have even the following s t r o n g e r solution. Bogdanov T h e o r e m [56, 62]. E v e r y linear system is asymptotically equivalent to a system with piecewise constant coefficients that take only two values. The following a s s e r t i o n establishes a conclusive and p r e c i s e r e s u l t in the t h e o r y of the asymptotic equivalence of an initial and p e r t u r b e d system. Millionshchikov T h e o r e m ([376]; cf. also Sec. 5). A p e r t u r b e d s y s t e m is asymptotically equivalent to the initial s y s t e m for e v e r y perturbation (5.1) (cf. Sec. 5) with a > a M. M o r e o v e r t h e r e e x i s t s a p e r t u r bation (5.1) with a < aM, such that the p e r t u r b e d system is no longer asymptotically equivalent to the initial system. The equivalence of a linear s y s t e m r e l a t i v e to the t r a n s f ~ Y (t) = X (t) S (t) has been studied by I. N. Zboichik [231, 234], O. A. K a s t r i t s a , and G. N. P e t r o v s k i i [268, 269]. The investigations of Yu. S. Bogdanov [58] and V. I. Arnol'd [20] on the analytic s t r u c t u r e of a m a t r i x that t r a n s f o r m s a variable m a t r i x to normal Jordan f o r m turns out to be useful in c o n s t r u c t i n g a triangulation t r a n s f o r m a t i o n . We r e f e r the r e a d e r also to Secs. 2, 4-7, and 11 and~to Refs. 1, 2, 52, 54, 93, 95, 100, 106, 188, 1 9 9 , 2 1 3 , 2 3 9 , 252, 258-260, 274, 280, 281,310, 338, 349,351, 356, 396, 402, 409, 424, 426, 427, 432, 442, 472, 482, 499, 500, 519, 521-523, 535, 603, 673, 676, 678, 692, 696, 724, 725, 746, 756, 757, 768, 775, 780, 790, 801, 828, 848, 853, 864,and 869. 13.
LINEAR
APPROXIMATION
STABILITY
Together with the initial linear s y s t e m with highest exponent X, we consider the s y s t e m
~=A(t)y+f(t,
y)
(13.1)
with m - p e r t u r b a t i o n s f , [If(t, Y)[I -~ N [I yll m, where m > I and t ~- 0 n e a r y = 0. In f i r s t approximation stability t h e o r y we have the following initial and fundamental a s s e r t i o n .
61
Lyapunov T h e o r e m (A. M. Lyapunov [349], pp.52-55, and M a s s e r a [363]). The zero solution of the s y s t e m (13.1) is asymptotically stable when
(m-- I) k-saL< 0.
(13.2)
This theorem was subsequentlyrefined by D. M. Grobman [168], who asserted that the variable (rL in Eq. (13.2) can be replaced by a smaller variable ~G (cf. Sec. 5). As regards the replacement of crL by ap in Eq. (13.2) we refer the reader to N. E. Bol'shakov [72]. Necessary and sufficient conditions on the diagonal coefficients, under which no given perturbations violate the stability or asymptotic stability of the zero solution, have been indicated for the case of a first approximation triangular system by Perron [810]. K. P. Persidskii [435] obtained sufficient conditions for such stability in the ease of a regular first approximation system and for unbounded coefficients in the expansions of f(t, y) in series. I. G. Malkin ([358], p. 379) used the bound 1]X(t.s)If< D exp[a(t--s) +[3s] of the Cauchymatrix of a linear system to prove the asymptotic stability of the zero solution of the system (13.1) under the condition ( m - l ) ~ +fl < 0. First approximation instabilitytests were constructed by N. G. Chataev [564] and by N. P. Erugin (Prikl. Matem. i Mekh., 16, No. 3 (1952)).
The Lyapunov t h e o r e m and its r e f i n e m e n t s p e r m i t bounds to be constructed of the value of m0 (though not to calculate this value exactly), beginning at which stability is observed, a s well as to obtain not always r e a c h a b l e upper bounds on the highest exponent A(f) (cf. [253, 134] for its definition) of the s y s t e m (13.1) withan m - p e r t u r b a t i o n f . The construction of these exact c h a r a c t e r i s t i c s was c a r r i e d out by means of R. E. Vinograd's central m-exponents ([103], p. 233). We have the following a s s e r t i o n . T h e o r e m {R. E . Vinograd [132, 134] and N. A. Izobov [253]). The highest exponent A(f) of e v e r y s y s tem (1)with an m - p e r t u r b a t i o n f does not exceed the c e n t r a l m-exponent ~2m that can be calculated by the Cauchy m a t r i x of the linear s y s t e m and its zero solution is asymptotically stable when 12m < 0. T h e r e exist m - p e r t u r b a t i o n s f that r e a l i z e t2 m (A(f) = ~m). t
The f i r s t a s s e r t i o n of the t h e o r e m was proved by means of an evaluating exponent (R. E. Vinograd ([103], p. 233; [132]) ~m----infl~m -1 R(t), where inf is taken over all pairs (W, R) of numbers W < 0 and #-+oo
t
piecewise continuous functions R (t) that obey the bound in iIX (t, s)H ~< W (t-- s)-5 R (t)-- mR (s), where s <- t; if no such negative n u m b e r s W exist, we set 12m = + ~. The validity of the second a s s e r t i o n was established by Millionshchikov's method of turnings using the constructive exponent (N. A. Izobov [253]) 9,~ =
lim I-~-_I~~)
wherethe sequence {~(~)} is definedby the equalities ~(0~)=a, ~k(~)=max[In]IX(k'0~,<~ i)I[+
r,$~)I, and by successively proving the equivalence of those two definitions. The continuity of 12m as a function of m has also been established [134] and segments on which it is strictly monotonic have been determined. The desired value of m0 is inf m when t2m < 0. In the critical case A = 0 as well as for a completely regular first approximation system, B. P. Demidovich [187] proved the asymptotic stabilityof the zero solution of the system (13.1) under the condition P < - i/(m-l), where y is the highest exponent of the linear system (cf. Sec, I). B.P. Demidovich's results along these lines were developed in Refs. 553-555, 455, 4, and 73. The characteristic vectors, i.e., sets of the form (exponent, degree) are a concretization of the Anorms of Yu. S. Bogdanov (cf. Sec. I). A stabilitytest was obtained by Yu. S. Bogdanov [57] in terms of such characteristic vectors. Let us note the Shestakov theorem (V. V. Nemytskii and V. V. Stepanov [416], p. 267) on the set of exponents of solutions of the system (13.1) in the case A (t) = const that tend to zero. W e refer the reader also to Refs. 6, 7, 35, 51) 66, 70, 71, 93, 104, 126, 133, 135, 138, 169-178, 219, 220,
237, 254, 317-319, 354, 450, 452, 454, 457, 516-518, 557, 602, 645, 647, 648, 656, 657, 664, 675, 677, 687, 749, 750, 785, 802-806, 833, 835, 836, 890, and 891, 14.
EXTENDED
LYAPUNOV
CHARACTERISTIC
NUMBERS
One method for extending the method of Lyapunov characteristic numbers to systems with a leading nonlinear part has been described by Yu, S. Bogdanov [59, 61, 63-65]. W e consider the system
62
~c--/(x),/
0)=0,
(14.1)
defined for the sake of simplicity on the entire n-dimensional space R n and possessing existence and uniqueness properties extending without limit in both directions. W e denote the solutions and integrals of (14.1) by, respectively, x = x (t; xo, to), x (to: x0, to) = x ~ ,
x o = x o (t; x, to), xo[t; x (t: xo, to), tol=Xo. Let us a s s u m e that a continuous function d {~, /3) of positive independent variables ~ and /3 is such that: 1)d(fl, c~) = - d ( ~ , /3); 2) d(fl, Y) > d(~, Y) for fl > ~; and 3) d(6,/3) + d(fl, ~) -> d(6, ~) for 6> fl > ~>0.
Or
W e construct by means of a continuous positive-definite function v (x) for any pair of sets /Y/t,2d2cR% a (small) v d - n u m b e r (Yu. S. Bogdanov [65]),
UM2,
~(M,,M2)=max {lira - - Tl sup d ('o (x (t; xo' O))' 1), t-~+~ a-,~Ml I
-- lira T inf d(v(x(t;xo, O)), 1)}. t=u XoEM, Bogdanov T h e o r e m [64]. If ~ (x0, x0) < 0 for e v e r y x0 6 R n, where x 0 ,~ 0, the z e r o solution (14.1) is asymptotically stable. On the other hand, if it is asymptotically stable t h e r e exist functions v and d, such that ~ (x0, x0) < 0 for e v e r y x 0 6 R n, where x0 '~ 0. In addition to (14.1) we consider the p e r t u r b e d s y s t e m ~-=g(y), g(O)=O, y = y ( t ; Yo, to)6R~.
(14.2)
The r e c u r s i o n limit sets (14.2) a r e defined in t e r m s of (14.1) by m e a n s of m
A4=:(Xo)=Lira {Xo[t; y(t; x0, 0), 0]}. t--*•162
We r e p l a c e M i and M 2 in the definition of ~ (Mr, Nil) on the e-neighborhood of these sets, pass to the limit as e ---+ 0, thus obtaining the r e s i d u e vd-number(~2)(Mi, M2). Bogdanov T h e o r e m [65]. F o r the v d - n u m b e r s ~ (Y0, Y0) of the solutions of the s y s t e m (14.2) we have the bound
(Yo,Yo)~<(~){M+ (xo), M (Xo)],xo =yo. Similar r e s u l t s for nonautonomous s y s t e m s a r e m o r e awkward in f o r m [63] (cf. also [67, 122,125,293, and 563]). 15.
OTHER
PROBLEMS
In this section we p r e s e n t a bibliography on different lines of r e s e a r c h into linear s y s t e m s that w e r e not t r e a t e d for want of space in the survey. On the subject of s y s t e m s with constant coefficients we r e f e r the r e a d e r to Refs. 15, 17, 18, 25, 35, 68, 111, 235-238, 267, 275, 2 9 8 , 2 9 9 , 3 0 0 , 304, 357, 366, 415, 507, 508, 598, 628, 641, 653, 655, 665, 689, 712, 729, 730, 735, 770, 781, 792, 807, 815, 832, and 843. On the subject of s y s t e m s with periodic coefficients we r e f e r the r e a d e r to Refs. 6, 7, 14, 26, 32, 3638, 53, 75-80, 82-84, 107, 109, 1~2, 145, 150-161, 182, 186, 201, 206-208,210-212, 214, 218, 246, 270~ 287292,303, 321-324, 340, 341, 345, 349, 358-360, 363, 414, 418, 419, 440-442, 468, 469, 477, 485, 488-498, 520, 524, 540, 541,558, 559, 564, 567, 568, 570-573, 596, 606-626, 633, 636, 638, 652, 660, 661, 666, 679, 681, 684, 685, 688, 691, 697, 700, 702, 709, 711, 713, 717, 719, 720, 736, 755, 758, 759, 767, 788,, 789, 793, 794, 796, 845, 847, 855, 866, 867, 868, and 875-877.
63
On the subject of systems integrable in closed form we r e f e r the reader~to Refs. 21-23, 69, 160, 204, 229, 308, 309, 316, 331, 336, 368,369, 371,372, 374, 412, 433, 471, 475, 476, 479-481, 512, 530-533, 561, 631, 642, 650, 70I, 703-706, 820, 870, 883, and 892. On the subject of the analytic theory of linear systems we r e f e r the r e a d e r to Refs. 27-31, 147, 148, 166, 179, 192-194, 196-198, 200, 201, 205, 211, 215-218, 220, 232, 233, 282-285, 294, 305, 311, 429, 461, 503, 509-511, 590-592, 599, 600, 630, 634, 714, 715, 726-728, 731, 733, 737, 738, 740, 743, 748, 752-754, 771-774, 778: 795, 798-800, 812-814, 818, 837, 840, 841, 871-873, and 879-881. On the subject of asymptotic methods of linear systems we r e f e r the r e a d e r to Refs. 1, 2, 8, 9, 259, 279-281,334, 402, 410, 431, 460, 470, 499, 501, 502, 513, 514, 535-539, 560, 574-577, 579-589, 668, 716, 723,741, 823-825, 854, 861-863, 865. I wish to express my appreciation, for the assistance and advice provided me in writing this survey by Yu. S. Bogdanov, V. M. Millionshchikov, V. L. Novikov, and R. A. l~rokhorova. LITERATURE 1. 2. 3. 4. 5. 6.
7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
64
CITED
K.A. Abgaryan, "Asymptotic decomposition method for a system of linear differential equations," Izv. Akad. Nauk. ArmSSR, Matem., 1, No. 2, 126-137 (1966). K.A. Abgaryan, "An asymptotic transformation of a Linear differential system, ~ Izv. Akad. Nauk ArmSSR, Matem., 6, No. 5, 368-378 (1971). L. Ya. Adrianova, ~Reducibility of systems of n linear differential equations with quasi-periodic coefficients," Vest,. Leningr. Univ., No. 7, 14-24 (1962). A. Azamov, "Application of n-th order characteristic exponents to the study of asymptotic stability, ~ Differents. Uravnen. 7, No. 11, 2086-2090 (1971). N.V. Azbelev and Z. B. Tsalyuk, "Necessary and sufficient boundedness condition on solutions of a class of systems of linear differential equations," Prikl. Matem. i Mekh., 2.88,No. 1, 149-150 (1964). M.A. Aizerman and F. R. Gantmakher, "Linear approximation stability of periodic solutions of a system of differential equations with discontinuous right sides," Dokl. Akad. Nauk SSSR, 116, No. 4, 527-530 (1957). M.A. Aizerman and F. R. Gantmakher, "Linear approximation stability of a periodic solution of a system of differential equations with discontinuous right sides," Prikl. Matem. i Mekh., 2_~1,No. 5, 658-669 (1957). Valerii M. Alekseev, "Investigation of parametrically perturbed linear systems, w Dop. Akad. Nauk UkrSSR, A, No. 11, 1011-1013, 1055 (1972). Valerii M. Alekseev, "Some new applications of the I. Z. Shtokalo method," Matem. Fiz. Resp. Mezhved. Sb., No. 12, 3-5 (1972). Vladimir M. Alekseev, "Asymptotic behavior of solutions of weakly nonlinear systems of o r d i n a r y differential equations," Dokl. Akad. Nauk SSSR, 13.__4,No. 2, 247-250 (1960). Vladimir M. Alekseev, "An estimate of the perturbations of solutions of ordinary differential equations. I," Vest,. Mosk. Univ. ,. Ser. Matem., No. 2, 28-36 (1961). Vladimir M. Alekseev, "An estimate of the perturbations of solutions of ordinary differential equations. II," Vestn. Mosk. Univ., Set. Matem., No. 3, 3-10 (1961). Vladimir M. Alekseev and R. E. Vinograd, " ' F r e e z i n g ' method, w Vest,. Mosk. Univ. Matem., Mekh., No. 5, 30-35 (1966). A. Alimov, "New Schur-Arzhanykh stability inequalities, ~ Izv. Akad. Nauk UzSSR, Ser. Tekhn. Nauk, No. 5, 25-29 (1961). Yu. I. Alimov, "Problem of constructing Lyapunov functions for systems of linear differential equations with constant coefficients, ~ Sibirsk. Matem., Zh., 2, No. 1, 3-6 (1961). D.V. Anosov, "Geodesic flows on closed negative-curvature Riemannian manifolds," T r. Matem. Inst. Akad. Nauk SSSR, 90, 210s (1967). I . S . Arzhanykh, "New stability inequalities, ~ Dokl. Akad. Nauk UzSSR, No. 2, 3-6 (196!). I . S . Arzhanykh, "On new stability inequalities, ~ Avtomat. i Telemekhan., 22, No. 4, 436-442 (1961). V.I . Arnol'd,nSmall denominators and kinetic stability problems in classical and celestial mechanics, ~ Usp. Matem. Nauk, 18, No. 6, 91-192 (1963). V . I . Arnol'd,~Parameter-dependent m a t r i c e s , " Usp. Matem. Nauk, 26, No. 2, 101-114 (1971). A . G . Aslanyan and V. I. Burenkov, WRefinement of a V. V. Morozov theorem, ~ Differents. Uravnen., 3, No . 4, 687-691 (1967).
22. 23. 24.
25.
26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.
38. 39. 40. 41. 42. 43. 44.
45.
46. 47.
A . G . Aslanyan and V. I. Burenkov, "On N. P. Erugin's problem on integrability in quadratures of a system of ordinary differential equations," Differents. Uravnen., 3, No. 5, 811-819 (1967). A . G . A slanyan and V. t. Burenkov, "Integrability in quadratures of a given systems of linear differential equations," Differents. Uravnen., 4, No. 7, 1241-1249 (1968). P . V . Atrashenok, "Determination of arbitrariness in selecting a matrix that transforms a system of linear differential equations into a system with constant coefficients," Vest,. Leningr. Univ., Set. Matem., Fiz. i Khim., No. 2, 17-29 (1953). A . I . Balins'kii and L . M. Zorii, "A representation of a general relation between a system of linear differential equations with constant coefficients," Dop. Akad. Nauk UkrSSR, A_, No. 3, 195-197, 283 (1971). V . P . Basov, Investigation of Kinetic Stability for a Given Class of Periodic Systems, Author's Abstract of Candidate's Dissertation, Leningrad State University (1949). V . P . Basov, "Constructions of solutions of a class of systems of linear differential equations," Prikl. Matem. i Mekh., 1__.88,No. 3, 313-328 (1954). V . P . Basov, "Investigations into the behavior of solutions of systems of linear differential equations in a neighborhood o~ an irregular-type point," Ukrainsk. Matem. Zh., 8, No. 1, 97-109 (1956). V . P . Basov, "Behavior of solutions of systems of linear differential equations in the neighborhood of an irregular-type singular point," Matem. Sb., 40, No. 3,339-380 (1956). V . P . Basov, "Investigation into the behavior of solutions of systems of linear differential equations in the neighborhood of an irregular-type singular point," Usp. Matem. Nauk, 11, No. 4, 201-202 (1956). V . P . Basov, "Asymptotic behavior of solutions of systems of linear differential equations, n DokI. Akad. Nauk SSSR, 1__0.0, No. 6, 951-954(1956). S. Bakbaev and M. Yataev, "Reducibility problem f o r periodic systems, ~ Vestn. Akad. Nauk KazSSR, No. 7, 43-46 (1968). E . G . Belaga, "Reducibility of a system of ordinary differential equations in the neighborhood of conditionally periodic motion, ~ Dokl. Akad. Nauk SSSR, 143, No. 2, 255-258 (1962). E . P . Belan, "Stability of almost diagonal systems of linear differential equations, 't Ukrai.nsk. Matem. Zh., 20, No. 4, 449-459 (1968). R, Bellman, Stability Theory for Solutions of Differential Equations [Russian translation], Izd. Inostr. Lit., Moscow (1954), 216 pp. I . N . Blinov, "Linear differential equations with piecewise-constant periodic coefficients," Avtomat. i Telemekhan., 26, No. 1, 180-183 (1965). L N. ]31inov, "Analytic representation of a solution of a system of linear differential equations with a periodic slightly oscillating parameter-dependent matrix of coefficients," Vestn. Leningr. Univ., No. 1, 5-13 (1965). I . N . Blinov, "Analytic solution of a linear system of differential equations with periodic parameterdependent coefficients, n Differents. Uravuen., 1, No. 7, 880-895 (1965). I . N . Blinov, "Analytic representation of a solution of a system of linear differential equations with almost periodic parameter-dependent coefficients," Differents. Uravnen., 1, No. 8, 1042-1053 {1965), I . N . Blinov, "Reducibility problems for systems of linear differential equations with quasi-periodic coefficients, n Izv. Akad. Nauk SSSR, Set. Matem., 31, No. 2, 349-354 (1967). I . N . Blinov, "Reducibility of a class of systems with almost periodic coefficients," l~atem. Zametki, 2, No. 4, 395-400 (1967). I . N . Blinov, "Regularity of a class of linear systems with almost periodic coefficients," Differents. Uravnen., 3, No. 9, 1461-1470 (1967). I.N. Blinov, "On a class of irregular systems," Differents. Uravnen., 4, No. 5, 949-951 (1968). I . N . Blinov, "Application of method of arbitrarily rapid convergence to the solution of the reducibili~, problem for linear systems with odd, almost periodic coefficients, ~ Matem. Zametki, 4, No. 6, 729740 (1968). I . N . Blinov, "Rapid upper-convergence method and its application to the reducibility problem for systems with almost periodic coefficients," Papers of the 5th International Conference of Nonlinear Oscillations [in Russian], Vol. 1, Kiev (1970), pp. 66-71. I . N . Blinov, "Vanishing of reducibility property for systems of linear differential equations with quasi-periodic coefficients," Matem. Zametki, 8, No. 1, 115-120 (1970). I.N. Blinov, "Violation of reducibility for systems with quasi-periodic coefficients induced by insufficiently rapid decrease of Fourier coefficients or of the nonalgebraicity of the frequency b~asis," Differents. Uravnen., 6, No. 2, 253-259 (1970).
65
48. 49. 50. 51.
Yu. S. Bogdanov, non normal Lyapunov systems, w Dokl. Akad. Nauk SSSR, 57, No. 3, 215-217 (1947). Yu. S. Bogdanov, aTheory of systems of linear differential equations, w Dokl. Akad. Nauk SSSR, 104, No. 6, 813-814 (1955). Yu. S. Bogdanov, "Remark to Section 81 of the I. G. Malkin monograph 'Kinetic stability theory, t (Gostekhizdat, 1952)," Prikl. Matem. i Mekh., 20, No. 3, 448 (1956). Yu. S. Bogdanov, "Lyapunov norms in linear spaces," Dokl. Akad. Nauk SSSR, 113, No. 2, 255-257
(1957). 52. 53. 54. 55. 56. 57. 58. 59. 60.
61. 62, 63. 64. 65. 66. 67. 68. 69. 70. 7"1. 72. 73. 74.
75.
66
Yu. S. Bogdanov, wCharacteristic numbers of systems of linear differential equations, w Matem. Sb., 41, No. 4, 481-498 (1957). Yu. S. Bogdanov, ~Remark to a method of finding periodic solutions, ~ PriM. Matem. i Mekh., 21, No. 5, 714 (1957). Yu. S. Bogdanov, "Structure of the solution of a given linear differential system," Dokl. Akad. Nauk SSSR, .129, No. 4, 719-721 (1959). Yu. S. Bogdanov, "Existence of approximating sequence for regular linear differential systems," Usp. Matem. Nauk, 15, No. 1, 177-179 (1960). Yu. S. Bogdanov, "Asymptotic characteristics of solutions of linear differential systems, ~ Papers of the 4th All-Union Mathematical Conference [in Russian], Vo!. 2, Nauka, Leningrad (1964}, pp. 424-432. Yu. S. Bogdanov, WLyapunov norms in the theory of linear differential systems," in: Functional Analysis and Its Applications [in Russian], Akad. Nauk AzerbSSR, Baku (1961), pp. 19-22. Yu. S. Bogdanov, nTransformation of variable matrix to canonical form," Dokl. Akad. Nauk BSSR, 7, No. 3, 152-154 (1963). Yu. S. Bogdanov, ~Application of generalized characteristic numbers for studying the stability of a quiescent point," Dokl. Akad. Nauk SSSR, 158, No. 1, 9-12 (1964). Yu. S. Bogdanov, ~The asymptotic characteristics of nonlinear systems," All-Union Symposium on Qualitative Theory of Differential Equations and Its Applications (Summaries of Reports} [in Russian], Samarkand (1964}, pp. 10-11. Yu. S. Bogdanov, "Asymptotic characteristics of nonlinear differential systems," Differents. Uravnen., 1, No. 1, 41-52 (1965). Yu. S. Bogdanov, "Asymptoticof equivalent linear differential systems, n Differents. Uravnen. 1, No. 6, 707-716 (1965). Yu. S. Bogdanov, nGeneralizedcharacteristic numbers of nonautonomous systems," Differents. Uravnen., 1, No. 9, 1140-1148 (1965). Yu. S. Bogdanov, "Clarifying asymptotic stability by means of small vd-numbers,n Differents. Uravnen., 2, No. 3, 309-313 (1966). Yu. S. Bogdanov, WEstimateof generalized characteristic numbers of differential systems, n Differents. Uravnen., 2, No. 7, 927-933 (1966). Yu. S. Bogdanov, "Methodof invariance in asymptotic theory of differential equations," Vestn. Belorus.Univ., Ser., 1, No. I, 10-14 (1969). Yu. S. Bogdanovand M. P. Bogdanova, "Nonlinearanalog of Lyapunovtransformation," Differents. Uravnen., 3, No. 5, 742-748 (1967). Yu. S. Bogdanovand A. F. Naumovich, WRelativepositions of two stationary linear differential systems, a Differents. Uravnen., 6, No. 3, 552-554 (1970). Yu. S. Bogdanovand G. N. Che.botarev, "Matrices that commute with their derivatives," Izv. Vyssh. Ucheb. Zaved. Matem., No. 4, 27-37 (1959). N. E. Bol' shakov,"Estimates of the radii of continuabilitydomains and attraction of perturbed systems," Vestn. Belorus. Univ.,Ser., 1, No. 2, 7-11 (1971). N. E. Bol'shakov,~First approximation stability for g-perturbations," Differents. Uravnen., 8, No. 7, 1143-1152 (1972). N. E. Bol'shakov,'rRole of Perron irregularity coefficient for explainingthe stability of two-dimensional first-order systems," Differents. Uravnen., 9, No. 2, 363-365 (1973). N. E. Bol'shakov,nPower irregularity coefficients and first-order stability," Vestn. Belorus. Univ., Ser. I, No. 2, 23-27 (1973). L. E. Borukhov, "Almost periodic solutions of given systems of linear differential equations with almost periodic coefficients," ScientificYearbook for 1954 of Saratov University [in Russian], Saratov (1955), pp. 659-660. K. A. Breus, "Solution of linear differential equations with rapidly varying periodic coefficients,, Dokl. Akad. Nauk SSSR, 108, No. 6, 997-1000 (1956).
76. 77. 78. 79. 80. 81. 82.
83. 84.
85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99, 100.
101. 102. 103. 104.
K.A. Breus, "Reducibility of a canonical system of differential equations with periodic coefficients~" Dokl. Akad. Nauk SSSR, 123, No. 1, 21-24 (1958). K, A. Breus, "A class of linear differential equations with periodic coefficients," Ukrainsk. Matem. Zh., 1_~2,No. 4, 463-471 (1960). K.A. Breus, "Canonical systems of differential equations with rapidly varying periodic coefficients," Ukrainsk. Matem. Zh., 18, No. 1, 3-10 (1966). V.I. Burdina, "Boundedness criterion on solutions of a system of second-order differential equations with periodic coefficients," Dokl. Akad. Nauk SSSR, 9_0_0,No. 3, 329-332 (1953). V.I. Burdina, "Boundedness of solutions of a system of differential equations," Dokl. Akad. Nauk SSSR, 93, No. 4, 603-606 (1953). V.V. Bushuev, "A method of finding the stability limit of a linear system, n Izv. Vyssho Ucheb. Zaved. ~nergetika, No. 3, 18-23 (1968). Ya. V. Bykov, "Characteristic exponents of a class of systems of differential and difference equations with periodic coefficients," Teachers of the Faculty of Physics and Mathematics of Kirghiz University, Mathematics Section [in Russian], Frunze (1965), pp. 3-6. Ya. V. Bykov, "A class of systems of ordinary differential equations," Differents. Uravnen., I , No. 11~ 1449-1476 (1965). Ya. V. Bykov and V. P. Misnik, "Linear systems of differential equations with periodic coefficients," in: Investigations in Integrodifferential Equations in Khirghizia,Issue 2 [in Russian], Akad. Nauk KirgSSR, Frunze (1962), pp. 267-270. B. F. Bylov, Stability of Characteristic Exponents of Systems of Linear Differential Equations, Author's Abstract of Candidate's Dissertation, Moscow (1954). B. F. Bylov, "Upper stability of greatest characteristic exponent," DokL Akad. Nauk SSSR, 103, No. 2, 181-184 (1955). B. F. Bylov, "Upper stability of greatest characteristic exponent of a system of linear differential equations with a l m o s t p e r i o d i c coefficients," Matem. Sb., 48, No. 1, 117-128 (1959). B. F. Bylov, "Almost reducible systems of linear differential equations," Sibirsk. Matem. Zh., 3~ No. 3, 333-359 (1962). B. F. Bylov, "Almost reducibility of a system of linear differential equations possessing different characteristic exponents," Sibirsk. Matem. Zh., 4, No. 6, 1241-1262 (1963). B. F. Bylov, "Almost reducibility criterion," Sibirsk. Matem. Zh., 6, No. 1, 38-43 (1965), B. F. Bylov, "Structure of solutions of a system of linear differential equations with almost periodic coefficients," Matem. Sb., 66, No. 2, 215-229 (1965). B. F. Bylov, "Reduction of a system of linear equations to diagonal form," Matem~ Sb., 67, No. 3, 338-344 (1965). B. F. Bylov, "Time-transformation in first-order stabil~ty problems," Differents. Uravnen.~ 1, No. 9, 1149-1154 (1965). B. F. Bylov, "Generalization of P e r r o n theorem," Differents. Uravaen., 1, No. 12, 1597-1600 (1965). B. F. Bylov, "Almost reducible systems," Sibirsk. Matem. Zh., 7, No. 4, 751-784 (1966). B. F. Bylov, "Geometric location and estimate of growth of solutions of perturbed systems," Differents. Uravnen., 2, No. 7, 882-897 (1966). B. F. Bylov, "Geometric location and estimate of growth of solutions of perturbed systems," Differeats. Uravnen., 2, No. 8, 1003-1017 (1966). B. F. Bylov, Almost Reducible Systems, Author's Abstract of Candidate's Dissertation, Acad. of Sci. of the BSSR, Minsk (1966). B. F. Bylov, "Symmetry property Of the concept of almost reducibility," Differents. Uravnen., 5, No. 3, 556-559 (1969). B. F. Bylov, "Reductions to block-triangular form and necessary and sufficient stability conditions on characteristic components of a linear system of differential equations," Differents. Uravuen., 6, No. 2, 243-252 (1970). B. F. t3ylov, "Simultaneous stability of characteristic exponents of self-conjugate systems of linear differential equations," Differents. Uravnen., 6, No. 6, 943-947 (1970). B. F. ]Bylov, "Generalized regular systems," Differents. Uravnen., 7, No. 4, 575-591 (1971). I B. F. Bylov, R. E. Vinograd, D~ ~I. Grobman, and V. V. Nemytskii, Theory of Lyapunov Exponents and Its Applications to Stability Problems [in Russian], Nauka, Moscow (1966), 576 pp. B. F. Bylov and D. M. Grobmaa, "Linear inclusion principles for systems of differential equations," Usp. Matem. Nauk, 17, No. 3, 159-161 (1962).
67
105. B. F. Bylov and N. A. Izobov, "Necessary and sufficient stability conditions on characteristic exponents of a diagonal system," Differents. Uravnen., _5, No. 10, 1785-1793 (1969). 106. B. F. Bylov and N. A. Izobov, "Necessary and sufficient stability conditions on characteristic exponents of linear systems," Differents. Uravnen., 5, No. 10, 1794-1803 (1969). 107. K.G. Valeev, "Solution and characteristic exponents of solutions of given systems of linear differential equations with periodic coefficients," Prikl. Matem. i Mekh., 2__4,No. 4, 585-602 (1960). 108. K.G. Valeev, WAmethod for solving systems of linear differential equations with sinusoidal coefficients," Izv. Vyssh. Ucheb. Zaved. Radiofiz., 3, No. 6, 1113-1126 (1960). 109. K.G. Valeev, "Stability of solutions of a system of two linear first-0rder differential equations with periodic coefficients in the resonance case," Prikl. Matem. i Mekh., 25, 794-796 (1961). 110. K.G. Valeev, "Investigation into the stability of solutions ofquasi-stationarysystems of linear differential equations with almost periodic coefficients," Izv. Vyssh. Ucheb. Zaved. Radiofiz.,_5, No. 6, 1206-1219 (1962). 111. K.G. Valeev, "A generalization of the I. Z. Shtokalo operational method," Dop. Akad. Nauk UkrSSR, A, No. 1, 5-10 (1972). 112. K. G. Valeev and V. S. Yakovenko, "Finite solutions of a system of linear differential equations with periodic coefficients," Izv. Vyssh. Ucheb. Zaved. Radiofiz., 14, No. 3, 399-402 (1971). 113. Wang Tan-Chih, "Stability of the zero solution of a system of linear differential equations with variable coefficients," Vestn. Mosk. Univ., Matem., Mekh., No. 2, 86-.94 (1969). 114. Ya. F. Vizel' and T, Sabirov, "Almost periodicity of an integral of an almost periodic function and reducibility of linear systems with a l m o s t p e r i o d i c coefficients," Dokl. Akad. Nauk TadzhSSR, 15, No. 7, 11-14 (1972). 115. R . E . Vinograd, "Boundedness of solutions of regular systems of differential equations with small components," Usp. Matem. Nauk, 8, No. 1, 115-120 (1953). 116. R. ~. Vinograd, "Stability of characteristic exponents of regular systems," Dokl. Akad. Nauk SSSR, 9_.!1,No. 5, 999-1002 (1953). 117. R. E. Vinograd, "Negative solution of stability problem for characteristic exponents of regular systems," Prikl. Matem. i Mekh., 17, No. 6, 645-650 (1953). / 118. R. E. Vinograd, ~New proof of Perron theorem and certain properties of regular systems," Usp. Mate m. Nauk, 9, No. 2, 129-136 (1954). 119. R. ~. Vinograd, nAn assertion of K. P. P e r s i d s k i i in his dissertation, 'Characteristic Numbers of 9Differential Equations'," Usp. Matem. Nauk, 9, No. 2, 125-128 (1954). 120. R. ~. Vinograd, "Instability of least characteristic exponents of a regular system," Dokl. Akad. Nauk SSSR, 103, No. "4, 541-544 (1955). 121. R. E. Vinograd, "Necessary and sufficient tests of the behavior of solutions of a regular system," Matem. Sb., 38, No. 1, 23-50 (1956). 122. R. l~.. Vinograd, "Insufficiencyof the method of characteristic exponents as applied to nonlinear equations," Dokl. Akad. Nauk SSSR, 114, No. 2, 239-240 (1957). 123. R . E . Vinograd, "Central characteristic exponent of a system of differential equaiions," Matem. Sb., 42, No. 2, 207-222 (1957). 124. R.E. Vinograd, "Estimate of the jump of a characteristic exponents for small perturbations," Dokl. Akad. Nauk SSSR, 11__~4,No. 3, 459-461 (1957). 125. R.E. Vinograd, "Inapplicabilityof method of characteristic exponents for studyingnonlinear differential equations," Matem. Sb., 4i, No. 4, 431-438 (1957). 126. R. E. Vinograd, "Conjugate Lyapunovnorms, n Dokl. Akad. Nauk SSSR, 119, No. 3, 415-417 (1958). 127. R.E. Vinograd, "General stability case for characteristic exponents and existence of leading coordinates,~ Dokl. Akad. Nauk SSSR, 119, No. 4, 633-635 (1958). 128. R.E. Vinograd, "Upper limit of characteristic exponents for small perturbations," in: Papers of the 3rd All-Union Mathematical Congress, 1956 [in Russian], Vol. 4, Akad. Nauk SSSR, Moscow (1959), p. 14. / 129. R . E . Vinograd, Theory of Characteristic Lyapunov Exponents, Author's Abstract of Candidate's Dissertation, Moscow State University, Moscow (1959). 130. R . E . Vinograd, "Freezing method," Uch. Zap. Kabardino-Balkarsk. Univ., Ser. Fiz.-Matem., No. 22, 41-42 (1964). g 131. R. E. Vinograd, ~Reachability of the central exponent," Differents. Uravnen., 4, No. 7, 1212-1217 (1968). 132. R. ~.. Vinograd, "Necessary and sufficient criterion and exact first approximation stability asymptotic," Differents. Uravnen., 5, No. 5, 800-813 (1969).
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