Oscillations in Planar Dynamic Systems

„„, (<>™)

provided that — cos tb — a—— sinib = 0, (3.10) Y dt dt where xp(t) = t +
("■*■*■)

Substituting Eqs. (3.7), (3.8), and (3.11) into Eq. (3.1) gives da . dd> . — sin ib + a — cos y> = —er [acos ip, —asiarp]. dt dt

,

. (3-12)

Equations (3.10) and (3.12) are both linear in the derivatives da/dt and ad(j>/dt. Solving for them gives the following expressions: da — = — eF(acosip, — as'mip)smip

(3.13)

d<j> ft\ —r- = — ( - \F(a cos rp, — a simp) cos ip

(3-14)

i>{t)=t +
(3.15)

Note that these are the exact equations for a(t, e) and
METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

89

The derivation of the first approximation of Krylov and Bogoliubov begins with the Fourier expansion of Fsinip and Fcosip.

This can, in general, be done because

both of these functions are periodic functions of xp with period 2n. This follows from an examination of the right-sides of Eqs. (3.13) and (3.14). Therefore, oo

Fsin^> = K0(a)+

^ [Km(a) cos(mip) + Lm(a)sin(mip)},

(3.16a)

m=l oo

Fcost/> = P 0 (a) + J2 [Pm(a) cos(mi/>) + Q m (a)sin(rm/>)],

(3.16b)

m=l

where K0{a)=f^-jf Km(a)=(-j

FsinV>#,

(3.17a)

Fsimp cos(mip) dip,

(3.17b)

p0(a)=/J_j/"

F cosiP dtp,

(3.17c)

Pm(a) = [ - ) /

F cos ipcos(mip) dip,

(3.17d)

Lm(a)=(-j

F simp sin(mip) dtp,

(3.17e)

Qm(a) = ( - ) /

FcosV>sin(m^)#.

(3.17f)

Thus, Eqs. (3.13) and (3.14) can be written — = -eK0(a) dt

d4 dt

- e^2[Km(a)cos(mip)

- Q P 0 (o) - Q

+ Lm(a)sin(mip)],

f^ [Pm(a) cas(rmp) + Q m ( a ) sin(m^)].

(3.18a)

(3.18b)

The first approximation of Krylov and Bogoliubov consists of neglecting all the terms on the right-side of Eqs. (3.18) except for the first; that is — = — eKo(a) = — ( — I / dt \2irJ J0

F(acosxp,— a simp) simp dtp,

(3.19a)

90

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS ^

= -(~\p0(a)

= - ( 5 7 - ) / ' F(acosi>, -asin ij>) cos ipdrfr.

(3.19b)

A heuristic justification for the first approximation is as follows. The righthand sides of Eqs. (3.13) and (3.14) are periodic functions of ip with period 2TT. If F(w,u)

is a polynomial function of w and u, then over any interval of 2ir in ip, F

is bounded. Therefore, l - O W ,

g=0(«>,

(3.20)

and for 0 < e interval of 2ir. Averaging the right-side of Eqs. (3.13) and (3.14) over the interval 2n in ip, for which a and are taken to be constants, the Eqs. (3.19) are obtained. A rigorous justification of the first approximation of Krylov and Bogoliubov can be given [2]. It follows naturally from the general method of Krylov, Bogoliubov, and Mitropolsky which is discussed in Section 3.4. In summary, the first approximation of Krylov and Bogoliubov to the oscillatory solution of cPy

J

+y

eF (y

dy\

= { >i)'

0<£<<1

'

< 3-21)

is y(t, e) = a(t, e) cos[t + cj>(t, e)],

(3.22)

where a(t, e) and (t, e) are solutions to the following system of first order differential equations:

da

fe\

f2n „ ,

-77 = ~ ( 2 ^ ) / d<j>

~dl

[-z—)

/

nacosVs-asint/>)sinV>#,

(3.23)

F(a cos ip,-a simp) cos ip dip.

(3.24)

These two equations can be written as da — = eA1(a),

(3.25a)

dd> -£ = eBi{a).

(3.25b)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

91

The general procedure consists of solving Eq. (3.25a) for a — a(t, e) and substituting this result into Eq. (3.25b) and solving for <j> = (f>(t,e). 3.2.2 Two Special Cases Let F be a function only of y, i.e.,

F 2/

( 'f) =i?l(!/) -

(326)

For this case, the equation for a(t) is da ( e \ f2* — = - ( — ] / Fi(acosV>)sin#.

(3.27)

The integrand is an odd function of ip and thus the integral is zero. Consequently, a{t,e) = A,

(3.28)

where A is a constant, and <j>(t, e) is given by the expression 4>{t,e) = eQ.{A)t + 4>o,

(3.29)

where (f>o is a constant and ft(A) = - ( - ! — ) /

Fi(Acos!/>)cosV>#.

(3.30)

Thus, for the case where F depends only on y, the first order solution is y = Acos{[l + ett(A)]t + <j>0}.

(3.31)

This case corresponds to a conservative oscillator. The effect of the nonlinearity is seen in the fact that the frequency of the oscillation, w = 1 + efl(A), depends on the amplitude A of the motion. For the second case, let F depend only on dy/dt, i.e.,

4SH(sy

^

92

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Thus, -^ = - ( - ^ - ) / F2(-a sinVO cos i/> dip. (3.33) at \2naJ J0 If F2(v) is an even function of v, then this case reduces to that of the previous situation discussed above. If F2{v) is an odd function of v, then

^ = 0 = > t f * , e ) = *o, j - = -(Y)

j 'F2(-asin4>)smil>dxl>

(3-34) = eAx{a).

(3.35)

Real, positive solutions of Ai(a) = 0 give the amplitudes of the limit cycles. If these are denoted by o;, then the first approximation to the i-th limit cycle is yW = a< cos(i + <£„),

t = l,2,...,J,

(3.36)

where I is the total number of positive solutions to A\(a) = 0. For arbitrary initial conditions, the oscillation has a variable amplitude and a frequency aj(e) = l + 0 ( e 2 ) . 3.2.3 Stability Properties of Limit Cycles In the first approximation of Krylov and Bogoliubov, the variation of the am­ plitude with time is given by the first order differential equation da — = eA1(a).

(3.37)

Theoretically, this differential equation can always be integrated, but the integra­ tion can be quite difficult in certain cases. However, valuable information can be obtained concerning the behavior of the solution a(t) by considering the properties of Ai(a).

In the material to follow, it is assumed that e > 0.

Since the purpose of this book is to discuss techniques for constructing ap­ proximations to the solutions of differential equations modeling "physical" systems, the following assumption is generally understood to hold: There exists no positive value a = a* such that A\(a) > 0 for all a > a*. If such a value of a existed, then

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

93

selecting the initial value of the amplitude to be greater than a* would lead to an unlimited increase in the amplitude. From a physical point of view, it is expected that systems of interest be bounded in space [10]. Of course, mathematical models can be constructed that violate this requirement. However, even for these cases, the general results to follow hold. Inspection of Eq. (3.37) shows that when Ai(a) > 0 the amplitude increases, and when Ai(a) < 0, the amplitude decreases. Stationary values of the amplitude are determined by the condition Ai(o) = 0.

(3.38)

Thus if the initial amplitude, a(0) = oo, is not stationary [i.e., Ai(ao) ^ 0], then the amplitude increases monotonically if A\(a0)

> 0 and decreases monotonically

if Ai(ao) < 0. With increase in time, the amplitude will tend to a stationary value. This means that a nonstationary oscillation approaches a stationary one with the passage of time. If, for example, the function F(y,dy/dt),

in Eq. (3.1), depends only on y,

then Ai(a) is identically zero. For this case, any oscillation is stationary. Physical systems modeled by these types of functions ^

+

y =

eFl(y),

(3.39)

are called conservative and correspond to the total energy being constant [11]. A detailed discussion of conservative oscillatory systems is given in the book by Andronov, Vitt, and Khaikin [11]. Assume that A\(a) is not identically equal to zero and let a\ be one of the positive roots of Eq. (3.38), i.e., A 1 (a 1 ) = 0,

a!>0.

(3.40)

This value for a corresponds to a stationary state. An issue of major importance is the stability of this stationary state. The following discussion shows how to

94

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

determine the stability of the stationary states using a so-called linear stability analysis. To proceed, let a(t) be a solution close to a\, i.e., a{t) =

0l

(3-41)

+ j3(t),

where j3(t) is "perturbation" and satisfies the condition |/3(0)| « 1.

(3.42)

Thus, substituting Eq. (3.41) into Eq. (3.37) gives d

Jt = eAy (a, + /?) = eA1 ( a , ) + e ^

^

/? + 0(/? 2 ).

(3.43)

The linear stability equation is

Note that this result follows from the fact that Ai(a) = 0 and the neglect of higher order terms in /9. The solution to Eq. (3.44) is /?(*) = /3(0)e eRt .

(3.45)

Observe that the small initial perturbation /?(0) increases with time if R > 0, while it decreases with time if R < 0. Thus, the stationary amplitude nj has the following (linear) stability properties: a(t, e) = a! a(t, e) = a\

is linearly stable if R < 0;

(3.46)

is linearly nonstable if R > 0.

(3-47)

Equation (3.38) always has at = 0 as a root. If AAi(O) > 0, da

(3.48)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

95

then the state of equilibrium (i.e., a = 0) is unstable. If

da

< 0,

(3.49)

the state of equilibrium is stable. Equation (3.48) represents the condition for selfexcited oscillations if one or more limit cycles exist. If the condition of Eq. (3.48) holds, then a small perturbation away from equilibrium causes the amplitude to increase to finite values with increase of time. The condition of self-excitation is not essential for the existence of stable stationary oscillations. All that is required is the existence of at least one root to Aj(a) = 0 that satisfies Eq. (3.49). 3.2.4 Equivalent

Linearization

Krylov and Bogoliubov [1] developed a method of equivalent linearization in which a nonlinear differential equation, of the form Eq. (3.1), is replaced by an equivalent linear differential equation with the property that the stationary solutions of the two equations differ from each other by terms of order e2 This method and its generalization, often called the describing function method, have found important uses in the theory of control systems [13, 14]. Consider the nonlinear differential equation

= ? + ' - * (»■!)•

<"•>

In the first approximation, the solution takes the form y = acosip,

(3.51)

where o and ip satisfy the equations: j

/ , \

/•2T

—- = — ( — I / dt \2nJ J0

F(acostp, — asini/>)sinip dip

^

= «,(«),

(3.52)

(3.53)

96

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

and

Ma)] 2

1—1-—| / \2traJ Jo - ( — ) /

F(a cos tp,— a sin ip) cos ip dip F(acos ip,-a sin$) cos ip dip +

0(e2).

(3.54)

Now define two new functions, Ke(a) and A e (a), as follows: Ke{a) = 1

(3.55a)

cos ib dip,

W Jo

Ae(a)=( — j /

(3.55b)

Fsinxpdip.

Thus, the equations of the first approximation can be written "Ae(a)

da

(3.56a)

di
ue(a) =

\Ke{a)]ll\

(3.56b)

Differentiate Eq. (3.51) twice and use Eqs. (3.56) to obtain

dy

.

fX.\

(3.57)

— = — ame sin ip — I — I a cos \b and

A2\

d?y

-J-T = —au>e cos ip + \eauje sin ip + I —■ 1 a cos ip

+ (T>°^+(£)(T>!^ dy

©

du>e

fdX\ (V

ay.

(3.58)

This last equation can be rewritten to the form (3.59)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

97

where the following have been used, based on Eqs. (3.54) and (3.55b): Ae(a) = 0(e),

^

= 0(e),

^

= 0(e).

(3.60)

Thus, the first approximation of Krylov and Bogoliubov gives a solution to Eq. (3.50) that also satisfies the above linear differential equation to terms of order e 2 . Note that this is precisely the accuracy with which the first approximation determines the solution to Eq. (3.50). From this viewpoint, the first approximation to the solution of nonlinear Eq. (3.50) and the solution to the linear Eq. (3.59) are equivalent. The function \e(a) is called the equivalent coefficient of damping, while Ke(a) is the equivalent coefficient of elasticity. Comparing Eqs. (3.50) and (3.59), it is clearly seen that the latter equation can be obtained from the former one by replacing the nonlinear term with the following linear term:

eF V

( ' f ) ■" " {[K'{a) ~ 1]V + Ae(a) f } '

(36

where Ke(a) and A e (a) are given by Eqs. (3.55). In the remainder of this section, two examples are given to illustrate the method of equivalent linearization. Additional details of the procedure and examples are given in the book by Minorsky [12]. Consider the differential equation ^ Here F = -y3

+ y + ey3=0.

(3.62)

and I " cos4 ^ # = ~ ,

Ke(a) - 1 = (~)

\e(a)

= - (—\

f

cos3 $ sin ^ d$ = 0.

The equivalent linear equation corresponding to Eq. (3.62) is

g + (i + «fi,.o, )v = o,

(3.63)

(3.64)

98

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

with w.(a) = [Ke(a)\^

+ 0(e2).

= 1 + *f

(3.66)

Note that Eq. (3.62) is a conservative oscillator, hence the amplitude a(t) is a constant, i.e., a(t) = A. With this requirement the solution to Eq. (3.65) is y = A cos

(1+ !£)

t + o

(3.67)

This agrees with previous results for the perturbation solution to Eq. (3.62); see Section 2.4.3. Consider now the van der Pol oscillator equation ^

+ !/ = e ( l - y ) - .

(3.68)

For this case

and the functions Ke(a) — 1 and A e (a) are given by (1 - a 2 cos if>)(-a sin ) cos ipdip = 0,

Ke(a) - 1 = -

(3.70)

and Xt(a)=(-J

(l-a2cos2V)(-asint/))sinV'# = - e ( l - — ) .

(3.71)

Thus, the linearized equation corresponding to the van der Pol differential equation is

e 1(*

a2\

J~ { -T)i+y d?y

dv

= 0-

,

< 3 - 72 )

Stationary solutions are obtained by requiring that the coefficient of the first deriva­ tive be zero. This gives e(l-

j j

= 0 = ^ a = 2.

(3.73)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

99

Note also that the frequency, from Eq. (3.70), is w(e) = 1 + 0(e2).

(3.74)

Thus, the stationary solution of both Eqs. (3.68) and (3.72) is y = 2 cos(t+

„).

(3.75)

3.3 Worked E x a m p l e s Using the M e t h o d of Krylov and Bogoliubov 3.3.1 Example A Let F = — y 2 , then the corresponding differential equation is ^

+ y + ey2=0.

(3.76)

Applying the results of Section 3.2.2 gives

and a(t,e)

= A,

(t,e) = 0,

(3.78)

where A and o are constants. Consequently, the solution of Eq. (3.76), using the first approximation of Krylov and Bogoliubov, is y = Acos(i + 0).

(3.79)

Thus, the first approximation gives exactly the same solution as the linear equation obtained by letting e = 0. The amplitude is constant and the frequency is 1, that is, co = 1.

100

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

3.3.2 Example B Consider the differential equation

§ + , + .(§)'-.. For this case F = —(dy/dt)2.

(,80)

In the first approximation

and the first approximation solution is y = Acos(t + 4>0),

(3.82)

where A and 0 are constants. This is exactly the same-result obtained in Exam­ ple A. Thus, when the nonlinear function is either — y2 or —(dy/dt)2

or a linear

combination of them, the first approximation is the same as in the linear case, i.e., e = 0. The effect of the nonlinearity therefore shows up only in higher order approximations to the solution. 3.3.3 Example C The conservative differential equation

-J. + y + ey 3 = o

(3.83)

has F = —y3. For this case a(t) = A = constant,

(3.84)

and 4>(t) is determined by the differential equation

Solving for gives

(t,e)=(^-\t

+ 0.

(3.86)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

101

Therefore, to first approximation, the solution to Eq. (3.83) is y = A cos

3eA2N

[('♦¥) t +

0

(3.87)

3.3.4 Example D The linear damped oscillator is modeled by the following differential equation
(3.88)

where F = —2dy/dt. The equations determining the phase and amplitude are

d dt

(3.89)

= 0,

da

fe\ f2*

-^ = — I — I / The solutions are

.a

(3.90)

a sin w #

<*< w . A > (t,e) = tQ,

Ae-et.

a(t,e) =

(3.91)

Therefore, the first approximation of Krylov and Bogoliubov gives the following solution to Eq. (3.88): y = Ae~et cos(t + <j>0).

(3.92)

This is to be compared to the exact solution y = Ae

tl

cos

[(*

i

.2 \ 1/2

t + to

(3.93)

I =^

(3.94)

-J

3.3.5 Example E An oscillator with nonlinear damping is


dy dt

d

corresponding to dy dy F == — dt dt'

(3.95)

102

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

For this case dcf>

(3.96)

— = 0 = ^ ( t , e ) = ^o, and _2\

da

r2w

—— I / K2* J Jo

~dl

= — I —— 1

/

| sin ^ | sin2 i/> dtp sin 3 ip di/> — /

sin 3 ip dxj)

(3.97) 3TT

This last equation can be integrated to give a0

a(t,e) =

(3.98)

Thus, the first approximation to the solution to Eq. (3.94) is _ a 0 cos(t + p)

(3.99)

It is of interest to compare this solution for F given by Eq. (3.95) to that of —(dy/dt)2.

Example B for which F = 3.3.6 Example F

For Coulomb damping [15], the equation of motion is

*y , —

+ y

„

fdy\

= -eCsgn(-

,

C>0,

(3.100)

where

— (I)'

(3.101)

+1, -1,

(3.102)

and sgn(x):

for x > 0, for x < 0.

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

103

The phase (i, e) is constant for this system and the amplitude equation is f2nI [-C sgn(-a simp)] simp dip

da ( t \ ~dt~~~\2~)

eC\ f2" — ) / sgn(—a simp) sin ip dip 27I

7 Jo

I*2TV

I J0

2n

--a?)-

f2n

sin ip dip — I Jn

sin ip dip (3.103)

if a / 0;

and da

if a = 0.

(3.104)

|<,

(3.105)

Integrating Eq. (3.103), for a ^ 0, gives a{t) = a0 - I

a 0 > 0.

The motion will continue as long as a(t) > 0 and will cease for time fi = 7ra 0 /2eC. Thus, the oscillation lasts for a finite time and the solution is given by the expression

y(*,0

(2eC a0 - [ it cos(i + <j>0),

(3.106a)

irao 2lC'

(3.106b)

for 0 < t < and

tart> wa0 2lC'

y(t,e) = 0,

(3.107)

3.3.7 Example G The nonlinear oscillator differential equation dy

cPy

^

+ y

+

= 0,

€

{dt+ar

(3.108)

has the function F equal to the expression

dt

y

'

(3.109)

104

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

where a is a constant of order one. The amplitude and phase equations are da fea\ = ~dt ~\2J'

(3.110)

d(j> 3eaa2 ~dl ~ 8

(3.111)

and

The corresponding solutions to these equations are a(t,e) =

Ae-et?2,

et 0-(^f-y- ,

(3.112) (3.113)

where A and o are constants. Thus, in the first approximation of Krylov and Bogoliubov, Eq. (3.108) has the solution y = Ae~^2

cos t -

(3aA2^ f ^ p V ' * +
(3.114)

Observe that the frequency depends on the time, i.e., (3.115)

3.3.8 Example H The last example is the van der Pol equation -nr2 + y = dt

e(l-y2)—, dt

(3.116)

where

'-<>-.■£■

(3.117)

The amplitude and phase satisfy the equations -37=(—J

/

a(l — a2 cos2 xj>) sin2 ij> dip,

(3.118)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

105

-jr = ( — ) / (1 — a cos2 ip)sinip COST/' d%l>. dt \2nJ J0 The integrals can be easily evaluated to give

Hf)K).

(3.119)

<»2»>

dd>

(3-121>

i = °-

The solution to Eq. (3.121) is (j>{t,t) = <j>0. Multiplying Eq. (3.120) by 2a and denning z = a2, the following equation is obtained dz ez

(l - £) .

(3.122)

This is a separable equation whose solution is

•-.+(«£-.)•

(3123)

where z0 = [a(0)] 2 = A 2 . Therefore, the first approximation to the solution of the van der Pol equation, according to the method of Krylov and Bogoliubov, is ^ 0 e 6 t / 2 c o s ( t +
e(

[l+(f)(e -l)]

1/2"

(3.124)

This function represents an analytic approximation to the unique limit cycle of the van der Pol equation. Examination of Eq. (3.124) shows that for any initial conditions, the solution y(t, e) approaches the function y = 2 cos(i + <^o), i.e., the function of Eq. (3.124) has the property Limy = 2cos(t + 0o)

(3.125)

t—►OO

where in Eq. (3.124) 0 < A0 < oo,

0 < (j>a < 2TT.

(3.126)

106

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

In more detail, if the initial amplitude is zero, i.e., Ao = 0, then a(t) = 0 for all t, and y(t) = 0; this is the equilibrium solution of the van der Pol equation. From Eq. (3.120), it follows that the solution a(t) = 2 is (linearly) stable since ^ ( o ) = ( f ) ( l - j ) ,

^

}

(3-127.)

= -e<0.

(3.127b)

da (See the discussion of Section 3.2.3.) Likewise, the equilibrium solution is (linearly) unstable since ^ l M da

=

l

> 0

.

(3.127c)

2

In summary, the stationary periodic solution of the van der Pol equation does not depend on the initial conditions. Its functional form depends uniquely on the parameters of the differential equation itself. In the (y, dy/dt)

phase plane, this

periodic solution or limit cycle corresponds to a closed curve. If 0 < e
S+^^'D'

°<£<
(3i28)

In the course of improving the first approximation, Krylov and Bogoliubov devel­ oped a technique for determining the solution to any order in e. The technique was then generalized and justified mathematically by Bogoliubov and Mitropolsky [2]. This section gives the details of this method applied to Eq. (3.128) and its extension

§+,_.„(, *) + , 0 (,*) where both F and G are polynomial functions of their arguments.

(,129)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY 3.4.1 y + y =

107

eF(y,y)

Assume that the solution to Eq. (3.128) takes the form [2] y = acosip + eu1(a,ip) + e2u2(a,ip) -\

,

(3.130)

where the Ui(a, $) are periodic functions of , with period 2n, and the quantities a and ij> are functions of time defined by the following equations: da — =eAi(a)

« + (2A2(a)---,

(3.131)

dib _ -^ = 1 + eB^a) + e2B2{a) ■ ■ ■.

(3.132)

Note that the functions Ai(a) and B,(a) depend only on the ''amplitude" a. The functions u;(a), A,(a), and B,-(a) are to be chosen in such a way that Eq. (3.130), after replacing a and ip by the functions defined in Eqs. (3.131) and (3.132), is a solution of Eq. (3.128). As soon as explicit expressions for the coef­ ficients on the right-side of Eqs. (3.131) and (3.132) are obtained, the problem of finding a solution to Eq. (3.128) is reduced to that of integrating Eqs. (3.131) and (3.132). These equations have separable variables and if the Ai(a) and JB,(O) are polynomial functions of a, then the integration of Eqs. (3.131) and (3.132) can be done using standard integration techniques. However, if the Ai(a) are polynomial functions of order greater than two, the resulting integrations do not lead generally to solutions for a(t) that are expressible in terms of the elementary functions. In any case, once a(t) is obtained, it can be substituted into Eq. (3.132) to give i/>(t). In general, the determination of the coefficients of the expansions in Eqs. (3.130), (3.131), and (3.132) do not present any theoretical difficulties. How­ ever, the formulas for the coefficients rapidly become complex, and in practice only a limited number of terms are used. To terms of order em, these expansions are y = acosrp + eu1(a,^)

+

\- tmum(a,ip),

(3.133)

108

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS % = eAi(a) + e2A2{a) + ■■■ + e m A m ( a ) , at

(3.134)

= 1 + efl,(a) + e2B2(a) + ■■■ + tmBm(a),

(3.135)

5 at where m = 1,2,

The practical applicability of these procedures is not deter­

mined by the convergence (or nonconvergence) of the series given by Eqs. (3.133), (3.134), and (3.135) a s m - t o o , but by their asymptotic properties for a given fixed value of m when e -> 0. Thus, Eqs. (3.130), (3.131), and (3.132) are to be treated as formal expansions necessary for determining the asymptotic expressions given by Eqs. (3.133), (3.134), and (3.135). There will be an arbitrariness in the definitions of the functions Ui(a,ip), since no restrictions are placed on choosing the functions Ak(a) and Bk(a) that generate the functions Ui(a,ip) [17]. To remove the arbitrariness, the following additional conditions are imposed:

/o

U{(a, ip)s'mtpdTp = 0,

/

Ui(a, ip) cos xp dip = 0,

(3.136) (3.137)

Jo for i — 1,2, . . . , m .

These conditions remove the fundamental harmonic in the

functions Ui(a,tj)) and, in addition, guarantee the absence of secular terms in the successive approximations. Therefore, a(t) is the full amplitude of the fundamental harmonic of the oscillation. The remainder of this section gives the procedures for determining the u;(a, t/>), A;(a), and Bi(a). Let H(a,tp)

be a function of the variables a and ip. The first and second

derivatives of H(a,i{>), with respect to time t, are

dH(a,\l>) _ da dH dtp dH Jr It ~~dl~da~ ltlhp'

(3 138)

-

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

109

and cPH(a,rj>) _<PadH dt2 dt2 da

(PjP_dH_ (da\2d2l \dt) da2 dt2 dtp

■ dt»**££+(*) dt dadip \dtj dip S2

Therefore, the first and second derivatives of y are dy _ da f dt dt \ +

du\ da
2du2

e

\ da ) dux , du2 \ - l + *-± + ..ji

(3.140)

and

fy

(fa (

,

dui

-,du2

\

__._,,

dui

2du2

2

dt

cPtp (

/da\ f d2u1 \dt) \e~daT da dip I dt dt \ + 1^)

(_aCos^

d2u\ dadip ^

+ e

2

+ e

d2u2 dadip ^

+ ...j

(3.141)

Using Eqs. (3.134) and (3.135), the following expressions are obtained: i\ da Y ^ cPa d fda\ da d /da^ t)~dt^e lit2 ~ dlVdi) ~ 'didaVdtj 2

da

n=l

'

= t

ndAn

A ^ + 0{e%

(3.142)

(Pip d fdip\ da d fdip\ da s—^ ndBn H2 ~ dt VdTj ~ dida\dt) ~ di ^ e "da" x

, 2i ,,

v

^ 1

= e A1—-

, / , / .3 3

+ 0(e ),

'

n=l

(3.143)

110

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS 2

(3.144)

(S) -A«+<*>), da dip = e ^ 1 + e 2 ( ^ 2 + ^ 1 B 1 ) + 0(e3), ~di~dt

(3.145)

2

= l + 2 e B 1 + e 2 ( B 1 2 + 2B 2 ) + 0 ( e 3 ) .

f^)

(3.146)

If Eqs. (3.142) to (3.146) are substituted into Eqs. (3.140) and (3.141), and terms of the same power of e are collected together, then y and dy/dt take the forms: dy . , / , . , dux —- = — a sin ip + e [ Ax cos w — aBx sin w + -=— dt \ dtp 2 + e

^ 2 cos^a5 2 sin^ + ^ —

+

Bl

— +

— j

+ 0(« 3 ),

(3.147)

and -j-j = - a cos ip + e ( - 2 A i sin i/> - 2 a S ! cos ip + —j- 1

+ e< + 2Ai

dAX dBt Ax - ~ - aB{ - 2aB2 ) cosip - [ 2A2 + 2AiB] + Aj - p a J sinip 2 d2 ux nn d ux + 2B dadip " dtp2

d2 « 2 dip2

+

0(e3).

(3.148)

With these results, the left-side of Eq. (3.128) can be written as -— + y = e( -2 Ax sin ip - 2aBx cos ip + —-~ + ux \ + e<

Ax —- - aB\ - 2aB2 ) cos ip

- (2A2+2AXBX 4-2A

d Ul

'

, ,

+Ax—±a) a R

V

smip

a V ^

+ 0(e3).

(3.149)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

111

Similarly, the right-side of Eq. (3.128) can be rewritten to the form:

(

d

v\

eF I y, — 1 = eF(acostp,— asmrp) + e

uiFy(a cos ip, —a sin ip)

+ I Ax cosip — aBx sinxp + -jry ) Fy>(acostp, — ashn/>) + 0(e3),

(3.150)

where the following notation is used: F , ( , , « )

S

^ M

* , ( , , » ) = «*&!£>.

(3.151)

Substituting the results of Eqs. (3.149) and (3.150) into Eq. (3.128), collecting together the terms with like powers of e, and setting them to zero, gives d2ux

dV>2 dip2

+ ui = -F0(a,i/>)-|-2J41sini/> + 2a.B1cosV>,

(3.152)

+ u 2 = Jri(a,V>) + 2 A 2 s i n ^ - | - 2 a B 2 C o s ^ ,

(3.153)

d2 u -g-j^- + um = Fm-1{a,il>) + 2Amsmip

+ 2aB,ncosi>,

(3.154)

where F0(a,ip) = F(a cos xp,—asm ip),

Fi(a,rp) = uiFy+

(3.155)

lAi cosip - aB\ simp + -=— j Fy>

+ (aBf-Ai

— M cosi/> + (zA^Bx

+ Aj — ^ - a j sinV>

In the above two equations, F y and F y ' , stand for the functions =

aFtacos^-asin^ Of

112

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS p , = d-F(acosV>,-asin^)

(3.157b)

dw See Eq. (3.151). It is clear that the functions JFjt(a, VO are periodic functions of the variable ip with period 2ir. They also depend on the amplitude a. The explicit expression for Fk(a,if>) is determined as soon as the functions A{(a), #;(«), and Ui(a,rp) are known to the k-th order. The functions A1(a),

B\(a),

and ui(a,ip)

can be found by doing a series of

calculations. First, expand F0(a,ip) and U\{a,tp) into Fourier series: oo

F0(a, ip) = g0(a) + ^[gn(a)

cos(nip) + ft„(a) sin(m/>)],

(3.158)

n=l oo

ui(a, ij>) = vo(a) + 2_][vn(a) cos(nrp) + wn(a) sm(nif>)].

(3.159)

71=1

Note that F0(a,xj>) is a known function, see Eq. (3.155), and, consequently, its Fourier coefficients gn(a) and hn(a) can be determined. Substituting Eqs. (3.158) and (3.159) into Eq. (3.152) gives OO

vo(a) + / ^ ( l — n )[vn(a) cos{ntj)) + wn(a)sm(nip)]

= go(a) + [gi(a) + 2aB\] COST/J

n=l oo

+ ^ (a) + 2A1]smip + ^[ff„(a)cos(raV) +

ft„(a)sin(nV)]-

(3.160)

n=2

Imposing the conditions of Eqs. (3.136) and (3.137) gives t>i(a) = 0,

uii(a) = 0.

(3.161)

Equating coefficients of the harmonics of the same order in Eq. (3.160) gives Ma)

= 9o(a),

vn(a) = £&,

(3.162a) (3.162b)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY hn(a) 1-n2'

!>n(«0

Bi(a) = -

113 (3.163)

hi(a) 2 '

(3.164)

gi(g)

(3.165)

2a

Therefore,

Ui(a,V>) = ffo(a) +

0„(a) cos(nxf>) + hn(a) sin(nip) 1-n2

^

(3.166)

/ 1 \ f2"

(3.167)

Ai(a) = — ( 2— 1 / F(acosxp,—a sin ip) sin ip dip, V *"/ Jo / i \ A2* 2?i(a) = — I - — I / F(acosrp,—asinip)cosrp dip, \2-naJ J0

(3.168)

where

M

"-G)jf

F ( a cos i/), —a sin ip) cos(nip)dip,

(3.169a)

F(acostp, —asinip) sin(nip)dip.

(3.169b)

Now that ui(a,ip), A\(a), and Bi(a) have been calculated, the function

F\(a,ip)

from Eq. (3.156) can be determined. Expanding Fi(a,ip) in a Fourier series gives oo

Fi(a, *) = ff^(a) + £ X > ( a )

cos

("
sin(m/>)].

(3.170)

n=l

Likewise, expanding u2(a, ip) in a Fourier series, substituting it and Eq. (3.170) into Eq. (3.153), and imposing the conditions of Eqs. (3.136) and (3.137), gives

M°) =

'

2a

-

(3.172)

o„ (a) cos(n*/>) + A„ (a) sin(n^) 1-n2

(3.173)

*(«) = u,(a,rP) = g{01)(a) + Y/

(3.171)

2

114

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

In detail, the functions A2(a) and #2(0) are given by the expressions:

/•2JT

V2W /

B2(a) = _

-[-

( d b ) i

where Alt

UlF

v+

( A! cos ip-aB!

simp + - ^ )

Fy, smipdiP,

(3.174)

\ a ) da

[ « 1 ^ + ( ^ c o s V - a 5 1 s i n V . + ^ ) ^ costp dip,

(3.175)

Bi, Ui, i ^ , and Fyi are given, respectively, by Eqs. (3.167), (3.168),

(3.166), (3.157a), and (3.157b). While the evaluation of higher approximations is complicated from an algebraic viewpoint, the method is straightforward. Thus, the evaluation of Ui(a,ip),

Ai(a),

and Bi(a) can be done systematically for any value of i. In summary, the first approximation to Eqs. (3.1) and (3.128), according to the Krylov-Bogoliubov-Mitropolsky asymptotic method, (see Problem 3.9) is [2] y = acosip, da

~dl dip ~dt

eAi(a),

1 + eB^a),

(3.176a) (3.176b) (3.176c)

where Ai(a) and JE?i(a) are given by Eqs. (3.167) and (3.168). This is exactly the same as the first approximation of Krylov and Bogoliubov given in Section 3.2. Likewise, the second approximation is y = a cos ip + eu\ (a,ip),

(3.177a)

j

— = eA1(a) + ^

e2A2(a),

= ! + €£?! (a) + e 2 B 2 (a).

(3.177b) (3.177c)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

115

In general, the m-th order approximation is given by the following system of equa­ tions: y = a cos t/> + eui(a, ) + • • ■ + e m _ 1 u m _ 1 ( a , i/>),

(3.178a)

-^ = eA^a) + e2A2(a) + ■■■ + emAm(a),

(3.178b)

-^ = 1 + eBi(a) + e2B2(a) + ■•■ + emBm(a).

(3.178c)

3.4.2 y + y = eF{y,y) +

e2G{y,y)

This section generalizes the method of Krylov, Bogoliubov, and Mitropolsky to nonlinear oscillatory differential equations having the form

*y2 +V = *U%)+*G(y*). dt ' " V dtJ \ dt.

(3.179)

The right-side of Eq. (3.179) contains a nonlinear term of order e2 There do exist nonlinear dynamic systems that can be modeled by an equation having this form. An example is the light variation of certain pulsating stars [18]. To begin, the solution to terms of order e2 is written as y = a cos t/> + e U l (o, i>) + e2u2{a, I/J) + 0{e3),

(3.180)

where da

dtfc dt

=eA1(a)

+ e2A2(a),

= l + eBl(a) + e2B2{a).

(3.181) (3.182)

The functions u-i(a,ip) and u2(a,ip) satisfy the equations: r\2

- ^ + ui = F0(a, xp) + 24i sin ip + 1aBx cos ip, dtp2

(3.183a)

r*2

1 4. u2 = F\(a,tp) + G{acostp, —asin*/>) + 2^2 sin V> + 2ai?2 cost/>,

di/>2

(3.183b)

116

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

where F0(a,ip) and Fx(a,ip) are given by Eqs. (3.155) and (3.156). The Fourier ex­ pansion of F0(a, ip) is given by Eq. (3.158). In terms of the coefficients of the Fourier expansion of F0(a, ), the function «i(a, xj>) is given by Eq. (3.166). Likewise, Ai(a) and B\{a) are determined by Eqs. (3.167) and (3.168). Define Fi(a,tp) to be Fi(a,»/>) = Jpi(a,^) + G(acos */>,-a sin ),

(3.184)

where F^(a,xp) is given by Eq. (3.156). Since G(a cos «/>,—a sin ), ui(a, r/>), -Ai(a), and Bi(a) are known, the Fourier expansion of F\(a,ip) can be determined. Write it as CO

A(«,) = ^ ( a ) + X ) [^ x ) («)coe(n^) + /#>(a)sin(m/,)]

(3.185)

n=2

The functions ^ ( a ) and £2(1) of Eqs. (3.181) and (3.182) are given by the relations A a (a) = - ^ ) ,

(3.186)

% ) = -

(3-187)

^

,

and U2(a; VO ' s

u2(a,V) = 5o1V) + X^

r< =7n ( i )(a) /-i

^ . / A +, /i„ 1(1) (a) sin(rn/>) cos(nt/i) 2 1-n

(3.188)

This completes the task of calculating a uniformly valid solution to Eq. (3.179) correct to terms of order e2. It should be kept in mind that to this order of e, the u 2 (a,V0 term is not retained in Eq. (3.180). (See Problem 3.6 and the above discussion after Eqs. (3.176).)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

117

3.5 Worked Examples Using t h e M e t h o d of Krylov, Bogoliubov, and Mitropolsky 3.5.1 Example A Consider the linear damped oscillator; its equation of motion is g

+ , = -2e|.

(3.189)

In Section 3.3.4, it was found that Ax(a) = -a,

5 j ( a ) = 0.

(3.190)

The function F(a cos t/>, — asinip) is F = 2asin,

(3.191)

and the corresponding Fourier coefficients are ff„(o) = 0,

n = 0,1,2,...,

/ii(a) = 2a, hn(a) = 0,

(3.192a) (3.192b)

n = 2,3,4,....

(3.192c)

Putting these results into Eq. (3.166) gives u1(a,rp) = 0.

(3.193)

Next Ai(a) and i?2(a) can be calculated from Eqs. (3.171) and (3.172); they are A2(a) = 0,

52(a) = - Q ) .

(3.194)

Thus, the second approximation equations are y = acosip, da = —ea,

~dl

~dt

-f

(3.195a) (3.195b) (3.195c)

118

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

The latter two equations have solutions a(t) = Ae~(\

(3.196a)

m=(l-j)t

+ Tfio,

(3.196b)

where A and tpo are arbitrary constants. Putting these results in Eq. (3.195a) gives the solution y = At

cos

K)

(3.197)

which is to be compared to the exact solution of Eq. (3.189) y = Ae~et cos [(1 -
(3.198)

Comparing Eqs. (3.197) and' (3.198) shows that they have the same amplitude and damping factor, and the frequencies agree with each other to terms of order e2 3.5.2 Example B The differential equation ^f- + y + ey2 = 0,

(3.199)

F = -y2 = -a2 cos2 ^ = - (y J - (~ j cos 2T/>.

(3.200)

has

The Fourier coefficients are

<7o(a) = - ( y J ,

Si(0) = 0,

g2(a) = ~(~\

(3.201a)

gn(a) = 0,

n = 3,4,5,...,

(3.201b)

hn(a) = 0,

n= 1 , 2 , 3 , . . . .

(3.201c)

It was found in Section 3.3.1 that A1(a) = 0,

B,(a) = 0.

(3.202)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

119

Putting these results in Eq. (3.166) gives Ul(a,V)

= - f y j + (~\

(3.203)

cos2^.

Substituting Eqs. (3.202) and (3.203) into Eqs. (3.171) and (3.172) gives A a (a) = 0,

B2(a) =

(3.204)

- ( ^ .

Thus, in the second approximation (3.205a)

y = a cos ij> + e I — 1 [cos 2i/» — 3], da = 0, dt dip _ dt~ _

(3.205b)

5e 2 a 2 ~ 12

(3.205c)

Solving the second and third of these equations, and substituting their results in Eq. (3.205a) gives y = A cos ( Y ) {

C O S

5e2A2 2 ( l - ^ ) i

+ 2^o

-3}

(3.206)

where A and ^0 a r e integration constants. 3.5.3 Example C For the van der Pol equation cPy dt2+y

=

„2^dV



(3.207)

the function F(a cos i/>, —o sin ip) is iT1 = (1 — a 2 cos2 tl>){—asinip) = —all

— I smijj + I — I sin3t/>.

(3.208)

120

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Its Fourier coefficients are gn(a)

= 0,

n = 0,1,2,...,

(3.209a)

A,(a) = - a f1 - T ) '

(3 2 9b)

h2(a) = 0,

(3.209c)

h3(a) = j ,

(3.209d)

hn(a) = 0,

'°

n = 4,5,6,....

(3.209e)

Thus, Aj(a), Bi(a), A 2 (a), 5 2 ( a ) , and u-i(a,tp) are given by the expressions

M«) = (|) (l " x ) '

A

^°) = °>

B2(a) = - Q ) ( l - a 2 + | Q ,

B,(a) = 0,

«1(a^) = -(^)sin3V'.

( 3 - 210a ) (3.210b) (3.210c)

In the second approximation, the solution to the van der Pol differential equation is y = acosV>-ef^jsin3V>, where da

~di

f)K).

2-i-G)(i-,+£)-

(3.211)

(3.212)

(»">

The solution for the amplitude a(t) is given by Eq. (3.123), i.e a{t)

= r- T^rrz r t ,,,!/,. [l + (f)(e -l)]

(3-214)

where A = a(0) is a constant. Note that since Lim a(i) = 2,

t->oo

(3.215)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

121

for large values of t, y(t) approaches the function y(t) where y(t) = 2 cos

I^H-tCHK1-^)

3 ( 1 - — ) t + 3V-o

(3.216)

3.5.4 Example D [19] The following differential equation arises in the modeling of certain phenomena in the oscillation of the light output of stars [18]:
2

)'
ea

+'&)

+

c-

[©

2^dV

(it

y2

(3.217)

0<e
(3.218)

where the parameters satisfy the conditions a = 0(1),

This is a generahzed form of the van der Pol differential equation [19], and, in fact, reduces to the van der Pol equation when a = 0. Observe that Eq. (3.217) has a nonlinear term that contains expressions of orders e and e2. Comparison with Eq. (3.179) gives

Fly,

2

dy_ dt

> . i->\

y^fi

, /

M

_

2)^/

+ ilr -* ( 1

<*f)-(x' '^'+

2 /

(3.219)

(3.220)

dt

A rather long, but direct, calculation gives

* - S - T

(3.221a)

A a (o) = 0,

(3.221b)

B2(a) = - ( f

S i ( o ) = 0,

l + (13a 2 - l ) a 2 +

32

(3.221c)

122

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS „3^

ui(a,ip) = aa2 - [ ^ J sinSV1-

(3.221d)

The equations for determining the amplitude and phase are da dt

£-'-'-

(3.222)

-T). 1 +

d8^-iy + (iy

(3.223)

The solutions to Eq. (3.222) is provided by Eq. (3.214). This function can be substituted into Eq. (3.223), which can then be integrated to give ip(t). However, the large time behavior of the solution y can be obtained by replacing a(t) by its asymptotic value 2; see Eq. (3.215). Doing this gives, for large t, the results y(t) = 2 cost/) + e 4a

cos(3V>)

14a\ . , „ 7 s /51-16a2\ ,„ T. — J sin(2) + ^ ~ j cos(3t/>)

+

©

(3.224)

sin(4i/>) + ( — ) cos(5V>)

where

$ = l-

V

(3.225)

13a - ^ 2

Examination of Eq. (3.222) leads to the conclusion that the original differential equation has a unique, stable limit cycle. Observe that Eq. (3.217) has, in the second approximation, exactly the same amplitude equation as the van der Pol oscillator, i.e., Eq. (3.222). It is of interest to find the class of generalized van der Pol differential equations that have this prop­ erty, i.e., determine functions f{y,dy/dt)

and g(y,dy/dt)

such that the nonlinear

equation

fy dt2

+ y

*-*%+'(>■$)]+«(>.%)•

<»»>

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

123

has the amplitude equation

3.6 Spurious Limit Cycles The van der Pol differential equation [3]

is known to have a single, stable limit cycle [20]. Now consider the modified van der Pol equation fl„
r

. A,1

-^3 + (l-y2)|

(3.229)

where e > 0 and /3 > 0. The application of the Lienard-Levinson-Smith theorem (see Appendix G, Section 2) leads to the following conclusion: The above modified van der Pol differential equation has a single, stable limit cycle for all values of e and j3 such that e > 0 and ft > 0. Now assume that the parameters e and j3, in Eq. (3.229), satisfy the conditions 0 < e < 1,

p = 0(1).

(3.230)

A first approximation to the solution to Eq. (3.229) can be found by using the method of Krylov and Bogoliubov. Doing this gives the following results:

S-•*<«>-■©('-?)•

(3 231

f = 1 +
(3-232)

Examination of these two equations leads to the following conclusion:

- >

124

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS In the first approximation, the modified van der Pol equation has a single, stable limit cycle. The amplitude of the limit cycle is (to terms of order e) equal to 2 in value, and the frequency is (to terms of order e) equal to

w= 1 +

3/?

(3.233)

These results are in agreement with the conclusions reached by means of the Lienard-Levinson-Smith theorem. In the second approximation, a direct calculation gives A2(a)

B2{a) = -

«i(o,^)

16

3+

©"

(3.234)

1 . 0 . + + ( 3 532^ l . «

©

(3.235) (3.236)

(/?cos3V> — sin3V>)-

Therefore, to this level of approximation, the variation of the amplitude with time is given by da

~dl

= eAi(a) + e2 A2{a)

2 , 1 16

•+(T)'

i-G . - * v +

196,9 64

4

(3.237)

Let z = a 2 , then Eq. (3.237) becomes


'"© - ^ ' + ( ^ 1 -

(3.238)

The positive solutions of Eq. (3.238) give the limit cycles in the second approxima­ tion.

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

125

At this point, a problem arises. To see this, consider the following cubic equation:

■[e-SO-'-GX'-^H

0.

(3.239)

The root (3.240) corresponds to an unstable limit point. The other two roots are located at

^ i j■G t e M d - 3 * ) * l-44s+

1/2

(jjx2

(3.241)

where

* = f>0.

(3.242)

Using the expansion /Q\

I i/2

(3.243) l-44i+(-jx2

= 1 - 22x +

0{x2),

the two roots take the following form for small x: z2 = 4 + 0(x),

'Z3

=

T9-x-l9+0^-

(3.244) (3.245)

Figure 3.6 gives the behavior of these two roots as a function of x for small x. In addition, it shows a plot of A as a function of z, where A is A = ez

-@(-¥>+(^>

(3.246)

These plots show that in the second approximation, see Eq. (3.237) or Eq. (3.228), the modified van der Pol differential equation has two limit cycles: a stable limit cycle with amplitude z2 and an unstable limit cycle having amplitude z3. But, ac­ cording to the Lienard-Levinson-Smith theorem, the modified van der Pol equation can have only one stable limit and no others! How is this paradox to be resolved?

126

(a)

(b)

Figure 3.6. (a) Plots of z2 and z3 as functions of x for x small, (b) Plot of A versus z.

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

127

First, observe that z2 = 4 + O(e) and corresponds to the known limit cycle of am­ plitude a = 2 + 0(e). This result was obtained in both the Lindstedt-Poincare and Krylov-Bogoliubov methods (see Sections 2.4.4 and 3.3.8). Second, note that z 3 has the following properties:

*=(w)l + 0(1) > Limz 3 = oo.

(3 24?)

-

(3.248)

£-1-0

Consequently, for small e, z 3 is very large. The limit cycle corresponding to this root will be called a spurious limit cycle. It appears in the second approximation of the method of Krylov-Bogoliubov-Mitropolsky, but, is an artifact of the method, i.e., it does not correspond to an actual hmit cycle of the modified van der Pol differential equation. The mathematical reason for the appearance of the spurious limit is the fact that A2{a) is a fifth order polynomial in a, while Ai(a) is a cubic polynomial. Consequently, the amplitude equation ^ = eA1{a) + eA2(a) at

(3.249)

has more stationary solutions than the first order result ^

= eA,(a).

(3.250)

The next higher order calculation for the amplitude equation takes the form ^

= eA^a) + e2A2(a) + t3A3{a).

(3.251)

at It can be expected that the polynomial order of Az{a) is larger than that of ^ ( a ) Thus, additional spurious limit cycles will exist. In the "limit" where all the con­ tributions are included, i.e.,

128

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

the expectation is that all of the spurious limit cycles will disappear and only the actual one, located at a = 2 + 0(e), will remain. The above calculations and discussion lead to the following conclusions: The application of the method of Krylov-Bogoliubov-Mitropolsky to the calculation of the periodic solutions of

<3 - 253 >

S+»-"(».S)-

may lead to spurious limit cycles. These are limit cycles whose amplitudes have the following behavior for 0 < e
(3.254)

where k and 7 are constants, with 7 > 0. For any finite order of the calculation, they should be ignored. The actual limit cycles have finite limits as e —» 0. 3.7 y +

y3=eF(y,y)

Mickens and Oyedeji [20] studied a new class of nonlinear oscillator differential equation, namely g

+ y3 = ^ ( y , | ) ,

0<e«l,

where F is a polynomial function of its arguments.

(3.255)

Just as for the differential

equation
^(

dy\

Eq. (3.255) may possess limit cycles. Mickens and Oyedeji obtained approximations to the solutions of Eq. (3.255) by extending the method of Krylov and Bogoliubov. To begin, assume that the exact solution to Eq. (3.255) takes the following form y(t) = A(t)co8[wt+ #*)],

(3.257)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

129

where, for the moment, A(t) and <^(t) are unknown functions of time and u> is an (unspecified) constant. Taking the derivative of Eq. (3.257) and letting ip = ut + <j>(t),

(3.258)

dy dA d<j> — = — LjAsmip + — cosi/> — A — sinV", at dt dt

(3.259)

and requiring that - ^ = -wAshu/>, dt

(3.260)

—- cosi/>- A-j- sin«/> = 0. dt dt

(3.261)

gives the relation

The second derivative is tPy

dA

.

d

i

2 A

i

m «™>

-7-5- = — u> —— siny> — u A -7- cos y> — w Acos^>. dt1

dt

(3.262)

dt

Substitution of Eqs. (3.257) and (3.262) into Eq. (3.255) gives dA . , d+ I —— cos y> dt dt \ 4u ) + (—) cos 3xp - ( - ) F(A cos ip, -LJA sin ip).

(3.263)

The Eqs. (3.261) and (3.263) are two linear relations for dA/dt and d/dt; solving them gives the following expressions for these derivatives: dA , . , {3A3\ . . . A — = — u! A cos t/> sin rp + I —— I cos tp sin ip dt + (—)

cos3V-sint/> - (-)Fsin«/>,

(3.264)

u>

A =

T

(izr ~ wA ) cos2 ^ + (z~)cos 3^cos ^ ~ («) F cos ^'

where F = F(ACOSJ/J, — uiAsmip).

(3 265)

'

Thus far these are exact expressions for the

derivatives. To obtain the equations needed to calculate an approximation to the

130

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

solutions of Eqs. (3.264) and (3.265), the right-sides of both equations are averaged over the period 2% in the variable ij>. Doing this gives the averaged equations: dA ~dt f

=

e I /

F(Acosip,— uA sin 4>) sin ip dtp,

2-Kb)

3A2 -(2^4) I

n^cosV,-a.Asin^)cosV-#+

Q)

ALO

(3.266)

(3.267)

This is the required generalized of the Krylov and Bogoliubov method to the dif­ ferential equation given by Eq. (3.255). How is the "constant" u> to be selected? Consider the case where F = 0, i.e., (3.268a) and select the initial conditions to be dy(0) = 0. dt

y(0) = An,

(3.268b)

An excellent first approximation to the solution of this equation is [21] 1/2

y(t) = A0 cos

A0t

(3.269)

A(t) = A0,

(3.270a)

Setting F = 0 in Eqs. (3.266) and (3.267) gives dA dt

0

or

and 3A*

(3.270b)

dt whose solution is

«<> = te

2 "\A -"-0

o

2

t + <j>o-

(3.271)

Comparison of Eqs. (3.257) with Eqs. (3.269) and with the result of Eq. (3.271) shows that for the case

u= —

A0,


(3.272)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

131

Consider now a conservative oscillator whose equation of motion is -^+y3=eFi(y),

dy(0) = 0. dt

y(0) = A 0 ,

(3.273)

The amplitude and phase equations take the following forms (see Eqs. (3.266) and (3.267)): dA

~dl

= 0,

2-*<•>+(£)

(3.274) 3A2

(3.275)

The solution to Eq. (3.274) is A(t) = A0, where A0 is a constant. Thus, the phase equation becomes d<j>

(3.276)

One possible choice for the constant u> is to let it satisfy the relation 1A2 2

(3.277)

"-"-a

If this is done, then the first approximation to the solution to Eq. (3.273) is given by the expressions: 1/2

A0t + 4>(t)

y{t) = Ao cos

(3.278a)

where 64 = -( dt

J

A

j

Fi(A0cosi>)cosj>dj>

=

eBi(A0).

(3.278b)

This last equation has the solution ^(t) = eB1(Ao)< + <^o.

(3.279)

The requirement <^(0) = 0 gives (f>0 = 0. Thus, y(t) can be written as 1/2

y(t) — A0 cos

A0+efli(A0)

(3.280)

132

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS Consider a van der Pol type oscillator whose equation of motion is ^

+y

(3.281)

=e(i-y)^-

The application of the Lienard-Levinson-Smith theorem shows that a unique, stable limit cycle exist for this equation. The F for this case is F = (1 — a 2 cos2 ip)(—aw sin */>).

(3.282)

Substitution of this function into the right-sides of Eqs. (3.266) and (3.267), and doing the integration gives dA _ dt


(3.283)

>-$ 3A 2

d<j>

(3.284)

~dl ~ \2^y

The limit cycle parameters are determined by the fixed-points of Eqs. (3.283) and (3.284). The nontrivial solution is A(t) = 2,

(3.285)

u = V3.

Based on these considerations, u should be selected to have the value given by Eq. (3.285). The solution to Eq. (3.283) has been given several times in this chapter and is

A\t)

=

AAl

(3.286)

Al+(4-Al)e-
Replacing w by \/3 in Eq. (3.284) and integrating gives

m

-®-

A2e" + ( 4 - A 2 )

*,

(3.287)

where the integration constant was determined by the requirement 4>(Qi) = 0. Note that for large t, the function <j>(t) goes to a constant, i.e., Lim (j>{t) = <j>* = constant. t—>oo

(3.288)

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

133

Hence, the asymptotic behavior of the first approximation can be expressed as follows: y(t) ~ 2 cos (Vzt + *\ .

(3.289)

For finite times, the solution is y(t) = A(t)cos U/3t + (t)\ ,

(3.290)

where A(t) is given by Eq. (3.286) and 4>(t) by Eq. (3.287).

The above discussed generalization of the Krylov-Bogoliubov method by Mickens and Oyedeji [20] to the class of differential equations represented by Eq. (3.255) is based on the use of circular or trigonometric functions, i.e., shn/> and cosi/'. In a series of papers [22, 23, 24], Bejarano and Yuste have further extended this method to include the use of Jacobi elliptic functions. The basis of their procedures is the fact that the differential equation §

+ ciy + c2y3=0

(3.291)

can be solved in terms of the Jacobi elliptic functions. They investigate the appli­ cation of these methods to the following general case of differential equations

g

+

,i + „+ «•«,(.. *),

(3.»2)

where 6 > 0 and 6 = 0(1),

Cl

= 0(1),

c2 = 0(1),

0 < e < 1.

(3.293)

They demonstrate, by numerically integrating the differential equations of interest, that the use of the Jacobi elliptic functions provide a more accurate approximation in comparisons with the use of trigonometric functions.

134

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Problems 3.1 In the first approximation of Krylov and Bogoliubov, explain why the first derivative of the assumed solution can be taken as the form given by Eq. (3.8). 3.2 Verify that the coefficients Kn{a), Ln(a), given by the expressions of Eqs. (3.17).

Pn(a),

and Qn(a) of Eq. (3.16) are

3.3 Work out the consequences for the amplitude and phase relations if the non­ linear function takes the form

VD-

Fx(y)

dy dt'

Use the analysis of Section 3.2.2. 3.4 Use the Krylov-Bogoliubov method to determine approximate solutions for the following equations:


(i-v)-«-v

(a)

*5- + V = < l - | v l %

(b)

cPy

dy_ dt ,dy

cPy

-d¥ +

y =

-^Tf

dy_ dt

(c)

(d)

3.5 Suppose in the linear stability analysis that <Mi(
dt

= 0.

What can be done for this case? Analyze this situation both graphically and mathematically. (See Eq. (3.43).) 3.6 Use the method of equivalent linearization to construct the linear equations for the examples of Problem 3.4.

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

135

3.7 Prove that the conditions given in Eqs. (3.136) and (3.137) eliminate the fun­ damental harmonic in the solutions obtained by using the Krylov-BogoliubovMitropolsky technique. Show that this condition also guarantees the absence of secular terms in all successive approximations. 3.8 Verify the results of Eqs. (3.174) and (3.175). 3.9 Prove that in the Krylov-Bogoliubov-Mitropolsky method the first approxima­ tion to y is y = a cost/), rather than y = acosip + eui(a,ip). To do this, show that the "error" in the equation y = acosift + tu\{a, ip), as well as the "error" in the simplified equation y = a cos iji, is of order e; consequently, the first ap­ proximation should be taken as y = acosi/>- (See Bogoliubov and Mitropolsky [2], pps. 4 7 ^ 8 . ) 3.10 Use the Krylov-Bogoliubov-Mitropolsky method to determine the solution to second order for each of the equations of Problem 3.4. 3.11 Calculate the third approximation for the Krylov-Bogoliubov-Mitropolsky method and apply it to the van der Pol equation as well as Eq. (3.217). 3.12 Apply the results of Problem 3.11 to the equation

cPy

2\dy

/i

v

'~dt~ay

^

3

where a > 0 and a = 0 ( 1 ) . 3.13 Can Eq. (3.213) be integrated? If so, obtain xp(t). 3.14 Consider the differential equation §

+ (A, + \2y + W

+ A4y3 + A 5 y 4 ) ^ + y = 0,

where the coefficients A, = 0 ( 1 ) . Use the methods of Section 3.2.3 to investi­ gate the possible limit cycles and their stability properties [25]. 3.15 Show that the equation -^

+ y = e(l-y

)-,

where n is an integer, has a unique, stable limit cycle. Calculate its amplitude and frequency.

136

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

3.16 Use the method of Krylov-Bogoliubov-Mitropolsky to obtain, in the second approximation, a solution to the equation


d-rt*-.

where 6 is positive and 0(1). Show that if 6 = 1, then y = sint is an exact solution to this equation. 3.17 Consider the equation

dt2

+

~, dy

!+<:

-2k - f + a dt

cy

+ y = 0,

where k, a, 6, and c are positive and of order e. Does this equation have any stable periodic solutions? 3.18 Use the results from Problem 3.12 to investigate "spurious" limit cycles for this equation. 3.19 Consider the equation d?y

,

dy

Use the method of Section 3.7 to calculate a solution. Formulate criteria to select UJ. 3.20 Extend the method of Krylov-Bogoliubov-Mitropolsky to the case where there is finite linear damping in addition to small nonlinear terms [4, 5]. For this situation, the equation of motion is
rpf

dy

where 7 is assumed to satisfy the condition 0 < 7 < 1. This corresponds to the system being underdamped in the linear approximation, i.e., when e = 0. Apply the method to calculate the first approximation to the solutions to the following equations: cPy „ dy 3 (a) —2. + + 27 27 ^ — + y + ty = 0

dt2 ^ , o

2+

dt ^lI

7

d

y

dt 1

+v=

n

e{l v)

2\dy

- Tf

METHOD OF KRYLOV-BOGOLIUBOV-MITROPOLSKY

137

Show that for Eq. (a), the method requires that the amplitude satisfies the restriction

Also, prove that all solutions to Eq. (a) have the property Lim y(t) = 0. Do a similar analysis for Eq. (b). References 1. N. Krylov and N. Bogoliubov, Introduction to Nonlinear Mechanics (Princeton University Press; Princeton, NJ; 1943). 2. N. N. Bogoliubov and J. A. Mitropolsky, Asymptotical Methods in the Theory of Nonlinear Oscillations (in Russian, State Press for Physics and Mathemat­ ical Literature, Moscow, 1963. English translation: Hindustan Publishing Co.; Delhi, India). 3. B. van der Pol, Phil. Mag. 4 3 , 700 (1926). 4. E. P. Popov and I. P. Palitov, Approximate Methods for the Analysis of Non­ linear Automatic Systems (in Russian, State Press for Physics and Mathemat­ ical Literature, Moscow, 1960. English translation: Foreign Technical Division, AFSC, Wright-Patterson AFB, Ohio; Report FTD-II-62-910). 5. K. S. Mendelson, J. Math. Phys. 11, 3413 (1970). 6. R. A. Struble, Nonlinear 1962).

Differential

Equations (McGraw-Hill, New York,

7. J. E. Burnette, Jr., The Number of Limit-Cycles for the Generalized, Mixed Rayleigh-Lienard Oscillator Equation (Clark Atlanta University; Atlanta, GA; July 1995). Master of Science thesis in the Department of Physics. 8. J. Burnette and R. E. Mickens, "Spurious Limit-Cycles in Higher Order Aver­ aging Methods," unpublished. 9. S. L. Ross, Differential Equations (Blaisdell; Waltham, MA; 1964). See Chapter 10.

138

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

10. A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators (Pergamon, New York, 1966). See the Introduction, pp. xv-xxxii. 11. See reference 10, Chapter II. 12. N. Minorsky, Introduction to Non-Linear Mechanics (J. W. Edwards; Ann Ar­ bor, MI; 1947). See pp. 205-206. 13. D. Siljak, Nonlinear Systems: The Parameter Analysis and Design (Wiley, New York, 1969). 14. A. Blaquiere, Nonlinear System Analysis (Academic, New York, 1966). 15. N. W. McLachlan, Ordinary Nonlinear Differential Equations in Engineer­ ing and Physical Sciences (Oxford University Press, London, 1950). See Sec­ tion 5.18. 16. See reference 2, p. 42. 17. W. S. Krogdahl, Astrophysical J. 122, 43 (1955). 18. W. Addo-Asah, H. C. Akpati, and R. E. Mickens, J. Sound and Vibration 179, 733 (1995). 19. L. Perko, Differential Equations and Dynamical Systems (Springer-Verlag, New York, 1991). See Section 3.8. 20. R. E. Mickens and K. Oyedeji, J. Sound and Vibration 102, 579 (1985). 21. R. E. Mickens, J. Sound and Vibration 94, 456 (1984). 22. S. Bravo Yuste and J. Diaz Bejarano, J. Sound and Vibration 110, 347 (1986). 23. S. Bravo Yuste and J. Diaz Bejarano, J. Sound and Vibration 139, 151 (1990). 24. S. Bravo Yuste and J. Diaz Bejarano, J. Sound and Vibration 158, 267 (1992). 25. See reference 2, pp. 96-97.

Chapter 4 HARMONIC BALANCE 4.1 I n t r o d u c t i o n The method of harmonic balance is a procedure for determining analytic approximations to the periodic solutions of differential equations by using a truncated Fourier series representation. An important advantage of the method is that it can be applied to nonlinear oscillatory problems for which the nonlinear terms are not "small," i.e., no perturbation parameter need exist. In general, when the method is properly used, it gives excellent approximations to the periodic solutions. There exists a vast literature on the method of harmonic balance. A selected list of some of the important papers that analyze and apply this method to a variety of differential equations is given at the end of this chapter in the references [1-26]. The material of this chapter focuses primarily on the formulation of the method of harmonic balance as given by Mickens [13, 14, 15]. However, it should be indicated that various generalizations of the method of harmonic balance have been made by several investigators: an intrinsic method of harmonic analysis by Huseyin et al. [11, 12]; unrestricted harmonic balance by F. F. Seelig [23-26]; the use of Jacobi elliptic functions by Bejarano, Bravo Yuste, and Garcia-Margallo [18-22]; and, two time scales harmonic balance by Summers and Savage [16]. Section 4.2 gives the details of the general procedures for calculating solutions by means of the method of harmonic balance. The emphasis is on the first and second approximations. For these approximations, the required work can usually be done by "hand." Sections 4.3 and 4.5 illustrate the use of harmonic balance by applying it to a number of important nonlinear differential equations. Section 4.4 studies the relationship between the method of harmonic balance and the LindstedtPoincare perturbation procedure when the differential equation takes the form

139

140

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS The mathematical justification of harmonic balancing procedures has been con­

sidered by several investigators.

In particular, the papers by Borges et al. [4],

Leipholz [5], and Miletta [10] provide good introductions to various issues relating to convergence of the approximations and error bounds. See also references [27, 28]. 4.2 General M e t h o d 4.2.1 Bounds on the Fourier Coefficients Consider the following nonlinear differential equation

dt2 + F

( ! )

■•

- * >

,.\' •d(4\,D ! ) '

dy_ dt'

(4.2)

where the functions F and G are, in general, polynomial functions of their arguments and, a and /3 denote sets of parameters that define the system. The parameter A is taken to be non-negative. If A = 0, then Eq. (4.2) becomes

dt2

+F

dy

= 0,

(4.3)

and this equation corresponds to a generalized conservative oscillator. The case for which F is given by (4.4a)

F = y, i.e.,


5? + » = i G '■'Si •"

dy_ dt'

(4.4b)

represents an oscillatory system for which limit cycles may exist. Assume Eq. (4.2) has a periodic solution of period T = 2rr/uj, i.e., y(t) = y(t + T).

(4.5)

This function can be represented as a Fourier series [29] oo

y(t) = 2_^ [o-k cos(fcoii) + bk sin(fcu)t)], fc=i

(4.6)

HARMONIC BALANCE 141 where {ak, bk : k = 1,2,3,...} are the Fourier coefficients. (The equation of motion can always be written in such a form that no constant term appears.) The Weierstrass theorem [29] states that for any continuous periodic function y(t) and for any e > 0, a trigonometric polynomial

Y(t, N(e)) = ^ [ak cos(kujt) + bk sin(Jfcu;t)]

(4.7)

t=i

can be found such that for some N(e) \y(t)-Y{t,N(e))\<e.

(4.8)

The purpose of harmonic balancing techniques is to approximate the periodic solu­ tions of Eq. (4.2) by a trigonometric polynomial N y(t,

N) = ^2[ak cos(kivt) + bk sin(ifcu>t)]

and determine both the coefficients {ak,bk

(4.9)

: k = 1,2, ...,7V} and the angular

frequency ui. All of these quantities are to be expressed finally in terms of the initial conditions. General results on the existence and convergence of y(t, N) to y(<), as N —» oo, can be found in references [4, 5, 10, 27, 28]. In the following discussion, it is shown that very stringent bounds exist on the Fourier coefficients of the periodic solutions to Eq. (4.2). T h e o r e m 1. Ify(t)

has r derivatives, the Fourier coefficients satisfy the relations \ak\ + \b„\<~,

where M is a constant. Ifdry/dtr

(4.10)

is also of bounded variation, then M

M + lM^p+r

(4-n)

142

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

(Essentially, a function f(t),

defined in [0,T], is of bounded variation if the arc-

length of f(t) over this interval is bounded; see reference [30].) T h e o r e m 2. Let y(t) be a function oft, analytic in the interval [0, T). It is possible to £nd a 9, where 0 < B < 1,

(4.12a)

M + |6*|
(4.12b)

and a constant A > 0, such that

These theorems can be applied to Eq. (4.2). Many important dynamic sys­ tems can be modeled by this type of equation where the functions F[y, (y)2,«] 2

G[y,(y) ,/3]

and

are polynomial functions of their arguments. For this case, the condi­

tions of both theorems are satisfied. (Note that if Theorem 2 holds, then, Theorem 1 is also true.) This can be shown by the following argument: If y(t) is a real ana­ lytic function in [0,T], then all the derivatives of y(t) exist at an arbitrary point on this interval. The r-th derivative can be calculated by successively taking the r — 2 derivatives of Eq. (4.2) and evaluating the resulting expression at t = t 0 - This process is always possible since the functions F and G are polynomials in y and ij. Thus, the r-th derivative can be expressed in terms of the ^-th derivative, evaluated at t — to, for I — i— 1, r — 2 , . . . , 1,0. (Note that since Eq. (4.2) is a second-order differential equation, y(to) and y(t0) are known, i.e., they can be selected to have any desired finite values. Consequently, the second derivative is known in terms of y(t0) and y(t0); the third derivative can be expressed in terms of y(to), y(*o), and y(to); etc. It immediately follows that the coefficients of the Fourier series of the periodic solutions to Eq. (4.2) satisfy the inequality of Eqs. (4.12) if F and G are polynomial functions of their arguments.

HARMONIC BALANCE

143

It should be kept in mind that the existence of the bounds given by Eqs. (4.11) and (4.12) does not necessarily imply that the Fourier coefficients decrease monotonically with an increase in the index k. However, these results do imply that the Fourier coefficients decrease rapidly. While this book is generally concerned with autonomous second-order differ­ ential equations, i.e., the differential equations do not depend explicitly on t, the above results can be readily generalized to the non-autonomous case where

* » -L n (

d

y A

n

(4.13)

such that H is periodic in t, i.e., (4.14) If H is a polynomial function of y and y, and an analytic function of t, then the bounds of Eq. (4.12) still hold. The consequences of these theorems can be illustrated by means of two ex­ amples [31]. The first is a nonlinear conservative oscillator, the second is a forced nonlinear oscillator, and the third is an antisymmetric, constant force oscillator. Consider the equation d2v

■>

A >0.

(4.15)

For this case, the functions F and G are F = y + \y3,

G = 0.

(4.16)

The exact solution to this equation, see Section 1.4.4, is

V(*,A) =

' 2nA kK{k)

2 „ a2m- 1 cos

'27r(2m - l)t'

(4.17)

144

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

where the initial conditions are y (0)

= A,

where k =

XA2 2(1 + A ^ ) J

* g ) dt

= 0

1/2

(4.18)

,

AK(k) '

[l +

XA2]1'2'

(4.19)

and K(k) is the complete elliptic integral of the first kind [32]. The coefficients a2m-i are (4.20)

a

2m-l

where

l+g2m-l'

nm

q = exp ■

Note that

k' = (1 -

k2fl\

(4.21)

(4.22)

0 < q < 1, and from Eqs. (4.20) and (4.22), it follows that < 6 m~ ,

0 < a2m-i

(4.23)

where (4.24) The exponential decrease of the coefficients is expected since Eq. (4.16) is a poly­ nomial function of y. Now consider the equation
-d¥ +

y

3

■ . =sint

'

(4.25)

which represents a periodic forced nonlinear oscillator. The following trigonometric polynomial provides an approximation to a periodic solution [27] of Eq. (4.25):

y(t) = ^2 &2m-1 s i n ( 2 m _ !)*'

(4.26)

HARMONIC BALANCE

145

where the coefficients are &i = 1.431189037,

b9 = 0.000059845,

63 = -0.126911530,

&„ = -0.000004691,

65 = 0.009754734,

&13 = 0.000000368,

67 = -0.000763601,

615 = -0.000000029.

The ratio of neighboring coefficients can be calculated and are given by the following numbers:

J^-8.87X107 2 ,' u 65 ~ - -7.83 ~ , , u " x" i10-

^_7.69xl0-2) bz 7 0 , „ i n ^&9- ~_ -7.83 x 10

2

67

613 ^ 5

^ - 7 . 8 5 x 1 0 -

_

7

.

M x l 0

.

611

^

s- -7.90 x 10- 2

»13

Note that all the ratios are close to the value —7.84 x 1 0 - 2 . (The last ratio value probably reflects the effect of truncation of the infinite Fourier series, wliile the first two ratio values indicate the dominance of the lower harmonics in the periodic solution.) The coefficients are expected to satisfy a relationship having the form given by Eqs. (4.12). For m > 3, the following relation is consistent with the above coefficient ratios: 62m_] ~ ( - l ) m + 1 & 0 2 m - \

(4.27)

where 6 and b are constants. The constant 6 can be determined from Eq. (4.27) and the coefficient ratios, i.e., h

J^+l. „ _02 ~ _7.84 x 1 0 - 2 ,

(4.28)

9 ~ 0.28.

(4.29)

and

146

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

In summary, the trigonometric polynomial approximation to a periodic solution of Eq. (4.25), given by the expression of Eq. (4.26), has Fourier coefficients that appear to decrease in the manner expected from the prediction of Theorem 2. The final example is the antisymmetric, constant force oscillator whose equation of motion is ^ j f + s g n ( y ) = 0.

(4.30)

With the initial conditions V(0)=0,

*&

= A,

(4.31)

the exact solution is y(t)

m± 3

V 7T J ^ v

' m=0

1

. [(2m + l ) i

(2m + l )

v

3

(4.32)

1A

'

(See Section 1.4.3.) Observe, from Eq. (4.30), that y(i) has a bounded second derivative, and that y(t) is of bounded variation. Hence, from Theorem 1, it follows that the Fourier coefficients satisfy the bound M hk+r < p .

,

(4.33)

Comparison of this expression with Eq. (4.32) gives M = 2,4 2 .

(4.34)

The general conclusions of this section can be expressed as follows: If the functions F and G are polynomial functions of their arguments, then the Fourier coefficients of periodic solutions to Eq. (4.2) should decrease rapidly, i.e., these coefficients are expected to satisfy the bound given by Eqs. (4.12). Similar results are expected if F and G are rational functions of their arguments. If F and G are only piecewise continuous functions of their arguments, then a bound like that given by Eq. (4.33) is expected to hold. The above discussion implies that the use

HARMONIC BALANCE

147

of trigonometric polynomial approximations to the periodic solutions of Eq. (4.2) may provide excellent representations of the actual periodic solutions. These results form the basis for the use of the method of harmonic balance. 4.2.2 Direct Harmonic Balance [13, 15] In the following presentation of the direct method of harmonic balance, it will be assumed that the equation of motion takes the form:

g + /(,,«) = ***>£

(4-35)

where / and g have the properties f(-y,

a) = -f(y,

a),

g(-y, /}) = +g(y, 0).

(4.36)

The restriction to this form is no actual loss of generality since almost all of the non­ linear oscillators that are considered in this book have this structure. An important feature of systems that can be modeled by Eqs. (4.35) and (4.36) is the property that only the odd harmonics appear in the Fourier expansions of the solutions. First consider the situation for a conservative system, i.e., g = 0 in Eq. (4.35): ^ f + / ( y , a ) = 0.

(4.37)

The initial conditions can be selected as y(0)=A, « = 0 .

(4.38)

The periodic solution to Eqs. (4.37) and (4.38) can be written oo

y(t) = £

Am(a, A) cos[(2m - 1)wt],

(4.39)

m=l

where it is indicated that the Fourier coefficients depend on the parameters a and the initial position A. While not written out explicitly, the angular frequency u> also depends on a and A, i.e., u=u{a,A).

(4.40)

148

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

The N-th order method of direct harmonic balance approximates the exact solution y(t) by the expression N A

yN(t) =J2 %

cos

K 2m - ^Pfi]

(4-41)

m=l

where Am

and U>N are, respectively, approximations to Am and u>. Equation (4.41)

contains N + 1 unknowns: the N coefficients and the angular frequency. They can be determined as follows: (1) Substitute Eq. (4.41) into Eq. (4.37) and write the resulting expression as N

^ F m c o s [ ( 2 m - l ) w t ] + HOH = 0,

(4.42)

m=l

where HOH stands for higher-order harmonics, and Hm = Hm(A[N\AiN\...,A^;Q,a).

(4.43)

(2) Set the functions Hm equal to zero, i.e., Hm=0, and solve for A2 ,A3 (3) A\

m = l,2,...,iV,

(4.44)

, . . . , AN , and u> in terms of A\ .

can be expressed as a function of the initial condition, by use of the

relation N

VN(O) = A = £ A W

(445)

m=l

Note that for each N, it is required that the initial conditions of Eq. (4.38) be satisfied, i.e., „ W (0) = A,

^

= 0.

(4.46)

Now consider the full equation given by Eq. (4.35). For this case, limit cycles may exist and the initial conditions cannot, in general, be o priori specified. The

HARMONIC BALANCE

149

N-th order method of direct harmonic balance approximates the periodic solutions of Eq. (4.35) by the expression 2/jv(t) = A\ ' cos(uiNt) N

{^m]

+ Y,

«>s[(2m - IpNt]

+ B 2 ° sin[(2m - l)u,Nt]\

, (4.47)

m=2

where the 2N + 1 unknowns, f i W A(N) R(N)

r.

A(N) (4.48)

R
can be determined as follows: (1) Substitute Eq. (4.47) into Eq. (4.35) and write the resulting expression as N

] T {Hm cos[(2m - l)u>Nt] + Lm sin[(2m - l)w w <]} + HOH = 0.

(4.49)

m=l

(2) Set the 2N functions Hm and Lm equal to zero, i.e., ffm=0,

Lm=0,

m = l,2,...,JV,

(4.50)

and solve the quantities listed in Eq. (4.48). In general, these quantities will be expressed in terms of the system parameters a and /?. 4.2.3 Rational Representations

[14, 33]

The differential equation §

+ , 3 = 0,

(4.51)

has an exact solution that can be expressed in terms of a Jacobi elliptic function. These functions have the feature that they can be written as ratios of series of trigonometric functions [34]. This suggests that approximations to the periodic solutions of Eq. (4.35) can take the form E ^ = i {A(nN) cos(2m - 1)0 + &iN) sin(2m - 1)6>] 1 + E m = i [
sin(2m
150

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

where 6 = «5&t.

(4.53)

Note that y^ is essentially the approximation of the previous subsection. Instead of giving the general procedures for determining the various coeffi­ cients and the angular frequency, only the first and second approximations will be discussed.

For most cases, the use of the form given by Eq. (4.52) leads to

very complicated algebraic equations that have to be solved for third and higher order approximations. In Section 4.5, the worked examples show that the second approximation can generally be done satisfactorily by hand calculations. The first approximation is just the expression y>(t) = i ( 1 1 ) c o s ( ^ t ) ,

(4.54)

with all the other coefficients zero. There are several distinct forms that can be used for the second approximation; three of them are listed below: yl(t) = A\2) cos(ui^) + 4 2 ) cos(3<^r) + 5< 2 ) sin(3u>j^), A™ cos(Q{t)

= 1

l +

C,11)cos(2u>1It)

+ .D<1) sm(2u>\t)'

(4.55)

(4.56)

and = n J

A < ' > c o ^ ) + ff>co.(3u>;t) l + D^sm^lt)

V

'

The harmonic balancing procedure for the representations of Eqs. (4.54) and (4.55) has already been discussed in Section 4.2.2. Consider the form of Eq. (4.56) first for a conservative oscillator. For this case, Eq. (4.56) reduces to the expression l}

cos(u>\t) ^ "l1 (1) 1 ViW ~ .l +. Ci ^(i) ^~7T> cos(2w 1 t)' , 1

W

_

A[ ^1

ut-o\

(4-58)

HARMONIC BALANCE

151

where the initial conditions are those of Eq. (4.38). Substitution of Eq. (4.58) into Eq. (4.37) and expanding the resulting expression into a trigonometric series, the following result is obtained: Hi[A[l),C1:i),u>\}cos9

+ H2[A{11\C[1)

,U>J]COB30 + HOH = 0,

(4.59)

where 9 = uj^t. Setting the coefficients of the two lowest harmonics to zero gives #i =0, which can be solved for 0\

H2 = 0,

(4.60)

and w\ as functions of A\'.

The amplitude A\1' can

then be expressed in terms of the initial value, y(0) = A, by means of the relation Am

Consequently, A\

, C\

, and ui\ are given as functions of A and the parameters a.

For nonconservative oscillators, limit cycles can exist. Eq. (4.56) must be used.

The full form of

Its substitution into Eq. (4.35) gives a trigonometric

series having the form # i cos 9 + H2 cos 39 + Lx sin 9 + L2 sin 30 + HOH = 0, where (Hi,H2,Li,L2)

depend on A\ ', C[ , D\

(4.62)

, and a>|, and where 9 = u>}t.

The functions also depend on the parameters a and j3. Harmonic balancing the expression of Eq. (4.62) gives Hx = 0 ,

H2 = 0,

which are to be solved for A*1*, Cj (1) , D\n,

Lt = 0 ,

L2 = 0,

(4.63)

and u>\ in terms of the parameters a

and p. Similar considerations apply to the form given by Eq. (4.57).

152

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

4.3 Worked Examples: First Approximation This section gives the details of the calculations necessary to obtain a first approximation to the periodic solutions to eight nonlinear differential equations by means of the method of harmonic balance. An important advantage of this technique is that it can be applied to differential equations for which perturbation methods cannot be employed. In the material to follow, both in this section and the remaining sections of this chapter, bars and unnecessary subscripts and superscripts will not be placed on the amplitudes and angular frequency. They will be taken, respectively, to be A\ and u>. However, it should also be clear that the final quantities of the calculations are approximations to the actual values of the corresponding expressions in the exact solutions. For conservative systems

J f + /(y,a) = 0,

(4.64)

the initial conditions are always selected to be 3,(0) = A,

«

= 0.

(4.65)

This situation gives Ai = A. For nonconservative systems, no such a 'priori selection can be done. The functional form of the first approximation is yi(t) = Ai cosujt,

(4.66)

and the corresponding "approximate" expressions for the first and second derivatives are dy\ -~ = -u)Ax smut, ~£r=

-u2AlCOsut.

(4.67) (4.68)

HARMONIC BALANCE

153

4.3.1 Example A Consider the following nonlinear oscillator equation g

+ , 3 = 0.

(4.69)

All of its solutions are periodic. The substitution of Eqs. (4.66) and (4.68) into this equation gives - w 2 Ax cos ut + A\ cos3 ut ~ 0,

(4.70)

(-"' + ¥)Aicoswt + HOH = 0.

(4.71)

Setting the coefficient of cosuii equal to zero allows the determination of w in terms of Aj;

« = yf^-

(4-72)

Therefore, the first approximation to the solution of Eq. (4.69) is yi(t) = A1cos(J^A1t).

(4.73)

Imposing the initial conditions, of Eq. (4.65), gives Ai = A,

(4.74)

and yi(t)

= A cos U^

At)

(4.75)

It will be shown later that the expression given by Eq. (4.75) provides an accu­ rate approximation to the solution of Eq. (4.69). Note that perturbation methods cannot be applied to this equation. 4.3.2 Example B Another nonlinear conservative oscillator differential equation is §

+ y + ey3=0.

(4.76)

154

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

For e > 0, all the solutions are periodic. Substituting Eqs. (4.66) and (4.68) into this equation gives —ui2AcosLot + Acoswt + eA3(cosut)3

-w2 + 1 +

3eA2

~ 0,

coswt + HOH = 0.

(4.77)

(4.78)

Setting the coefficient of cos ut equal to zero and solving for u gives 3eA2

(4.79)

w=\ 1+

Hence, the first approximation to the solution of Eq. (4.76) is yi(t)

= Acos\

\/l +

-L-i

(4.80)

It should be emphasized that this expression provides a good representation for the solution for all e > 0 and arbitrary values of A. 4.3.3 Example C The van der Pol differential equation is

- f2+ j / = e(l-y 2 )-£, dt

dt'

e>0.

(4.81)

The approximate solution and its derivatives for this case are given by Eqs. (4.66), (4.67), and (4.68), where the initial conditions are unspecified. Therefore, ^ | + y = (-a; 2 + l)A 1 cosa;<,

(4.82a)

and (1 - y 2 ) - ^ = (1 - A\ cos2 wi)(-Aiw sin wi) dt A2 = — A\u l - - ~ )sina;i + HOH

(4.82b)

HARMONIC BALANCE

155

Substituting these expressions into Eq. (4.81) gives

,4-!)

( - u / + l)Aj coswt + eAuj ( 1 - -7- ) sinwt + HOH = 0.

(4.83)

Setting the coefficients of the cosuit and sinwt terms equal to zero and solving for u> and A\ gives two solutions; they are A\ = 0,

ui = arbitrary,

(4.84a)

Ai = 2,

w = 1.

(4.84b)

The first solution corresponds to the trivial solution or a limit point, i.e., yi 1 } (0 = 0,

(4.85)

while the second gives the approximate parameters of the limit cycle, i.e., y f ' ( t ) = 2 cost.

(4.86)

Based upon the knowledge gained from the application of the LindstedtPoincare perturbation method to the van der Pol equation, it is clearly seen that Eq. (4.86) reproduces exactly the first term in the perturbation series of Eq. (4.81) under the condition that 0 < e < 1. 4.3.4 Example D Another van der Pol type oscillator differential equation is

§+„• = «!-rtf,

«>0.

(4.87)

Substitution of Eqs. (4.66), (4.67), and (4.68) into this equation, yields the following expression: 12\

/

A2-

C_ w 2 + M i j A1 cosut + eAitJ (l - ^

J sinw* + HOH = 0.

(4.88)

156

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Setting the coefficients of cosojt and sinwt equal to zero gives

A 1 L2 - ^ )

uAl

(l - £\

= 0,

(4.89a)

= 0.

(4.89b)

The limit cycle parameters are Ax = 2,

w = \/3,

(4.90)

and the approximate solution to Eq. (4.87) is yi(t) = 2COB (S3t\.

(4.91)

In view of the results obtained in Example C, it is expected that the j/i(t), given in Eq. (4.91), holds only for 0 < e < 1. 4.3.5 Example E In dimensionless form, a mass attached to a stretched wire (see Section 1.2.2) has the equation of motion g

+ y -

7

^ f = 0 ,

This equation is conservative and has the

0
(4.92)

first-integral

G)(S) 2+ ^ )= ^°'

(4 93)

-

where the potential function is y2

v(y) = j - W i

+ y2 + K

(4.94)

and E is the "total energy" of the nonlinear oscillator. All the motions correspond­ ing to Eq. (4.92) are periodic.

HARMONIC BALANCE

157

Observe that for small and large y, the equation of motion takes the forms: y small: -^f- + (1 - X)y ~ 0,

(4.95a)

V large : — + y ~ 0.

(4.95b)

Consequently, the angular frequency increases as the initial value of y(0) = A increases. For small A, UJ ~ -\/l — A, while for large A, w ~ 1. To apply the method of harmonic balance to this problem, the differential equation must be written in a form that does not contain the square-root expression. Carrying out this procedure gives

fy

( i + y 2 ) dt2 + y

= AY.

(4.96)

Substituting the first order harmonic solution into this equation gives (1 + A2 cos2 uit) [(-u2

+l)Acosut]

= A2(A cosuit) 2 .

(4.97)

Expanding and simplifying this expression gives (-W2 + l)2

1+

+ HOH = 0.

^

(4.98)

There are two solutions for w; they are -ll/2

w± =

1±

(4.99)

f^f\

The discussion after Eqs. (4.95) shows that the solution with the negative sign should be selected, i.e., 1/2

1

A

f^4\

(4.100)

158

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Therefore, a first approximation to the periodic solutions of Eq. (4.92) is

\ yi(t) = A cos

"I 1/2

(4.101)

V^f

4.3.6 Example F [35] The nonlinear differential equation cPy dt2

+ lvlv = o,

(4.102)

corresponds to a conservative oscillator whose first-integral is

3)(£)+(>»°-^

(4.103)

where E is the constant total energy. The assumed approximate solution is j/i (£) = A cos u>t.

(4.104)

However, to proceed, ]y\ = A\cosu>t\ must be calculated. A Fourier analysis gives, for —7r < 9 < 7r, the result [35]

0

i=©[G) + G) cos2 ^-(^) cos4e +

(4.105)

Therefore, |y|y is given by the expression \y\y=

/8J42\

( — J c o s w i + HOH

(4.106)

and using this result in Eq. (4.102) gives

(-»' + £ j Acoswt + HOH = 0.

(4.107)

HARMONIC BALANCE

159

Setting the coefficient of cos ait to zero and solving for w gives for yi(<) the following expression yi{t)

= A cos U^tV

(4.108)

4.3.7 Example G The following nonlinear differential equation is a generalization of the Lewis equation [36]: g

+ y»-
(4-109)

Applying the Lienard-Levinson-Smith theorem to this equation shows that it has a single stable limit cycle. For this problem, it follows that ^ +

, , » - , ( _ « * + ? £ £ ) Ax coB«t + HOH

(4.110)

and using Eq. (4.105) «(! - \y\)^T -» -u*M at

fl - ^ J sinwt + HOH. \ on )

(4.111)

Under these substitutions, Eq. (4.109) becomes (_w2

+

3Ai_ j

Ai

cosojt

+ u)ef1_

i l l j A 1 smut + HOH = 0.

(4.112)

Applying the harmonic balance technique gives o_

A, = ^ ,

/ o \ 3/2

«=*^J

-

(4.113)

There is a second solution Ai = 0,

a» = arbitrary,

(4.114)

160

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

which corresponds to the fixed-point, yi(t) = 0. The solution for the values given in Eq. (4.113) is

/s.\

3 2 r //-A 3 \ 3 //2 1

(4.115)

Hi * This gives a first approximation to the limit cycle of Eq. (4.109). 4.3.8 Example H

Consider the following generalization of the van der Pol differential equation:

§ 2 + y + ( a - / 3 y 2 + 7 ^ ) $ = 0, dt

dt

(4.116)

where the parameters (a, j3,7) are all positive. One or more limit cycles are expected to exist for this equation. The exact number obtained depends on the values of the parameters. Substituting Eqs. (4.66), (4.67), and (4.68) into Eq. (4.116) gives, after simpli­ fication, the following result: (—10 + X)A\ cosut —

«-({k+m^

Aisinu;i + HOH = 0.

(4.117)

Applying harmonic balance gives for the angular frequency w = 1.

(4.118)

The amplitudes of possible limit cycles axe given as roots to the cubic equation A,

!>:-©*-

0.

(4.119)

The root Ax = A^

= 0,

(4.120)

corresponds to a fixed-point of Eq. (4.116), i.e.,

V(t\t) = 0.

(4.121)

HARMONIC BALANCE

161

The other two roots A ( 2 ) and A ( 3 ) are given by [A{2)? = Q

[ft - v / / ? 2 - 8 a 7 ] ,

(4.122a)

[A^f

[/J + V02 ~ Say] .

(4.122b)

= (~\

Note that >i (2) and A ( 3 ) are real and distinct if the following condition holds P2 > 8aj.

(4.123)

For this situation two limit cycles exist with i(2) < i(3).

(4.124)

The following argument indicates that the fixed-point at the origin is stable, the limit cycle with amplitude A ( 2 ) is unstable, and the limit cycle with amplitude

A^

is stable: (1) The coefficient of the dy/dt term can be considered a damping function, i.e., when it is positive this term causes the amplitude to decrease and when it is negative the amplitude increases. (2) For small values y and dy/dt, the damping function is positive. Therefore, in the neighborhood of the origin in phase space, the phase trajectory moves toward the origin. Hence, the origin, i.e., the fixed-point corresponding to AS1) = 0, is stable. (3) For large values of y, the damping function is also positive. This implies that there exist a value y* > 0 such that for \y\ > y*, the motion in phase space is such that the phase trajectory decreases its distance from the origin. (4) The only possibility that can lead to the situation described in (2) and (3) is for the limit cycle with amplitude A' 2 ' to be unstable, while the one with amplitude A^

be stable.

162

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS (5) In summary, for /32 > 8«7, there exists a stable fixed-point, an unstable

limit cycle, and a stable limit cycle. The corresponding solutions according to the first approximation of harmonic balance are yW(t)

= AW = 0,

(4.125a)

yf°(<) = A ( 2 ) c o s i ,

(4.125b)

y[3\t)

(4.125c)

= AW cost,

where AW and AW are given by Eqs. (4.122). If /?2 = 807, then the two limit cycles coincide. The limit point at AW = 0 remains. For this case, the "single'' limit cycle at AW = AW = i 7

(4.126)

is semi-stable. For P2 < 807, AW and AW are complex conjugates of each other and the fixed-point AW = 0 is globally stable, i.e., for any initial conditions the solution decays to the origin in phase space. Exactly the same conclusions are reached if the parameters (a,/?,7) are as­ sumed to be of order e and the method of Krylov and Bogoliubov is then used to obtain an approximation to the solution of Eq. (4.116); see Problem 4.7.

HARMONIC BALANCE

163

4.4 Comparison w i t h t h e Lindstedt-Poincare M e t h o d The eight worked examples of the previous section fall into four categories:

Category

Examples

(I)

B, E

(II)

C, H

Differential Equation y+y

= eFl(y)

(4.127a)

F j ( - y ) = -Fiiy)

(4.127b)

y + y = eF2(y)y

(4.128a)

F2(-y)

(III) (IV)

A, F D, G

= +F 2 (y)

(4.128b)

y + F 3 (y) = 0

(4.129a)

Lim y _ 0 F3(y)/y = 0

(4.129b)

y + F^y) = eF^y

(4.130a)

F 4 ( - y ) = +F 4 (y).

(4.130b)

Category I differential equations correspond to conservative systems, and, in general, all their solutions are periodic for arbitrary initial conditions. For small values of e (replace A in Example E by e and let |y| < 1) the harmonic balance first approximation solution agrees with that determined by means of the LindstedtPoincare perturbation technique.

Moreover, the first approximation appears to

provide good analytic approximations to the periodic solutions for large values of t and arbitrary initial conditions. Category II differential equations generally have limit cycles as the periodic solutions. A detailed examination of Examples C and H shows that the first approx­ imation of the harmonic balance method gives accurate values for the amplitude of the fundamental harmonic term and the angular frequency provided that e is small. As such, the calculated results are the same as those obtained by use of first-order perturbation techniques. Category III differential equations model conservative oscillatory systems for which the force has no linear term. For this class of equations perturbation proce­ dures cannot be used to construct approximations to the periodic solutions. The

164

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

method of harmonic balance seems to provide good analytic approximations to the periodic solutions for arbitrary values of the initial conditions. The differential equations that belong to Category IV have no linear force term and generally are expected to have limit cycles as the periodic solutions. Detailed examination of Examples D and G leads to the expectation that the calculated values of the limit cycle amplitudes and angular frequencies are accurate only for small values of e. The following statements summarize the above results: (1) The application of the first approximation harmonic balance method to Cat­ egories I and II differential equations should give accurate analytic approximations to the periodic solutions for (essentially) arbitrary values of the system parameters and initial conditions. Note that since Categories I and II problems are conservative, it is always possible, and, generally, convenient to use the following initial conditions y(0) = A,

«

= 0.

(4.131)

(2) The first approximation of the method of harmonic balance, when applied to Categories II and IV differential equations, generally gives the same result as first-order perturbation procedures. 4.5 Worked Examples: Second Approximation This section presents the details of how to construct approximations to the periodic solutions to two nonlinear conservative oscillators. Each equation is treated by a different second approximation expansion based on the method of harmonic balance. 4.5.1 Example A Section 4.3.1 gave the harmonic balance first approximation solution to the differential equation

-dr + v = °>

( 4 - 132 )

HARMONIC BALANCE

165

with initial conditions

rfO)-*

*f=0.

(4.133)

The expression obtained was Vl(t)

= A cos U^AtV

(4.134)

A possible second approximation form for the periodic solution to Eq. (4.132) is y2(<) = Ai cos 0 + A2 cos 30,

(4.135)

9 = ut.

(4.136)

where

Substitution of Eq. (4.135) into Eq. (4.132) and simplifying the resulting expression gives Hi(AuA2,u)cos8

+ H2(A1,A2,u)co836

+ KOK = 0,

(4.137)

where

Hl=Ax

\M-\7)MA 1 \4J * V2 H2 = _gA^ + Q A « + (^AlA2 + (^A K4J"

(4.138) (4.139)

The harmonic balancing condition leads to the equations Hl(A1,A2,u:)

= 0,

(4.140)

H2(A1,A2,u>)

= 0.

(4.141)

Solving Eq. (4.140) for u>(Ai,A2) gives (1.142a) where (1.142b)

166

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

If this value for CJ is substituted into Eq. (4.141), then, after some algebraic manip­ ulation, the following equation is obtained for x: 51a;3 + 27x 2 + 21z - 1 = 0. This equation is cubic and therefore has three solutions.

(4.143) However, the root of

interest is the one for which \x\ < 1,

(4.144)

for this requirement forms the basis of the harmonic balance method. To proceed, assume that such a solution, xs, exists, i.e., |x s |
(4.145)

To further improve this result, retain the quadratic term in Eq. (4.143) and solve this equation to obtain z' 2 ) = 0.0450,

(4.146)

as an improved value for xa. From the definition of x, it follows that Y

a 0.045.

(4.147)

Thus, the amplitude of the higher harmonic is about 5% the value of the fundamen­ tal amplitude. Comparison of the expressions for the angular frequency, UJ, given by Eqs. (4.134) and (4.142a) shows that the inclusion of the higher harmonic increases the value of u> by about 2%. The second approximation to the solution of Eq. (4.132) is Viit) = Ai[cos0 + x a cos 30}.

(4.148)

Since 2/2(0) = A, it follows that Ax = —-—

~ 0.9574,

(4.149)

HARMONIC BALANCE

167

and from Eq. (4.142a) w ~ 0.8507A.

(4.150)

The period is given by 2_n ^ 7 ^ 8 5 9 a; A

T =

Consequently, rj2(t) is y2(<) = (0.957),4[cosu;i + (0.045) cos Suit].

(4.152)

The exact solution to Eq. (4.132) is y(t) = Acn{At; 1/V2),

(4.153)

where en is the Jacobi elliptic function. The exact period of the oscillation is m

7.4163 A '

(4.154)

which can be compared to the periods of the first (Tj) and second approximations (T2) 7.2552 T, = — ^ — ,

7.3859 T2 = - j - .

m

(4.155)

A Fourier expansion of the exact solution, Eq. (4.153), gives a ratio of the amplitudes of the cos3w£ and coswi terms to be 0.0450778, as compared to the value given by Eq. (4.147). These results show that the method of harmonic balance provides an excellent approximation to the solution of Eq. (4.132). 4.5.2 Example B [35] Consider the nonlinear conservative oscillator differential equation (see Sec­ tion 4.3.6) §

+ h/|y = 0,

y(0) = A,

^

= 0.

(4.156)

168

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

A second approximation to the periodic solutions is , ^ = 1

A-i cos 0 + 5 ! cos20'

6 =

(4.157)

U;t

Using the fact that for — n < 0 < 7T

|COSe|=

©[© + © COS2e -(^) C ° S ^

+ ■

(4.158)

it follows, after some algebraic manipulation, that (1 + Bicos20) 3 |y|y

i+ i'

+

3Bi 5

i+

175!

cos0

cos30 + HOH

(4.159)

and (1 + B1COS20)3C^-

dt2

=

-L>2A1

+ 35j

1+ 5 35i

-1

115?

cos30 + HOH

(4.160)

Substituting of Eqs. (4.159) and (4.160) into Eq. (4.156), and equating the coeffi­ cients of cos 8 and cos 30 to zero gives the two equations

m 8A2 157T

i+

35i 5

1+

17^!

= w 2 A, 1 + B i -

95?

115?2 1

(4.161)

35j

(4.162)

These are three equations for the three unknowns A\, B\, and u>. To obtain B\, divide Eq. (4.161) by Eq. (4.162) and simplify the resulting expression. Doing this gives the result

HARMONIC BALANCE

169

An excellent first approximation to the ''small" root of Eq. (4.163) is 5 i ^ - ^

=-0.054.

(4.164)

The angular frequency u> can be determined from Eq. (4.162) and is given by the expression

)[ ( l + ^ ) / (£a_3 B l )

,15TT

Ai,

(4.165)

or, using the result of Eq. (4.164) u ; ~ 0.933 V^4i-

(4.166)

The amplitude A\ can be expressed in terms of the initial conditions. From Eqs. (4.156) and (4.157), the following result is obtained

A

1

'122\ ~ ^ ) - 4 = (0.946).4. ,129 y

(4.168)

Substitution of all of these results into Eq. (4.157) gives the following expression for the periodic solution of Eq. (4.156) =

(0.946)Acos(0.908v/ZQ 1 - (0.054) cos(1.816\/Zt)

Problems 4.1 Calculate the Fourier series for |cos#| for —7r < 6 < ir. Apply the theorems of Section 4.2 to predict bounds on the rate of decrease of the Fourier coefficients. 4.2 Prove that in the Fourier expansion of the solutions to Eq. (4.35) only odd harmonics appear if the conditions given by Eq. (4.36) hold.

170

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

4.3 Explain why for the first harmonic balance approximation the following form should always be selected yi(t) = A\ cos tut, rather than yi(i) = A\ coswt + B\ sinuit. 4.4 Construct the first approximation using harmonic balance for a conservative oscillator if the initial conditions are

4.5 Apply the results of Problem 4.4 to Examples A, B, and E of Section 4.3. 4.6 Construct an approximation to the solution of the differential equation -j± + y + ey3 = 0,

t > 0,

using the form !/2(0

=

-^1 cosuit + A2 cos 3wt.

See reference [15]. 4.7 Assume the parameters (a, j3,7) in the example of Section 4.3.8 are all of order e. Use the Krylov-Bogoliubov method to verify the results obtained by the method of harmonic balance. 4.8 Construct a first approximation to the solutions of the differential equations

,

w+y

3

=t

*-Th\

(a)

i-y2J

(b)

L

where 0 < /? < 1. 4.9 Determine all the roots to the cubic equation given by Eq. (4.143). 4.10 Provide a justification for the result of Eq. (4.145). 4.11 Show that the exact period of Eq. (4.132) is given by Eq. (4.154).

HARMONIC BALANCE

171

4.12 Work out the details leading to Eqs. (4.159) and (4.160). 4.13 Calculate the three roots of Eq. (4.163). 4.14 Determine first and second order approximations to the differential equation

§ + v+ {

r

^ = o,

*>0, g>0.

4.15 Analyze the number and type of limit cycles for the differential equations

+ (<*-/V+7y4)^| + y + W 3 = 0

^

(a)

Q, /3, 7, fi all positive, 2
... , ,. :> ay + by =

<-«r i

<»

a, 6, c, d all positive. See reference [37]. 4.16 Can the method of harmonic balance be applied to the systems discussed in Sections 1.4.2 and 1.4.3? If so, construct the first approximation solutions. 4.17 Harmonic balance, when used in lowest order, leads to inconsistencies for non­ linear conservative oscillators having mixed-parity terms. An example of such an equation is [38]

-^ + ay + by2 + cy3 = 0. Apply various harmonic balance approximations to this equation. What major conclusion comes out of this effort? References 1. J. C. West, Analytical Techniques for Nonlinear Control Systems (English Uni­ versity Press, London, 1960). 2. D. D. Siljak, Nonlinear Systems (Wiley, New York, 1969). 3. A. R. Bergen and R L. Franks, SIAM J. Control 9, 568 (1971). 4. C. A. Borges, L. Cesari, and D. A. Sanchez, Q. Appl. Math. 32, 457 (1975).

172

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

5. H. Leipholz, Direct Variational Methods and Eigenvalue Problems in Engineer­ ing (Noordhoff International Publishing, Leyden, 1975). See Section 4.2. 6. J. Kudrewicz, Int. J. Circuit Theory and Appl. 4, 161 (1976). 7. D. J. Allwright, Math. Proc. Camb. Phil. Soc. 82, 453 (1977). 8. P. E. Rapp and A. I. Mees, Int. J. Control 26, 821 (1977). 9. A. I. Mees and L. O. Chua, IEEE Trans. Circuits and Systems C A 5 - 2 6 , 235 (1979). 10. P. Miletta, in R. Chuagui, editor, Analysis, Geometry and Probability (Marcel Dekker, New York, 1985). See pp. 1-12. 11. A. S. Atadan and K. Huseyin, J. Sound and Vibration 9 5 , 525 (1984). 12. K. Huseyin and R. Lin, Int. J. Non-Linear Mechanics 26, 727 (1991). 13. R. E. Mickens, J. Sound and Vibration 94, 456 (1984). 14. R. E. Mickens, J. Sound and Vibration 111, 515 (1986). 15. R. E. Mickens, J. Sound and Vibration 118, 561 (1987). 16. J. L. Summers and M. D. Savage, Phil. Trans. R. Soc. London A 340, 473 (1992). 17. N. MacDonald, J. Phys. A: Math. Gen. 26, 6367 (1993). 18. J. Garcia-Margallo and J. Diaz Bejarano, J. Sound and Vibration 116, 591 (1987). 19. J. Garcia-Margallo, J. Diaz Bejarano, and S. Bravo Yuste, J. Sound and Vi­ bration 125, 13 (1988). 20. S. Bravo Yuste, J. Sound and Vibration 130, 33 (1989). 21. J. Garcia-Margallo and J. Diaz Bejarano, J. Sound and Vibration 136, 453 (1990). 22. S. Bravo Yuste, J. Sound and Vibration 145, 381 (1991).

HARMONIC BALANCE

173

23. F. F. Seelig, Z. Naturforsch 35a, 1054 (1980). 24. F. F. Seelig, J. Math. Biology 12, 187 (1981). 25. F. F. Seelig, Z. Naturforsch 38a, 636 (1983). 26. F. F. Seelig, Z. Naturforsch 38a, 729 (1983). 27. M. Urable, Arch. Rat. Mech. Analy. 20, 120 (1965). 28. A. Stokes, J. Diff. Eq. 12, 535 (1972). 29. N. K. Bary, A Treatise on Trigonometric Series, Volume 7 (MacMilland, New York, 1964). 30. R. C. Buck, Advanced Calculus (McGraw-Hill, New York, 1978). 31. R. E. Mickens, / . Sound and Vibration 124, 199 (1988). 32. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954). 33. D. Wood, IMA J. Appl. Math. 33, 229 (1984). 34. H. T. Davis, Introduction (Dover, New York, 1962).

to Nonlinear Differential

and Integral

Equations

35. R. E. Mickens and M. Mixon, J. Sound and Vibration 159, 546 (1992). 36. J. B. Lewis, Trans. Am. Institute of Electrical Engineers, Part 7772, 449 (1953). 37. G. Schmidt and A. Tondl, Non-Linear Vibrations (Cambridge University Press, Cambridge, 1986). 38. A. Venkateshwar Rao and B. Nageswara Rao, J. Sound and Vibration 170, 571 (1994).

Chapter 5 MULTI-TIME EXPANSIONS 5.1 Introduction In the previous chapters, approximate solutions to the nonlinear differential equation

*+,-«,(,*).

.<.«!.


were obtained by means of perturbation techniques that used a single time variable or scale. However, in the analysis of many dynamical systems, several different time scales may naturally exist. (Following the procedures of Chapter 1, the differential equations are always put in dimensionless forms. Thus, all "time" variables are dimensionless.) It is consequently of theoretical and practical convenience to consider perturbation methods that take into account a number of different time variables

To illustrate the basic concepts, consider the equation for a linear oscillator with small linear damping:
dy

0

+ 2 e

_

+ y = 0.

(5.2)

The exact solution to this equation is y = Ae-<* cos [(1 - e2)^H + ] ,

(5.3)

where A and <j> are arbitrary constants. Observe that the time variable t appears in either the combination et or (1 — e 2 ) 1/,2 t. The first combination determines the decay properties of the amplitude, while the second combination is related to the period of the free oscillations. Let t = et,

t+={l-e2yl\

then, since 0 < t
(5.4)

MULTI-TIME EXPANSIONS

175

These results have been used by Cole and Kevorkian [2] to formulate a method called the two-variable expansion procedure. It is assumed that a uniformly valid asymptotic solution to Eq. (5.1) can be written as m-l

y ( M ) = £ > * w ( M + ) + 0(em),

(5.5)

where t = et,

(5.6a)

t+ = (1 + e2u2 +
(5.6b)

with the wjt being constants. The time derivative is transformed according to the following equation:

| = e | + ( l + ^ 2 + ... + ^ m ) J F .

(5.7)

The use of Eqs. (5.5), (5.6), and (5.7) transforms the original ordinary differential equation into a partial differential equation. Note that for the linear damped harmonic oscillator, the exact solution, given by Eq. (5.3), depends explicitly on the variables to =t,t\

— et, t2 = e2t,...,

as well

as e itself. Thus, a second version of the multi-time method can be constructed by considering the time variables tk = ekt,

k = 0,1,2,....

(5.8)

In general, the time scale r„ is slower than i* where n > k. For this case, a uniformly valid asymptotic solution is assumed to take the form: m-l

y(t, e)=J2

e*Vt(*o, *i , . • • , * » ) + 0{em).

Applying the chain rule, the first derivative is

dt

dto

dti

dt2

dtm

(5.9)

176

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

This version of the method of multi-times is called the many-variable procedure or the derivative expansion procedure [1]. The proceeding two methods may be further generalized. The many-variable procedure can be generalized by using an asymptotic sequence {£*(e)} rather than powers of e: t, s S»(e)l,

(5-11.)

(5 Ilb)

-

S-I»Kk=0

"

These expressions may be further generalized by using tk = 6k(e)gk[nk(e)t],

(5.12a)

- = Ys^^g'MW-Qf,

(5-i2b)

where {/J*(e)} is another asymptotic sequence and {gk(x)} is a set of functions with continuous first and second derivatives. The transformations of Eq. (5.12) allow for linear as well as nonlinear time scales. The two-time expansion procedure can also be generalized.

For example,

Eqs. (5.6) and (5.7) can be generalized to the forms: t = fi(€)t,

(5.13a)

m

t+ =J2h(e)gMe)tl

(5.13b)

fc=o

k=0

d dt+'

(5.13c)

This chapter discusses the two-time expansion procedure in Section 5.2 and the derivative expansion procedure in Section 5.4. Worked examples illustrating the two methods are given in Sections 5.3 and 5.5. Chapter 6 of the book by Nayfeh [1] gives an excellent summary of the method of multi-times. It also contains many references to the research literature where

MULTI-TIME EXPANSIONS

177

these procedures are applied to a wide variety of problems in physics, engineering, and applied mathematics. See also references [2-8]. 5.2 T w o - T i m e Expansion Assume that the solution to Eq. (5.1) takes the following form: y(t,e) = y0(lt+)

+ ey1(t,t+)

+ e2y2(t,t+)

+ --- + em-1ym-1(t,t+)

+ O(cm),

(5.14)

where t - et,

(5.15a)

t+ = (1 + e2u>2 + e3u>3 + ■ • • + emujm)t,

(5.15b)

and the u>j are constants. The time derivatives are

^ = (1+ ^2 + ...+ ^ m ) J _ + £ | , d> df2

(5.16a)

d " ft+2

(l + e^ + .-. +■ e^f~mj y~ •

+ 2e(l + 62u,2 + • • • + emu>m)-£^

+ e2 J ^ .

(5.16b)

The basic procedure of the two-time expansion method is to substitute Eqs. (5.14) and (5.16) into Eq. (5.1), and set the coefficient of each power of e equal to zero. The resulting equations are then solved in succession. However, the solutions will contain arbitrary functions that depend on u>k, t, and t+. These arbi­ trary functions can be determined by imposing the condition that no secular terms appear to each order in e. To proceed with the method, the second derivative of y{t, t) must be calculated. This derivative is J2

r\2

- ^ | = (1 + e2u>2 + e3u>3 + • • -f-fc+2 (2/0 + «/i + e2y2 + ■ • •) d2 + 2e(l + £2w2 + • ■ . )+— _ ( y o + at at 2 2 d 2 + * £Z(yo+eyi+e y2 + ■••)•

ey,

+ e2y2 + • • •) (5.17)

178

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Expanding this expression and collecting together terms of the same order in e, gives

cPy2 = c^/o2 + / a 2y + d>y0 \ dt

a<+

\at+

dt+dtj

+ e2(^l + 2^2 *»L + 2 - ^ + ^ A + 0(e3). + + 2 \dt+* a*+2 + at+dt at 2 )

(5.18)

In a similar manner, the first derivative is

2-&+-(&+$)+-'(&+«&+$)+<*,>-

(5 19)

-

To terms of order e 2 , the nonlinear function F has the following expansion:

"(4H(»-IF) +< ! The differential equations satisfied by j/o(i + ,i), yi(t+,t),

and J/2(^ + ; ^)

can

be

obtained by substituting Eqs. (5.18), (5.19), and (5.20) into Eq. (5.1) and grouping terms of the same power of e together. Setting the coefficients of the various powers of e to zero, gives Q$ d2yi

+ Vo = 0, „ d2y0

(5.21) (

dy0\

d2V2 . „ d2y0 d2y0 d2Vl + y2 = -2u2 ^ - r r - —~- - 2 ■

+! Equation (5.21) has the solution y0(t+,t)

= A 0 (F)sint + + B0(t)cost+,

(5.24)

MULTI-TIME EXPANSIONS where, for the moment, A0(t)

and B0(t)

179

are arbitrary functions of t. Thus, the

second term on the right-side of Eq. (5.22) is _ 2 - ^ = _ 2 ^ c o s < + + 2 ^ s i n t + dt+dt dt dt

(5.25)

Let the original differential equation satisfy the initial conditions y(0,e) = A, where A is a constant.

^

= 0,

(5.26)

Using the results of Eqs. (5.14) and (5.19), the initial

conditions, in the new variables, are yo(0,0) = A,

yt(0,0) = 0,

k > 1,

dyo(0,0) = 0, dt+ ^ ^

^ OT

)

+

.

2

+

(5.27b)

^ | £ ) = 0 ,

+

^ 0 )

+

(5.27a)

^ l

+

(5.27c)

=

0 .

(5.27d)

at

OT

In the next section, the two-time expansion method is applied to six examples of differential equations having oscillatory solutions. Other worked examples can be found in references [1, 3, 4, 5, 6, 7]. 5.3 Worked Examples Using the Two-Time Expansion

5.3.1 Example A Consider the linear damped harmonic oscillation

§+*§+» = »•

<^8>

For this case, Eq. (5.22) becomes d2yi

dFi

d2y0 + y

>

=

2

dy0 2

- ol^- W-

(5 29)

"

180

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Substituting Eqs. (5.24) and (5.25) in Eq. (5.29) gives

g, + V l = _ 2 ( ^ + , 0 ) c o , i + (f + B o ), i D l + . (,30, Eliminating the terms that produce secular terms in this last equation gives ^

+ Ao=0,

(5.31a)

^ + Bo = 0. dt The solutions to these equations are

(5.31b)

at

A0(t) = a0e-7,

B0(t) =

faeA',

(5.32)

where a0 and /?o are integration constants. These constants can be determined from the initial conditions given by Eqs. (5.27). They are found to be <*o = 0 ,

A, = A.

(5.33)

Thus, j/o(< + ,i) is given by the following expression yo = Ae~(cost+.

(5.34)

With this result, Eq. (5.30) becomes Q^

+ Vi = 0 ,

(5.35)

whose solution is yi = Ai(t)smi+

+ Bx(?) cost + .

(5.36)

Substituting these expressions for t/0 and yi into the right-side of Eq. (5.23) gives

MULTI-TIME EXPANSIONS

181

or 32y2 + y2 = -2 dt+2

—~+Ai

- ( 2 + W 2 J Ae-*

+2

l<+.

cost* (5.38)

The absence of secular terms in the solution for j/2 gives the following two equations: ^

+ Ax = Q + u>2) Ae~T,

(5.39a)

+ S i = 0.

(5.39b)

dt The solutions to these equations are A,{t)

ai + Q+"2Mtje-',

(5.40a)

B1(t) = fae-

(5.40b)

where a\ and f}\ are integration constants.

If Eq. (5.40a) is substituted into

Eq. (5.36), a secular term is obtained unless

«2 = " 2 -

(5.41)

Consequently, Eq. (5.36) becomes !/! = a\ e _ t sin t + + /?i e - ' cos

(5.42)

The constants ot\ and j3t can be determined from the initial conditions given by Eqs. (5.27). They turn out to be ax =A,

px = 0,

(5.43)

and, consequently, yx is yx = Ae * s i n t + .

(5.44)

182

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS Therefore, to terms of order e 2 , the solution to Eq. (5.28) is y(t, e) = Ae~* [cos t+ + e sin t+] + 0(e2)

y(t,e) =

K)

Ae-«

2 t + e sin ( 1 - — | t + 0(e ).

(5.45)

(5.46)

This result is to be compared with the exact solution to Eq. (5.28), which is Ae-tt cos(l - e 2 ) 1 / 2 * + (l_€2)l/2

y(t,e) =

1(1 - 6 2 ) 1 ' 2 *

(5.47)

5.3.2 Example B For the differential equation (5.48) Eqs. (5.22) and (5.23) become d2yi dt+2

0

■ "

d2yo dt+dt

2

(5.49)

and d2 V2 d2y0 d2y0 „ d2y1 + 2/2 = - 2 w : —^--2 Qt+2 »< ' ft+2 Qp Qt+Qt

■2y 0 yi.

(5.50)

The substitution of Eq. (5.24) into Eq. (5.49) gives d2Vl dt+-

(A2+B2\

dA0

+ d5

+ 2 ^ °- -:-.+ s i n t + +,

Mg--Bo2 c o s 2 t + - y l o 5 o s i n 2 < + . (5.51)

The requirement of no secular terms gives (5.52)

dt whose solutions axe A0(t) = a0,

B0(t) = Po,

(5.53)

MULTI-TIME EXPANSIONS where a0 and 0O are integration constants.

183

The initial conditions, given in

Eqs. (5.27), can be satisfied if a0 = 0 ,

p0 = A.

(5.54)

Thus, the function yo is yo = >lcost + .

(5.55)

The corresponding solution to Eq. (5.51) is y\ = ~\YJ

cos2t+ + A1(7)smt+

+ (~r)

+ B1(7)cost+.

(5.56)

Substituting the functions y0 and y\ into Eq. (5.50) gives d2y2 + V2 = dt+2

ndAi

„

,

at

5A : D

cos t

+

. , ndBi + 2 —— sin V dt

— AB\ — AB\ cos 2t+ - AAi sin 2< — ( —— ) cos 3 i + . (5.57) The secular terms in the solution y2 can be eliminated if

~dT = A r+ IT) • ^

= 0.

(5 58a)

-

(5.58b)

These differential equations have the solutions /

5J42\

A1(t) = A\u2 + — Jt + ai,

(5.59a)

B,(t) = A ,

(5.59b)

where <*i and /?i are integration constants. The function yi will contain secular terms unless «-2 = - ^ - .

(5-60)

184 OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS Consequently, Eq. (5.56) can be written as yl = -

6)+ (g)

cos 2tf

+ a1 sin t+ +

cost+.

(5.61)

Imposing the initial conditions of Eqs. (5.27) on yl gives

Thus, yl is y1 = -

C2)

- (3 - 2cost+ - cos2t+),

&d the solution to Eq. (5.48) to terms of order e2 is y(t,r) =Aces [ I -

t-r($){3-2cos

("<)r2]

- cos2 [I -

[I-]2€) : (

(g)c2] + t}

0(e2).

(5.64)

To obtain Eq. (5.64), use was made of the following results:

-t = ~ t ,

t+ = [ I -12a):(

5.3.3 Example C Consider the nonlinear differential equation

for which Eqs. (5.22) and (5.23) are

and

t

t.

MULTI-TIME EXPANSIONS

185

Substituting Eq. (5.24) into Eq. (5.67) gives d2Vi , . dB0 . , odA0 ^ T T ? + !/i = - 2 —~r cos i^ + 2 —-— sin r

_ /A| + gg\ _ Mg~£g\ cos2t+ + Afl5o gin 2<+

(5 69)

The elimination of secular terms in the solution J/J requires A0(t) = a 0 , where ao and j30 are constants.

B0(t) = /?0,

(5.70)

The use of the initial conditions, given by

Eqs. (5.27), further requires that a0 = 0,

A, = A.

(5.71)

Consequently, the function yo is y0 = A c o s r + .

(5.72)

The differential equation for yi, with the conditions for elimination of secular terms satisfied, is

<"3>

0 — -(T) + ( T H + The general solution to this equation is j/i = - ( - - j - ( — ) cos 2t+ + Ax(?) sin t+ + B-L(?) cos t+.

(5.74)

Substituting Eqs. (5.72) and (5.74) into (5.68) gives

d2y2 + V2 = dt+2

d n M

n

,

A3

■

dB\

.

i

cos
- ABj + ABi cos 2t + + A Ai sin 2t+ - ( —- J cos 3t +

(5.75)

186

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Again, elimination of the secular terms requires A, dt

A2\

, (

AT*. dB

x

(5.76)

0.

These equations have the solutions (5.77a)

A1(t) = A[u>2 + — Jt + ai,

(5.77b) where a\ and fa are integration constants. The first of Eqs. (5.77) will lead to secular terms in y\ unless

(5-78)

"* = - ( ^ ) -

Applying the initial conditions allows the determination of ay and fa. They have the values

«i = 0 ,

fa

9 4 22 2A

= —.

(5.79)

Consequently, the function yi is yi = ( — ] ( - 3 + 4 cos t+ - cos 2t+).

(5.80)

Therefore, to the second approximation, the solution to Eq. (5.65) is

y(t, e) = A cos

cos 2

®'MS){-

■3 + 4 cos 1 -

1 - l-=r u t}+0(e<).

-=r e

(5.81)

5.3.4 Example D The function y\, corresponding to the differential equation cPy + y + eys = 0, dt2

(5.82)

MULTI-TIME EXPANSIONS

187

satisfies the following differential equation

d2y0

dV.

3

(5.83)

Substituting j / 0 , from Eq. (5.24), into the equation gives d2 Vi + Vi=dt+2

dAp dt

cost"1

+ (7)AIBQ + A3

+ + (-j(A30

- 3A0B2)sm3t+

sint

+ C^j(B3

- 3A20B0)cos3t+.

(5.84)

No secular terms will appear in yt if the following relations hold:

at

\o/

at

\o/

(5.85a) (5.85b)

The Eqs. (5.85) can be combined to give the following result:

±(Al+B20) = 0,

(5.86a)

A2+B2=A2

(5.86b)

where the initial conditions were used to evaluate the integration constant on the right-side of Eq. (5.86b). Using this result, Eqs. (5.85) become dAp

dBp

-UJIBP,

dt where

—Tzr

dt

=WlA0,

3A2

(5.87)

(5.88)

U>!

The solutions to Eqs. (5.87) are Ap = Acos(ujit

-f- <}>),

(5.89a)

188

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS (5.89b)

B 0 = Asm(ujit + (f>), where is a constant. Substituting Eqs. (5.89) into Eq. (5.24) gives yo = A cos(u>i£ + ) sin t+ + A s i n ^ j t + <j>) cos t+

(5.90)

Using the relation sin(#i + 62) = sin^i cos #2 + cos 81 sin 02,

(5.91)

yo = Asin(w 1 ( + ( + + ).

(5.92)

Eq. (5.90) becomes

Satisfaction of the initial conditions, as given by Eqs. (5.27), gives for (f> the value v

(5.93)

2

Consequently, to terms of order e, the solution to Eq. (5.82) is y(t,e) = y0(t+,t)

+ 0(e) = A cos

i

+

i ^ i . t + 0(e).

(5.94)

5.3.5 Example E An oscillator with nonlinear cubic damping has the following equation of mo­ tion: dt2

+ y+e

®f-

0.

(5.95)

The function yi is a solution to the differential equation 2 d2Vi , 0 d y0 2 + yi = - 2 dt+ ' •" " dt+dt

fdyo \dt+

(5.96)

Using y 0 , given by Eq. (5.24), this equation becomes d2yi dt+- ■ + Vi

cos< +

+

2

^r

(^j){Al

+

{VA°Bo + [vBo -3B2)cos3t+

sini+

+ (^f)(3Al

- B20)sin3t+.

(5.97)

MULTI-TIME EXPANSIONS

189

The elimination of secular terms, in the solution for j/i, gives the two equations dA

o

, fi\ . , ,2 . „2 +[^)Ao(Al + B20) = 0,

dJ

-J-+

- )B0(Al + B*) = 0.

(5.98a)

(5.98b)

at \o/ + The initial conditions for y0(t ,t) are Vo(0,0) = A,

^

^

= 0,

(5.99)

consequently, A0(0) = 0,

B 0 (0) = A.

(5.100)

Consider Eq. (5.98a) at i= 0. From the results of Eq. (5.100), it follows that ^ a - o .

(5.101,

This last result, along with the results of Eq. (5.100), implies that A0(t) = 0,

(5.102)

^ + ( | V = 0.

(5.103)

and at \o/ The solution to Eq. (5.103), with the condition given by Eq. (5.100), is B

° =

^TE-

(5-104)

Thus, in the first approximation, the solution to Eq. (5.95) is

y«,0 = -

A cost

/ r. 1 i/2+°^-

( 5 - 105 )

190

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

5.3.6 Example F As a last example, consider the Rayleigh differential equation dPy ^ + 2

dy _ (1\ fdyV /

= e

dt

(5.106)

\3j \dt)

For this case, Eq. (5.22) is

^ +yi = _2|!^+|^_^^Y Qt+2 ' « " -"gi+fi^dtf

(5 107)

\3j\dt+J

"

The substitution of y0, given by Eq. (5.24), into the right-side of Eq. (5.107) gives 9d2yi V

\dA0

fAl + A0Bj\]

dBQ

(AlB0

.

+ Bl

) - ( j f ) (Ag - 35 0 2 ) cos 3i+ + {^j

{ZAl - B%) sin3t+

(5.108)

The elimination of secular terms requires that AQ and B0 satisfy the following differential equations: (5.109a)

(5.109b) The initial conditions on j/o require that A o (0) = 0,

B0(0) = A.

(5.110)

Letting t = 0 in Eq. (5.109a) and using the first result of Eq. (5.110), gives dA0(0) dt

0.

(5.111)

From these facts, it follows that A0{t) = 0,

(5.112)

MULTI-TIME EXPANSIONS

191

and dB

« = (?±y-Bl). dt

(5.113)

Integration of Eq. (5.113) gives ~

1A 2

2

[A - (A - 4)e- T | Thus, the first approximation to the solution of the Rayleigh equation, accord­ ing to the two-time expansion method, is / ^ v(*,«) =

2A COS t

„,. 777 + 0 ( e ) .

, (5.115)

[A2 - (A 2 - 4 ) e - " ] 1 / 2 5.4 Derivative Expansion Procedure Assume that a solution [1, 5] to the differential equation §

+ y= ^ ( y , | ) ,

o<

£

«i,

(5.116)

which is uniformly valid for all times is given by an asymptotic expansion having the form m

y(t,e) = ^ e * y t ( * o , * i , . • • ,*m) + 0 ( e m + 1 ) , ifc=o

(5.117)

where tk = ekt.

(5.118)

a-I/a;+<*"'>■

(«")

The corresponding time derivative is

/t=o Using Eqs. (5.117) and (5.119) to calculate the first and second derivatives of y gives the expressions: ^ = D0y0 + e(A>Vi + Diy0) + e2(D0y2 + DlVl dt

+ D2y0) + 0(e3),

(5.120a)

192

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

dt2

= Dly<> + e(D%yi + 2

2D0Diyo)

2

+ e {D y2 + 2D0Diyi + D\Vo + 2D0D2y0) + 0(e 3 ),

(5.120b)

where Dk = -i-. dtk

(5.121)

The initial conditions = A,

m

« = 0 , dt

(5.122)

become, if the calculation is only done to order e 2 : yo(0,0,0) = A,

yt(0,0,0) = 0,

ayo(o,o,o)

0,

dt0 ayx (0,0,0)

at„ ay 2 (0,0,0)

at 0

(5.123a) (5.123b)

%o (0,0,0) +

at,

= 0

dt,

Aft (0,0,0) +

k>l,

(5 123c)

'

-

ayo(0,0,0) +

a*2

=0

-

(5 123d)

-

The basic procedure for the derivative expansion method is to substitute Eqs. (5.120) into Eq. (5.116), collect together terms of the same power of e and set them equal to zero, and solve the resulting differential equations. These solu­ tions will contain arbitrary functions of the variables i o , < i , . . . ,tm. However, these functions can be determined by requiring that no secular terms appear. The next section gives three examples to illustrate the derivative expansion method. Other worked examples are given in references [1, 5]. 5.5 Worked E x a m p l e s Using t h e Derivative Expansion P r o c e d u r e

5.5.1 Example A For the first example, consider the linear damped oscillator
(5.124)

MULTI-TIME EXPANSIONS

193

To terms of order e 2 , the solution has the representation 2

Substituting Eqs. (5.125) and (5.120) into Eq. (5.124), and equating the coefficient of each power of e to zero, gives the three equations Dly0+y0

= 0,

Dlyi + yi = -2D0Dm D20y2 + Vi = -2£> 0 yi - 2Dm

- 2D0Diyi

(5.126) - 2D0y0,

(5.127)

- D\y0 - 2D0D2yo.

(5.128)

The solution to Eq. (5.126) is yo(to,ti,t2)

= A0(ti,t2)cost0

+ B0(ti,t2)smt0.

(5.129)

Substituting this expression into the right-side of Eq. (5.127) gives Dlvi + yi = 2 f ^ -

+ AQ) smt0 - 2 (^-

+ B0) cost0.

(5.130)

Secular terms in the solution for y\ can be eliminated by requiring AQ and Bo to satisfy the following differential equations: ^ + A > = 0 , at i

(5.131a)

^

+ B 0 = 0.

(5.131b)

D20yi + yi = 0.

(5.132)

Thus, the equation for y\ is

The solutions to Eqs. (5.131) and (5.132), respectively, are A0 = a0(t2)e-tl,

(5.133a)

194

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS Bo = A>(<2)e-(1,

(5.133b)

and yi = J 4 i ( t i , t 2 ) c o s i 0 + Bi(ti,t2)

(5.134)

sin to-

Substituting Eqs. (5.133) into Eq. (5.129) gives Vo = ao(<2)e -<1 cos to + Po{t2)e~tx sini 0 -

(5.135)

If Eqs. (5.134) and (5.135) are substituted into the right-side of Eq. (5.128), then the following differential equation is obtained for y2:

Dly2 + y2 = - 2 ^ - - 2B1 + doe-'* Oil

+

2 ^ e - " dt2

Ft A dcto 2 -^- + 2AX + / V _ < 1 + 2 dti dt2

cos to sint 0 .

(5.136)

Secular terms can be eliminated in the solution for y2 if the following two equations are satisfied: dBl

dt

+B

-

dt2)e

'

r + * - ( f + £)•-■

(5.137a) (5.137b)

Note that yi will have secular terms unless the right-sides of Eqs. (5.137) are zero. Setting them equal to zero gives the following pair of differential equations for c*o and /30:

dp0 __ a 0

H^ ~ Y' dap _ _ / ? 0

dt2 ~

T*

(5.138a) (5.138b)

These equations have the solution a 0 ( t 2 ) = Cj cos f -% + A ,

(5.139a)

Po(t2) = C, sin ( f + ^V

(5.139b)

MULTI-TIME EXPANSIONS

195

where C\ and are constants. If Eqs. (5.139) are substituted into Eq. (5.135) and use is made of the relation cos(0! — 62) = cos(?i cos 02 + sin#i sin0 2 )

(5.140)

then the expression for t/o becomes yo('o,
(l

cos ( « . - ! ) - ♦ ■

(5.141)

The application of the initial conditions, expressed in Eq. (5.123a), allows C\ and <j> to be determined. They have the values Cj = A and <j> = 0. Therefore, the first approximation to the solution to Eq. (5.124) is ft y{t , e) = Ae~ cos ( l - ^-) t + 0(e).

(5.142)

5.5.2 Example B The second example is the van der Pol differential equation
d 2^ y

n

(5.143)

To terms of order e, yo and yi satisfy the following differential equations: (5.144a)

Dlyo + Vo = 0 OoVi + 2/1 = -ZDoDm

+ (1 -

vl)D0ya.

(5.144b)

The solution to the first equation is (5.145)

y0 = Ao(
LdA«

A [i

(Al + BlW\

,.(-.£ + * 1 +

Al+Bl

/ t e r m s that do not produce secular terms )

sin in

")]} cos to (5.146)

196

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

The elimination of secular terms requires that A0 and B0 satisfy the following equations:

,£-4-(4±*) .1 A

2•>

dBo

^ ~

The functions A0(h)

/ A'l

'

B -R 0

I D2\ 1

I _ [ " » ; ~» | .

(5.147a) (5.147b)

and B0(U) satisfy the following initial conditions 4 , ( 0 ) = A,

flo(0)

= 0.

(5.148)

Evaluating Eq. (5.147b) at
The results for B0(ti),

(5.149)

0.

given by Eqs. (5.148) and (5.149), imply that (5.150)

B 0 (*i) = 0. Consequently, Eq. (5.147a) becomes

ȣ-*(-3)-

(5.151)

The solution to this equation is 2A A0(t1)

= [A2 +

(4-A2)e~*i]1/2'

(5.152)

Thus, to terms of order e, the solution to Eq. (5.143) is

tf(*> 0 =

2 A cos t

m + °(e)-

(5.153)

5.5.3 Example C The last example is the Rayleigh equation

dt2

+ y =e

dy _ fl\ dt \3

(dy dyY dt)

(5.154)

MULTI-TIME EXPANSIONS

197

The equations for y 0 (*o,*i) and yi(< 0 ,*i) are

Dfoo + yo = 0, D2oVi +Vi=

+ D0y0 - Q ) ( A , V o ) 3 -

-2D0Dm

(5.155a) (5.155b)

The solution to Eq. (5.155a) is given by Eq. (5.145). Substituting this expression for y0 into the right-side of Eq. (5.155b) gives DQVI + Vi =

+ [-2 ^

+ B, - ( & W + B})| cosi, 2

3^)cos3t0- (-^)(Ag-3^)sin3to.

(5.156)

The requirement of no secular terms in the solution for yj gives the following two equations for Ag and B0: 2 ^

-An

9dB°

- R

Ag + Bg

1-

^ + -Bo2

(5.157a)

(5.157b)

Note that Eqs. (5.147) and (5.157) are identical. Therefore, Ag and Bo are given, respectively, by the results of Eqs. (5.152) and (5.150). Therefore, the first approximation to the solution of the Rayleigh equation is 2A cos t

y(t, e) = 2

[A + (4 -

r-p, + 0(e) A2)e-«}1/2

(5.158)

Problems 5.1 Verify the results given by Eqs. (5.7), (5.10), (5.12b), and (5.13c). 5.2 Show that in the two-time expansion method yo, yi, and yi satisfy Eqs. (5.21), (5.22) and (5.23).

198

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

5.3 Prove that the general solution to Eq. (5.21) is given by the result of Eq. (5.24). 5.4 Verify the conditions given by Eqs. (5.27). 5.5 Derive Eqs. (5.87) and show that they have the solutions given by Eqs. (5.89). 5.6 Show that the result given by Eq. (5.102) follows from Eqs. (5.98), (5.100), and (5.101). 5.7 Derive the results given by Eqs. (5.123). 5.8 Derive Eqs. (5.126), (5.127), and (5.128) from Eqs. (5.124) and (5.125). 5.9 Construct solutions to the following nonlinear differential equations, to terms of order e2, using both the two-time and derivative expansion procedures: d2y

—

dy

„

+y =

e(l-y)-

2

dy

-d¥

2

ey

+ y =

, ,dy

-d¥+y = -e^li d2

y

d2y

-dW +

_L

y = e

c\

i

(b)

dy

- H


(a)

wdy

»"+($)'

(c) (d) (e)

(f)

References 1. A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973). See Chapter 6. 2. J. D. Cole and J. Kevorkian, in Nonlinear Differential Equations and Nonlinear Mechanics, editors, J. P. LaSalle and S. Lefschetz (Academic, New York, 1963); pp. 113-120. 3. J. D. Cole, Perturbation Methods in Applied Mathematics (Blaisdell; Waltham, MA; 1968). See Chapter 3.

MULTI-TIME EXPANSIONS

199

4. J. Kevorkian, in Space Mathematics, Part 3, editor, J. B. Rosser (American Mathematical Society; Providence, RJ; 1966). 5. A. H. Nayfeh, J. Math, and Phys. 44, 368 (1965). 6. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for and Engineering (McGraw-Hill, New York, 1978). See Chapter 11.

Scientists

7. A. H. Nayfeh and B. Balachandran, Nonlinear Dynamics: Concepts and Appli­ cations (Wiley-Interscience, New York, 1993). 8. M. H. Holmes, Introduction to Perturbation York, 1995). See Chapter 3.

Methods (Springer-Verlag, New

Chapter 6 GENERAL SECOND-ORDER SYSTEMS 6.1 Introduction Thus far, the book has considered planar dynamical systems that are modeled by a second-order differential equation having the form

f+ . - * ( . . * ) .

dt2

(")

where F is a rational function of its arguments and e is a small positive parameter. However, a general planar dynamical system can be represented by two coupled first-order differential equations, i.e.,

J = /(*»y),

(6-2a)

% = 9ix,y).

(6.2b)

The elimination of one of the variables, in Eqs. (6.2), to give a single secondorder equation in the other variable usually leads to an expression that is rather complex in form. For example, the two coupled first-order equations dx — = x - xy, ^

= -y + xy,

(6.3a) (6.3b)

provides an elementary model of a predator-prey interaction. The variable x satisfies the following nonlinear, second-order differential equation: x ^ - - ( l - x + x2)^- + x2+ x3 =0. dt'dt

(6.4)

A direct comparison of Eqs. (6.3) and (6.4) indicates that the first-order system may be preferable to work with than the single, second-order equation.

®O0

GENERAL SECOND-ORDER SYSTEMS

201

For many dynamical systems [1—7], a single parameter plays an important role in characterizing the behavior of the solutions. In this case, the equations of motion take the form dx -fi = f{x,y,fi),

(6.5a)

-^ = 9{x,V,IJ),

(6.5b)

where /z is a parameter. The following questions arise in the analysis of Eqs. (6.5): (i) Where are the equilibrium states or fixed-points located and what are their (linear) stability properties? (ii) Do stable limit cycles exist and, if so, what are the approximate values for their amplitudes and frequencies? (iii) How do the properties of the fixed-points and the limit cycles change as the parameter n is varied? The main purpose of this chapter is to provide answers to these questions. The next section examines the differential equations that model the oscillations in the biochemical reaction of glycolysis. It is shown that a geometrical analysis of the phase plane of the variables gives only two possibilities for the solution behaviors to the differential equations. To proceed further in the analysis of these types of equations, additional mathematical tools have to be introduced. Section 6.3 states a form of the Hopf bifurcation theorem, which is then used to find the conditions for which the glycolysis reaction has periodic solutions, i.e., a stable limit cycle exists. In Section 6.4, a general procedure is given for determining the conditions under which Eqs. (6.5) have a ''small" amplitude limit cycle and, if such a limit cycle exists, its stability properties.

The procedure combines the results of the Hopf

bifurcation theorem with that of an averaging method. Section 6.5 gives a "recipe" for directly calculating the bifurcation parameter, and the amplitude and angular frequency of any limit cycles. Finally, in Section 6.6, a number of worked examples are given to illustrate the use of the procedure derived in Section 6.4.

202

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

6.2 Glycolytic Oscillator [11, 12] The model, dimensionless equations of motion for this biochemical reaction, are — = - i + ay + x2y, at

(6.6a)

^ = b-ay-x2y, (6.6b) at where a and 6 are positive parameters. The trajectories in phase space, y = y(x), are solutions to the following differential equation: dy_ _ b-ayx2y dx —x + ay + x2y

,g

The general behavior of these trajectories can be determined by finding the nullclines to Eq. (6.7). Observe that dy/dx is zero on the curve yo(x), where *(») = ~ ^ , and is unbounded on the curve y^x),

(6-8)

where

Voo = — £ - j . a + xl

(6.9)

Figure 6.2.1 gives a sketch of the nullclines along with an indication of the regions where dy/dx has a definite sign. Detailed examination of this figure shows that all trajectories that originate in the first quadrant remain in the first quadrant. Also, the Eqs. (6.6) have only a single fixed-point located at s

= 6'

v-

—na ■ a + bz

( 6 - 10 )

Figure 6.2.2 gives typical phase space trajectories that originate "far" from the fixedpoint. The behavior of these curves leads to the following conclusion: Trajectories in phase space that start sufficiently far from the fixed-point, located at (x, y), spiral in toward the fixed-point, i.e., no plane space trajectory that originates in a finite portion of the first quadrant becomes unbounded.

203

Figure 6.2.1. The nullclines for Eqs. (6.6).

204

Figure 6.2.2. Typical phase space trajectories for Eqs. (6.6) that originate far from the fixed-point.

GENERAL SECOND-ORDER SYSTEMS

205

To complete this analysis, the behavior of trajectories near the fixed-point must be determined. There are essentially two possibilities: (1) The fixed-point, (x,y),

is stable and all trajectories eventually end on it.

This corresponds to the fixed-point being a global attractor. (2) The fixed-point, (x,y),

is unstable. For this case one or more limit cycles

may exist. Just which case occurs depends on the values of the parameters a and b. Fig­ ure 6.2.3 illustrates these two situations. Figure 6.2.3a corresponds to the fixedpoint being stable; consequently, it is a global attractor. For Figure 6.2.3b, the fixed-point is unstable. The figure drawn is for the case of a single, stable limit cycle surrounding the fixed-point. If the limit cycles are nondegenerate, then, in general, an odd number of limit cycles can occur. This type of geometrical analysis will not allow the determination of the exact number of limit cycles. The stability of the fixed-point can be found by applying linear stability anal­ ysis. Let x(t) = x + rn(t),

(6.11a)

y(t) = y + r)2(t),

(6.11b)

where r]i(t) and ri2(t) are perturbations about (x,y) 0
and

0
(6.11c)

The rj's satisfy the equations (6.12) dt\n2)

\Ml

^ 2 2 / V?2

where An = _ l + - ^ _ , a + bl A

u

= — ^ , a + bl

+ b2 ,

(6.13a)

A 22 = - ( a + 6 2 ).

(6.13b)

Al2=a

206

Figure 6.2.3. (a) Fixed-point is a global attractor. (b) Fixed-point is unstable and there is one stable limit cycle.

GENERAL SECOND-ORDER SYSTEMS

207

The matrix A has the following properties: D = det A = a + b2 > 0, T = trace A = —

(6.14a)

&4 + ( 2 a - l ) + a ( l + a ) " a + b2

(6.14b)

The two eigenvalues of A are

X1(a,b) = f~\

[T+^T2-4D]

,

(6.15a)

A2(a, 6) = Q) [r - VT* - W] ,

(6.15b)

where the fact that Ai and A2 depend on the parameters a and 6 has been indicated. The solution for the perturbation functions, r?i(i) and rj2(t), takes the form r ?1 (t) = C 1 e Al< + C 2 e A 2 ',

(6.16a)

»72(i) = C 3 e A l ( - r C 4 e A 2 \

(6.16b)

where C3 and C\ are linear combinations of C\ and C 2 . Prom Eqs. (6.15), it follows that if T > 0, then the eigenvalues have ''real parts" that are positive. If this is true, then the perturbations, »?i(t) and ?72(i), grow with time. Consequently, the fixed-point (x,y) is unstable. Likewise, if T < 0, the fixed-point is stable since the magnitude of the perturbations goes to zero. Equation (6.14b) shows that T is a function of the parameters a and 6. In terms of a and b, the boundary curve separating the parameters for stable and unstable fixed-point behavior is given by T = 0,

(6.17)

& 4 - r ( 2 a - l ) & 2 + a ( l + a) = 0.

(6.18)

Solving for b2 gives

b\{a) = Q) [1 - 2a + VT^\

,

(6.19a)

208

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS b2_(a) = Q

[1 - 2a - VT=8a]

.

(6.19b)

These functions define the two branches of a double valued function b2(a) in the (a,b2) parameter plane. The function 6 2 (a) has the following properties: (1) 6 2 (a) is defined only for 0 < a < 0.125. (2) Everywhere in this interval of a values, except for a — 0.125, the function 2

6 (a) has two positive values. For example, at a = 0, 62+(0) = 1,

62_(0) = 0,

(6.20)

and at a = 0.125, 6^(0.125) = 6i(0.125) = 0.375.

(6.21)

(3) The slope of b\(a) and 61(a) are db\ da db2_ da

fl[ 1 ! |

2 1J < 0, ^1 - 8a 2

Jl-8a

> 0.

(6.22a)

(6.22b)

Thus, in the interval 0 < a < 0.125, 6+(a) has a positive slope, while b2_(a) has a negative slope. (4) Figure 6.2.4 provides a sketch of the general shape of the boundary curve given by Eq. (6.18) or Eqs. (6.19). Evaluation of b2(a) at any point inside the "fingertip" shaped region shows that T > 0 and consequently, the fixed-point is unstable in this region of parameter space. (5) If the fixed-point is unstable, then one or more limit cycles can exist. (See Appendices G and I.) However, this analysis will not determine the precise number of limit cycles or their approximate location in the phase space. To accomplish this goal, more powerful mathematical tools must be used. One such tool is the Hopf bifurcation theorem which is introduced in the next section.

209

Figure 6.2.4. General shape of the boundary curve given by Eq. (6.18). The fixedpoint is stable in the region outside of the "fingertip" shaped area.

210

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

6.3 Hopf Bifurcation T h e o r e m [13] Consider the following pair of first-order, coupled differential equations: f

= /(*,y,^),

(6.23a)

f

=
(6-23b)

where /i is a parameter whose value can be varied. The steady state or fixed-points are obtained from the solutions to the equations

/(2,»,/0 = 0,

s ( M , / i ) = 0,

(6.24)

y = y(fi).

(6.25)

and will, in general, depend on fj,, i.e., x = x{n),

The essential content of the Hopf bifurcation theorem is that it predicts the appear­ ance of a limit cycle about a fixed-point whenever the stability of the fixed-point changes as the parameter JJ. is varied. Thus the theorem is of value for investigating the conditions under which limit cycles can exist for Eqs. (6.23). The Hopf bifurcation theorem, as it will be stated below, can be generalized to N > 2 coupled, first-order differential equations. This generalization and the associated formal proofs are given in the references [14-17]. A rather informal, but excellent introduction to this theorem and some of its applications is presented in reference [18]. 6.3.1 The Theorem [19] Consider two coupled, first-order differential equations that depend on a pa­ rameter jj,: dx — = f(x,y,fj,),

(6.26a)

^=fl(x,y,/x).

(6.26b)

GENERAL SECOND-ORDER SYSTEMS Assume that f(x,y,fi)

and g(x,y,n)

are continuous functions of (x,y,fi)

211

and have

partial derivatives with respect to these variables. Assume that for each value of H the equations have a fixed-point, [x(fi),y(ft)],

that depends on fi. Define the

Jacobian matrix, evaluated at [x(^),y(/i)], to be

J00=(& \ dx

|S)

,

(6-27)

dy / (i,p)

and denote the eigenvalues of this matrix to be

Ai l2 00 = a(/0 ± »&(/*).

( 6 - 28 )

Let there exist a value of fj, = fi*, called the bifurcation value, such that a(/i*) = 0,

b(n*) ? 0,

(6.29)

and as fi is varied through /J,*, the real parts of the eigenvalues change signs, i.e.,

Given these conditions, the following three possibilities can occur: (1) When fi = fi", a center is created at the fixed-point and infinitely many neu­ trally stable concentric closed trajectories surround the fixed-point, [x(fi*),y({i*)], (See Figure 6.3.1.) (2) There exists an open interval of \i values, VTKIKIH, such that a single closed trajectory surrounds the fixed-point [x(fi),y(fi)].

(6.31) (See

Figure 6.3.2.) This closed trajectory is a limit cycle. As \x is varied, the size, R(/J,), of the limit cycle changes as

a0i)<x|/i-vi 1 / 2 .

(6.32)

212

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

There exists no other closed trajectories near

[x{fi),y(n)]-

This case is known as a supercritical bifurcation since the limit cycle occurs for

(3) There exists an open interval of /J. values fi2
//*,

(6.33)

such that a single closed trajectory surrounds the fixed-point at [x(fi), J/(AO]- Similar conclusions to those of case 2 hold for this situation. This case is known as a subcritical

bifurcation.

Comments (i) For many applications, the functions f(x,y,fi)

and g(x,y,fi)

are analytic

functions of (x, y, n). Consequently, all the partial derivatives, with respect to these variables, exist. (ii) The theorem does not provide any information on the values of n\ and \t-i- Consequently, the theorem, in general, holds only for values of fi close to the bifurcation value /J,*. (iii) In general, a(/j) ^ 0 and b(fj.) ^ 0, except for isolated values of (i. Thus, the trajectories in phase space near the fixed-point will be either stable spirals, for a((j.) < 0, or unstable spirals, for a((i) > 0. Consequently, the Hopf bifurcation theorem predicts the appearance of a limit cycle about a fixed-point that changes stability as the parameter fj. is varied. (iv) For either the supercritical or the subcritical situations, the size of the limit cycle increases continuously from zero for fi close to fi*\ see Eq. (6.32). (v) For n close to ^t*, the angular frequency of the motion on the limit cycle is given approximately by the expression w = b(fj.*).

(6.34)

213

Figure 6.3.1. At \i = fj,*, a center is created and infinitely many neutrally stable closed trajectories surround the fixed-point.

214

Figure 6.3.2. An isolated limit cycle exists for fi satisfying either of the conditions given by Eq. (6.31) or (6.33).

GENERAL SECOND-ORDER SYSTEMS

215

A more precise relation is

T

= W) = ^ ) + 0 i " ~ ^

(6 35)

-

where T(n) is the period. (vi) Figures 6.3.3 and 6.3.4 illustrate what happens, respectively, for the cases of supercritical and subcritical bifurcations when p, is near /J,*. (vii) Finally, note that the Hopf bifurcation theorem does not, in general, allow for the a priori prediction of the critical value for the bifurcation parameter. 6.3.2 Example Consider the following coupled differential equations — = fix -y at

+ xy2,

(6.36a)

-]- = x-ny + xy3. (6.36b) at There is a single fixed-point at (x,y) — (0,0). The associated Jacobian matrix is JM=('i

;'),

(6-37)

A 1 ) 2 00 = /*±»-

(6-38)

and its eigenvalues are

Therefore, the functions a{fj.) and b(fi) are a(/i) = p,

6(/i) = 1,

(6.39)

and a(0) = 0,

^

= 1^0.

(6.40)

Also, the real parts of the eigenvalues, a(/j), changes sign at \i = 0 and, for ft = 0, the linear part of Eqs. (6.36) has a fixed-point that corresponds to a center. Thus,

216

Figure 6.3.3. A supercritical bifurcation as ft is varied: (a) stable spiral; (b) neu­ trally stable closed trajectories; (c) stable limit cycle.

217

Figure 6.3.4. A subcritical bifurcation as y, is varied: (a) unstable spiral; (b) neu­ trally stable closed trajectories; (c) unstable limit cycle.

218

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

H* = 0, and it can be concluded that the system of equations undergoes a Hopf bifurcation at p, = 0. The next issue to investigate is whether the bifurcation is subcritical or super­ critical. To proceed, transform to polar coordinates x=rcos0,

y = rsin#.

(6-41)

A simple calculation gives

±=flr+ry2.

(6.42)

Since r > 0 and y2 > 0, then for p. > 0, r(t) obeys the condition r(t) > roe*".

(6.43)

This result implies that all trajectories are repelled from the origin and go out to infinity. Clearly, there are no closed trajectories for p > 0. Thus, the unstable fixed-point is not enclosed by a stable limit cycle and the bifurcation cannot be supercritical. The only other possibility is for the bifurcation to be subcritical, i.e., for p < 0, the stable fixed-point at (x,y) = (0,0) is enclosed by an unstable limit cycle. 6.4 General Procedure for Two-Variable S y s t e m s [8] This section investigates the possible periodic solutions to the following system of differential equations dx — =f(x,y,p),

, (6.44a)

^=g(x,y,P),

(6.44b)

where p denotes the A^-parameters that characterize the system. In particular, the conditions under which Eqs. (6.44) have a small amplitude limit cycle will be found. To do this, the relevant bifurcation parameter must be determined in terms of the

GENERAL SECOND-ORDER SYSTEMS

219

parameters p. In addition, the calculations will give the period and amplitude of the limit cycle along with its stability properties. To begin, assume that x = y = 0 is a fixed-point of Eqs. (6.44). It is also assumed that / and g are analytic about x = y = 0. This means that they can be expanded in Taylor series to give the following results for Eqs. (6.44): — = (ax + by) + (a2x2 + 2b2xy + c2y2) + (a3x3 + 3b3x2y + 3c3xy2 + d3y3) + • • •,

- £ = (ex + dy) + (A2x2 + 2B2xy + C2y2) at + (A3x3 + 3B3x2y + 3C3xy2 + D3y3) + ■■■,

(6.45a)

(6.45b)

where all the indicated constants are known when / ( x , y) and g(x,y) are given. The linear transformation [20] x = bu,

y = @u — Qv,

(6.46)

where 0=

(d -a)2 - v " ."' 4

-i 1/2

-bebe

,

(6.47a)

a+ d 2 ' d—a , 2 p= fl

(6.47b)

=

(6-47c)

when substituted into Eqs. (6.45) gives, after a long, but straightforward calculation, the result —- = (fiu — ilv) + (a2u2 + 2b2uv + c2v2) dt + (a3u3 + 3b3u2v + 3c3uv2 + d3v3) -\

,

~ = (ft« + pv) + (A2u2 + 2B2uv + C2v2) dt + (A3u3 + 3B3u2v + 3C3uv2 + D3v3) + ■■■,

(6.48a)

(6.48b)

220

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

where the barred parameters are given in terms of the parameters of Eqs. (6.45) by the following relations: a2=a2b

+

-62 = _ft(62 + c2 =

(6.49a)

2b20+(^-\

(6.49b)

^ ) ,

c 2 ft 2

a3 = a3b2 + 30b3b+3c302 63 = - n c3 = n 2

+ d30"<

636 + 2c3/3 + c3 +

(6.49c)

m

(6.49d) (6.49e)

d30

(6.49f) (6.49g)

<*3

c2/3

a20b + 20'b2 + -*£- - A2bl - 2bB20 - C202 B j = - ( /362 I C2

-»(x

C2^

6

6B 2 - C2/3

(6.49i)

)•

(6.49J)

C2

0[a3bz+30b3b+3c302

+

d303

- (A3b3 + 30B3b2 + 3C3b02 + D303) d30'

B3 = -0[b3b

+ 2c30 +

c 3 = n 0(c3

+ ^)-(C3b

d30

A-«■(*-&)

(6.49h)

+ (B3b2 + 2C3b0 + D302), + D30)

(6.49k) (6.49£) (6.49m) (6.49n)

GENERAL SECOND-ORDER SYSTEMS 221 The parameter fi, given by Eq. (6.47b), is the bifurcation parameter for the (u,v) differential equations [9]. Thus, ft and /? can be rewritten to show explicitly their functional dependence on n, i.e., n(fi)=[-(n-a)2-bcf/\ P{y.) = H-a.

(6.50) (6.51)

The transformation from (u,v) to polar coordinates (r, 0), u = r cos 0,

v = rsin8,

(6.52)

when substituted into Eqs. (6.48) gives -^ = pr + r2(j>3+r34 + ...,

(6.53a)

de — =ft+ rV>3 + r2V>4 + • • •,

(6.53b)

where 3 = [^2 cos 0 -f- 2&2 cos 0 sin 0 + c2 sin 6] cos 0 + [A2 cos 2 0 + 2B2 cos 0 sin 0 + C2 sin 2 0] sin 0,

(6.54a)

^>3 = — [a2 cos 2 6 + 262 cos 0 sin 0 + c2 sin 2 0] sin 0 + [A2 cos 2 0 + 2B2 cos 0 sin 0 + C2 sin 2 0} cos 6,

(6.54b)

<^4 = [a3 cos 3 0 + 363 cos 2 0 sin 0 + 3c~3 cos 0 sin 2 0 + d3 sin 3 0] cos 0 + [A3 cos 3 0 + 3 5 3 cos2 0 sin 0 + 3C3 cos 0 sin 2 0 + £>3 sin 3 0] sin 0, (6.54c) xpt = — [a3 cos 3 0 + 363 cos 2 0 sin 0 + 3c 3 cos 0 sin 2 0 + d3 sin 3 0] sin 0 + [A3 cos 3 0 + 3B3 cos2 0 sin 0 + 3C3 cos 0 sin 2 0 + 5 3 sin 3 0] cos 0. (6.54d) The differential equations for the (r, 0) variables are, in general, intractable since the right-sides of these equations depend on both r and 0. However, very useful equations can be obtained by applying an averaging procedure. To do this, first

222

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

note that the right-sides of Eqs. (6.53) are periodic in the variable 9 with period 2ir. Thus, averaging the right-sides over the 6 variable gives [10] % = fir + j>y + 0(r5), at ^

= n + Vi4r2 + 0 ( r 4 ) ,

(6.55a)

(6.55b)

at where / (f,3de = 0= / Jo Jo ( i \ y2" /3\

ipzdd,

^ = (2W / *4<w = I § J ^ + di + B*+ ^a)' ^4 = (h) J ^de = ( I ) ( ^ 3 + ^ - &3 - da)•

(6.56a)

(6 56b)

' (6.56c)

(The variables r and 6 in Eqs. (6.55) should be barred to indicate that they are ''averaged" quantities. However, the absence of bars should cause no difficulties.) From bifurcation theory [14-17], it follows that r = 0{^),

(6.57)

and from Taylor's expansion theorem & = *4(0) + &(0)/i + O(/x 2 ),

(6.58a)

^4 = & ( 0 ) + # ( 0 ) / i + 0(n2),

(6.58b)

noo = n(o) + n'(o)/i + o(n2).

(6.58c)

Using the result in Eq. (6.50) gives ft(0) = [-(6c + a 2 )] 1 / 2 , «(0)

= jfcy

(6.59a) (6.59b)

GENERAL SECOND-ORDER SYSTEMS

223

Multiply Eq. (6.55a) by 2r and using the definition z = r2 ,

(6.60)

-^ = 2fiz + 2faz2 + 0 ( z 3 ) .

(6.61)

this equation becomes

By introducing the new variable z, where z = nz,

z = 0(l),

(6.62)

a consistent first approximation can be obtained by use of the equation dz

r

-

n

— = 2fiz [1 + 4>4(p)s] ,

(6.63)

where the parameter {i is assume to be small, i.e., M < 1.

(6.64)

This expression is correct to terms of order ft. Likewise, under the same assump­ tions, Eq. (6.55b) becomes

d =m+

^

W)+tlMo)"-

(6 65)

-

Consider now the differential equation given by Eq. (6.63). This equation has two fixed-points; they are h = 0,

z2 = - j

— .

(6.66)

The equilibrium state, z.\ = 0, is unstable for ft > 0 and stable for ft < 0. With this information, all of the possible solution behaviors for z{i) can be determined: Case 1: \x > 0, <£4(0) > 0. For this situation, the derivative of z{t) is positive for all z > 0; therefore, the amplitude of the motion increases without bound.

224

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Case 2: /* > 0, <^4(0) < 0. The equilibrium state, z\ = 0, is unstable. However, there is a stable limit cycle having the amplitude value 1 zo = - &(0)'

(6.67a)

1/2

ro =

(6.67b) .(0)

Case 3: // < 0, <£4(0) > 0. For this case, the derivative of z(t) is everywhere negative; hence, all solutions decrease to zero, i.e., the origin is globally stable. Case 4: n < 0, <^4(0) < 0. This situation corresponds to the equilibrium state being locally stable and the limit cycle unstable. A stable limit cycle exists only for Case 2. Consequently, the steady-state periodic motion has the amplitude given by Eq. (6.67b) and an angular frequency tj(ju) that can be determined by substitution of Eq. (6.67a) into Eq. (6.65), i.e., w(/x) = ft(0) + fi

a

&(0)"

fi(0) fc(0)

(6.68)

where, the steady-state phase is $(t) = w(/i)( + 6Q,

0O = constant.

(6.69)

In summary, if Eqs. (6.45) have a limit cycle solution, then a first approximation to this solution is x(t) = br0 cos(ut + 60)

(6.70a)

y(t) = —ar0 cos(u;t + 0O) — ft(0)ro sin(wt + 0O),

(6.70b)

GENERAL SECOND-ORDER SYSTEMS

225

where r 0 and u are given, respectively, by Eqs. (6.67b) and (6.68). Observe that all of these quantities are directly determined by the parameters in the original pair of differential equations, namely, Eqs. (6.45). 6.5 T h e R e c i p e This section gives a recipe for obtaining the value of all the factors in Eqs. (6.70) directly from the parameters expressed in the original system of differential equa­ tions, i.e., Eqs. (6.45). It should be emphasized that in the following procedure, at each step, only terms to lowest order in fj, or its powers should be retained. Steps in the Recipe (1) From Eqs. (6.45), obtain the values of the relevant parameters: (a,b, c,d), ( a 2 , 6 2 , c 2 ) , (A2,B2,C2),

( a 3 , 6 3 , c 3 , d 3 ) , and {A3,B3,

C 3 , D3).

(2) Calculate fi, /? and ft(0):

t*

P

a+d 2 '

(6.71)

d—a

(6.72)

2 '

n(0) = [-(6c + a 2 )] 1 / 2

(6.73)

(3) Calculate <£4(0) and ^ ( 0 ) : & ( 0 ) = ( I ) {b2a3 - 2a&&3 + b2B3 - 2abC3 + [a2 + fi2(0)](d3 + D3)} ,

Mo) =

8fi(0)

-b3A3+ab2{3B3-a3)

- 3a2&(C3 - 6s) + a\D3 3O(0)

+

(6.74)

- 3C 3 ) +

2a2
,„ 3 +

bC

aD

3

(6.75)

226

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS (4) Calculate r0 and w: 1/2

ro

&(0)J

w = fi(0) + \i

(6.76)

'

a

&(0)

|fl(0)

^4(0)J

(6.77)

(5) The solution, to this level of approximation, is (6.78)

i ( i ) = bro cos(o;< + #o) y(t) = — aro cos(u>t + 6Q) — n(0)r o sin(u>t + 00),

(6.79)

where 00 is a constant. 6.6 Worked Examples This section presents a number of examples that illustrate the application of the "recipe'' rules of the previous section. All of the pairs of first order differen­ tial equations model important systems that arise in the natural and engineering sciences. 6.6.1 van der Pol Equation [21] T h e van der Pol e q u a t i o n w r i t t e n in s y s t e m form is

dx

dy_ dt

(6.80a)

(-x + ey) -

ex2y,

(6.80b)

where e > 0. Comparison with Eqs. (6.45) gives a = 0,

6=1,

c = —1,

02 = 02 = C2 = 0 ,

A2=B2

= C2= 0,

d = e,

(6.81a) (6.81b) (6.81c)

GENERAL SECOND-ORDER SYSTEMS a3 = b3 = c3 = d3 = 0, A0 = 0,

353 = -e,

C3=D3

227

(6.81d) = 0.

(6.81e)

Substitution of these values into Eqs. (6.71) to (6.79) gives /i = ^ ,

fl(0)

= l,

r 0 = 2,

n'(0) = 0,

(6.82a)

w = 1,

(6.82b)

x(t) = 2cos(t + 90),

(6.83a)

and

y{t) = -2sm(t

+ 0o).

(6.83b)

This is the usual (first order in the small parameter e) result. 6.6.2 WCM Oscillator [22] This equation is a large amplitude generalization of the van der Pol differential equation. In system form, the equations are dx -^ = y,

(6.84a)

x e M==-x - ++ 4i-^)y> [T^

( 6 - 84b )

where e and fi are positive parameters that satisfy the conditions 0<e
0 < p. < 1.

(6.84c)

The phase space of this system of equation is restricted to the region |z| < 1,

\y\ < oo.

(6.85)

The Taylor series of the right-side of Eq. (6.84b) is ^

= [-x + (ep)y] + e(fl - l)x2y + ■■■

(6.86)

228

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Comparison of Eqs. (6.84a) and (6.86) with Eqs. (6.45) gives a = 0,

6=1)

c = —1,

d

(6.87a)

= £/i,

(6.87b)

a2 = b2 = c2 = 0, A2=B2

(6.87c)

= C2 = 0,

(6.87d)

Q3 = h = Ci = di = 0,

A3 = 0,

3B 3 = e(ji - 1),

(6.87e)

C 3 = D3=0.

Therefore, from Eqs. (6.71) to (6.79), it follows that H= f,

tt(0)

= 1,

r„ = 2 - ^ ,

(6.88a)

n'(0) = 0,

(6.88b)

w = 1,

and x(t) = 2i//Icos(t + #o),

(6.89a)

y(t) = -2y/£sm(t

(6.89b)

+ 60).

This approximate solution agrees with the result of other calculations [22]. 6.6.3 Batch Fermentation [2] A model for batch fermentation processes is given by the following system of differential equations: dx — = -{AaP)y

J

- [(Aa)xy + (Ap)y2} - Axy2,

= [(Ba/3)x + (Byl32)y} + [(aB)x2 + (2B07)xy + [(Bj)x2y

- (2B(3e)xy2] - (Be)x2y2,

-

(6.90a)

(B/32e)y2] (6.90b)

GENERAL SECOND-ORDER SYSTEMS where (.A, B,a,/3,7,

229

e) are the system parameters. Again, comparison of these equa­

tions with Eqs. (6.45) gives a = 0,

b=-Aa/3, a2 = 0,

262 = -Aa,

A2 = ofl, a3 = 0 , A3 = 0,

3 S 3 = Bj,

(6.91a)

c2 = -Afi,

(6.91b)

C2 = -Bp2e,

(6.91c)

2B2 = 2 5 / 3 7 , 63 = 0,

d = By/32,

c = Bap,

3c3 = -A, 3C 3 = -2B/3e,

d3 = 0,

(6.91d)

D3 = 0.

(6.91e)

£1'(0) = 0,

(6.92a)

— U,

(6.92b)

Prom these parameter values it follows that By82 H = —J-,

fi(O)

T

o = ~ ,

. = afiVAB,

u = ap*VAB-

OJ./I

[23eJ

\

y A 1

and x(r) = -2/3^/7 cos(wt + 0„),

(6.93a)

y(r) = -23 J — sin(o>< + 0O)V "

(6.93b)

6.6.4 Brusselator Model [23] This simple model of a hypothetical set of chemical reactions that lead to oscillations is described by the system of equations: ^ = 1 - (1 + 3)X + aX2Y, dt

(6.94a)

~ = 3X- aX2Y, (6.94b) dt where a and ft are positive parameters. This system has one fixed-point located at X = 1,

Y = ^. a

(6.95)

230

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

The transformation X = 1 + x(t),

Y = £ + y(t), a

(6.96)

when substituted into Eqs. (6.94), gives § = [(/? - l)x + ay] + (px2 + 2axy) + ax2y, at

(6.97a)

§ = - ( / ? * + ay) - (fix2 + 2axy) - ax2y. at

(6.97b)

Comparison of these equations with Eqs. (6.45) gives the following identifications: a = /3-l,

b = a, a2 =0,

A2 = -0, a3 = 0, A3 = 0,

c = -p,

b2= a,

c2 = 0,

B2 = -a, 363 = a,

d=-a,

3B3 = -a,

(6.98b)

C2 = 0,

c3= 0,

(6.98c)

d3 = 0,

C3 = 0 ,

(6.98a)

D3 = 0.

(6.98d) (6.98e)

In contrast to the previous three examples, a detailed accounting will be given of the calculations of the various quantities needed to determine a first approximation to the solutions x(t) and y(t). The bifurcation parameter, fi, is _ a + d _ /?-(! +a) ^~ 2 2 '

(6.99)

The quantity ft(0) is given by the expression

fl(0) = [-(6c + a 2 ) p / ^ = o = [-a(-P) -(P= [ap-p2+2p-l]\ii=0.

l)2]1/2!^ (6.100)

Prom Eq. (6.99), it follows that P = 2fj, + (l + a) = l + a + 0(fi).

(6.101)

GENERAL SECOND-ORDER SYSTEMS

231

Consequently, ft(0) is (6.102)

fi(0) = y/a + 0(n). The quantity <^4(0) is given by the expression

&(0)= Q ) [-20?-1) (£)+<«*(-«) ,1=0

= -(y)[2(/3-l)-3a]|M=0.

(6.103)

Using the result of Eq. (6.101), in this last equation, gives

**<°>—(T)

<0.

(6.104)

From the discussions given in Section 6.4, it follows that Eqs. (6.97) have a stable, small amplitude limit cycle if

,-'-<

1 +

">>o,

(6.105)

with fi "small." For this case, the equilibrium state, x(t) = 0 and y(t) = 0, is unstable. The quantity r 0 is i/2 i"o

=

.(0)

/ o \ / . . \ 1/2

=(M

(6.106)

Therefore, using the parameter values of Eqs. (6.98) and the above ro, the solutions x(t) and j/(t) are determined from Eqs. (6.78) and (6.79). They are given by the following expressions: 1/2

c(t) = 2 1 1/2

y{t) = - 2

cos (y/at + 0O) ,

cos (v'crf + 0O) - ( - ] (2/z) 1/2 sin (y/Zt + 0O) •

(6.107a)

(6.107b)

232

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

In terms of the variables in the original differential equations, i.e., Eqs. (6.94), the solution can be written as X{t)=l Y(t) = (£)

+ x(t),

(6.108a)

+ y(t).

(6.108b)

Problems 6.1 Show that Eq. (6.4) is equivalent to the two coupled first-order equations given by Eq. (6.3). 6.2 Prove that the nullclines divide the first quadrant of the (x, y) plane as indicated in Figure 6.2.1 and that in each region the derivative, dy/dx, has the indicated sign. 6.3 State the Hopf bifurcation theorem for iV-coupled first-order differential equa­ tions [24, 25]. 6.4 Derive Eq. (6.42). 6.5 Show that the linear transformation of Eq. (6.46) when substituted into Eqs. (6.45) gives the results of Eqs. (6.48) and (6.49). 6.6 Derive Eqs. (6.53) and (6.54) from the transformation of Eq. (6.52). 6.7 Evaluate the integrals to verify the results of Eqs. (6.56). 6.8 Can the order \x term on the right-side of Eq. (6.68) be justified at this level of the calculation? If yes, then why can it be included? If no, then why should it not be included? 6.9 Derive the results for the first approximations to x(t) Eqs. (6.70).

and y(t) given by

6.10 Why is the phase space for the system of Eqs. (6.84) restricted to the region given by Eq. (6.85). 6.11 Rewrite the solutions for x(t) and y(t), given by Eqs. (6.93), in terms of the bifurcation parameter (JL of Eq. (6.92a).

GENERAL SECOND-ORDER SYSTEMS

233

6.12 Apply the Hopf bifurcation theorem to the glycolytic oscillator modeled by Eqs. (6.6). Can a stable limit cycle exist? If so, what are the conditions for it to exist? 6.13 The following equations model the chlorine dioxide-iodine-malonic acid reaction [26]: dx 4iy = a — x — dt 1 + x2 $ = &* 1 dt 1 + x2 where a and 6 are positive. Show that this system has limit cycle in the first quadrant (i.e., x > 0, y > 0) provided a and 6 satisfy certain restrictions. What is the bifurcation parameter \i1 Calculate the limit cycle for small values of fi. 6.14 The following system of equations models a predator-prey interaction [18]: — = x[x(l

-x)-y]

!Tt=y{x-a) where x > 0, y > 0 and a > 0. Apply both geometrical techniques and the Hopf bifurcation theorem to analyze the behavior of the solutions to this system. References 1. R. E. Mickens, Nonlinear Oscillations (Cambridge University Press, New York, 1981). 2. Y. Lenbury, P. S. Crooke, and R. D. Tanner, BioSystems 19, 15 (1986). 3. 0 . Sporns and F. F. Seelig, BioSystems 19, 83 (1986). 4. E. Beltrami, Mathematics for Dynamic Modeling (Academic, Boston, 1987). 5. L. Edelstein-Keshet, Mathematical Birkhaiiser, New York, 1988).

Models in Biology (Random House/

6. P. N. V. Tu, Dynamical Systems: An Introduction with Applications in Eco­ nomics and Biology (Springer-Verlag, Berlin, 1992).

234

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

7. S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley; Reading, MA; 1994). 8. R. E. Mickens, BioSystems 24, 31 (1990). 9. T. V. Davies and E. M. James, Nonlinear Differential Equations (AddisonWesley; Reading, MA; 1966). See Section 3.8. 10. J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamic Sys­ tems (Springer-Verlag, New York, 1985). 11. See reference 7, pp. 205-208. 12. E. E. Sel'kov, Eur. J. Biochem. 4, 79 (1968). 13. E. Hopf, Ber. Verh. Sachs. Akad. Wiss. Leipzig, Math.-Nat., 95, 3 (1942). 14. B. D. Hassard, N. D. KazarinofF, and U. Y. H. Wan, Theory and Applications of Hopf Bifurcation (Cambridge University Press, New York, 1981). 15. J. E. Maxsden and M. McCracken, The Hopf Bifurcations and Its (Springer-Verlag, Heidelberg, 1976).

Applications

16. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, Heidelberg, 1986, 2nd edi­ tion). 17. D. K. Arrowsmith and C. M. Place, An Introduction (Cambridge University Press, Cambridge, 1990).

to Dynamical

Systems

18. G. M. Odell, Appendix A3, in Mathematical Models in Molecular and Cellular Biology, L. A. Segel, editor (Cambridge University Press, Cambridge, 1980). 19. The statement of this theorem and the discussion that follows is based on reference [5]; see pp. 341-344. 20. R. H. Rand and D. Armbruster, Perturbation Methods, Bifurcation Theory and Computer Algebra (Springer-Verlag, New York, 1987). See pp. 20-22. 21. S. van der Pol and J. van der Mark, Nature 120, 363 (1927). 22. R. E. Mickens, J. Sound and Vibration 130, 513 (1989).

GENERAL SECOND-ORDER SYSTEMS

235

23. S. K. Scott, Chemical Chaos (Clarendon Press, Oxford, 1991). See pp. 60-61. 24. See reference [5], p. 344. 25. M. Farkas, Periodic Motions (Springer-Verlag, New York, 1994). See Section 7.2. 26. I. Lengyel, G. Rabai, and I. R. Epstein, J. Am. Chem. Soc. 112, 9104 (1990).

Appendix A MATHEMATICAL RELATIONS This appendix gives various mathematical relations that are used regularly in the calculations of the text. The references listed at the end of this appendix contain extensive tables of other useful mathematical relations and analytic expressions. A . l Trigonometric Relations A.1.1 Exponential Definitions of Trigonometric

sinx=

Functions

e'z — e~'x — e'i xi +I e„ - n

(A.l) (A.2)

A.1.2 Functions of Sums of Angles

sin(x ± y) = sin i cos y ± cos x sin y

(A.3)

cos(i ± y) = cosx cosy ^f sin a; siny.

(A.4)

A.1.3 Powers of Trigonometric

Functions

sin2 x = f - j ( l - cos2i) cos-1 x=

Ift- 1 (1 + cos 2x) v2.

sin 3 x = I - I (3 sin i — sin3x)

(A.5) (A.6) (A.7)

cos J x = ft I - I (3 cos x + cos 3x)

(A.8)

sin 4 x = I - ) (3 — 4 cos 2x + cos 4x)

(A.9)

SK

APPENDICES cos4 X =

1

(3 + 4 cos 2x + cos 4x)

237 A.10

sin 5 x = I — j (10 sin x — 5 sin 3x + sin 5x)

A.ll

cos 5 x = ( — ] (10 cos x + 5cos3x + cos5x)

A.12

sin 6 x = I — I (10 — 15 cos 2x + 6 cos 4x — cos 6x)

A.13

cos6 x = I — J (10 + 15cos 2x + 6 cos4x + cos6x).

A.14

A.1.4 Other Trigonometric

Relations

fx±y\

sin x ± sin y = 2 sin I

(x=fy\

cos —-—

v 2 ;

\ 2 J

(x + y\

(x — y\

o ■ fx + y\ . (x - y\

cos x — cos y = —2 sin I

sin (

sin x cos y = I - 1 [sin(x + y) + sin(x — y)]

G)

A.15 A.16 A.17 A.18

cos x sin y = I - J [sin(x + y) — sin(x — y)]

A.19

cos x cos y = ( - I [cos(x + y) + cos(x — y)]

A.20

sin x sin y = I - j [cos(x — y) — cos(x + y)]

A.21

sin 2 x — sin 2 y = sin(x + y) sin(x — y)

A.22

cos2 x — cos2 y = — sin(x + y) sin(x — y)

A.23

cos2 x — sin 2 y = cos(x + y) cos(x — y)

A.24

sin 2 x cos x = I — J (cos x — cos 3x)

A.25

sin x cos2 x = I - J (sin x + sin 3x)

A.26

sin 3 x cos x — I - ) (2sin 2x — sin4x)

A.27

238

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS (A.28)

sin xcos x — 1 — 1(1 — cos4x) sin x cos 3 x = f ^ ) (2sin 2x + sin4x)

(A.29)

sin x cos x = I — j (2 cos i — 3 cos 3x + cos 5x)

(A.30)

= I — I (2 sin x + sin 3x — sin 5x)

(A.31)

= — I — 1(2 cos x + cos 3a; + cos5x)

(A.32)

)(2sinx + 3sin3x + sin5x). sin x cos x = ( 1— 6

\

A.1.5 Derivatives and Integrals of Trigonometric

Functions

d_ dx cos x = — sin x d dx■ sin x = cos x

I I

(A.34) (A.35)

cos x dx = sin x

(A.36)

sin i
J 'sin

xdx = I — I x —

"

\2J~ \4j

(A.37) sin2x

(A.38)

/ cos2 x dx = I - I x + I - ) sin 2x

/ /

I

sin rax sin kx dx =

sin(ra — k)x

2(m-k)

sin(m + k)x

2(m + k)

, , sin(m — k)x sin(m + k)x cos rax cosfcxdx = — ; -r-—| 7 f— 2(ra - k) 2(ra + ifc) , , sin rax cos kx dx =

I

(A.33)

/

cos(m — k)x — rr 2(m-k)

cos(m + k)x r r + k) rr— 2(m

cos rax cos kx dx = n6mk',

(A.39)

m2^k2

(A.40)

, , , m Jir k

(A.41)

, , ,9 m ^ k

m, k integers

(A.42) (A.43)

APPENDICES /

/

sin mx cos kx dx = 0;

m, k integers

sinmx sin kx dx = 7r£m*;

239 (A.44)

m, k integers

(A.45)

-IT

/ x s i n x dx = sinx — xcosx jx

(A.46)

sin i dx = 2x sin a: — (x 2 — 2) cos x

(A.47)

/ x c o s x d x = cosx + xsinx

(A.48)

/ x2 cos x dx = 2x cos x + (x 2 — 2) sin x.

(A.49)

A.2 Factors and Expansions

(a ± b)2 =a2± (a±b)3

=a3±3a2b

2ab + b2

(A.50)

+ 3ab2±b3

(A.51)

(a + b + c)2 = a? +b2 + c2 + 2{ab + ac + be)

(A.52)

(a + b + c) 3 = a 3 + 63 + c 3 + 3a 2 (6 + c) + 36 2 (a + c) + 3c 2 (o + 6) + 6a6c a2 - 62 = (a - 6)(a + 6) a2 + b2 = (a + ib)(a - ib),

a3 + b3 =(a + b)(a2-ab

(A.54)

i = %/^T

a 3 - 63 = (a - b)(a2 + ab + + b2).

(A.53)

(A.55) ft2)

(A.56) (A.57)

A . 3 Quadratic Equations The quadratic equation ax2 + bx + c = 0

(A.58)

240

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

has the two solutions - £ac -b + -JW-

Xi

(A.59)

2a

-b-sftf -- 4ac

x2 =

(A.60)

2a

A.4 Cubic Equations

The cube equation

z 3 + pz 2 + qz ■+- r = 0

(A.61)

can be reduced to the form

x3 + ax + b = 0

(A.62)

P *= *--.

(A.63)

by substituting for z the value

The constants a and 6 are given by the expressions 3
(A.64)

, _ 3p 3 - Qpq + 27r 27

(A.65)

Let A and B be denned as

A =

B

'b\ ,2)

+

\2J

/ft 2

3 1/2 + a ^

\4

+

1/3

U 27J

(A.66)

(A.67)

27)

The three roots of Eq. (A.62) are given by the following expressions xj =A + B, 'A + B\ x2 = - I — ^ —

,— +V=3

(A.68) (A-B

(A.69)

APPENDICES 241

Let

If A > 0, then there will be one real root and two complex conjugate roots. If A = 0, there will be three real roots, of which at least two are equal. If A < 0, there will be three real and unequal roots. A.5 Differentiation of a Definite Integral with R e s p e c t t o a P a r a m e t e r Let f(x,t)

be continuous and have a continuous derivative df/dt,

in a domain

in the x-t plane that includes the rectangle V>(t) < x < (t),

t\
(A.72)

In addition, let ip{i) and <j>(t) be defined and have continuous first derivatives for ti < t < <2- Then, for t\ < t < t2, we have

-J

^ f{x,t)dx = f[mi]-£-fW),t]-£

+J

t

Qjf{*,i)dx.

(A.73)

A.6 Eigenvalues of a 2 x 2 Matrix The eigenvalues of a matrix A are given by the solutions to the characteristic equation det(A - XI) = 0,

(A.74)

where / is the identity or unit matrix. If A is an n x n matrix, then there exists n eigenvalues Aj, where i = 1 , 2 , . . . , n. Consider the 2 x 2 matrix

A

<° J)

(A.75)

242

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

The characteristic equation is d e tr-

A

AUO.

(A.76)

A2 - TX + D = 0,

(A.77)

c

d — X

Evaluating the determinant gives

where T = trace(A) = a + d, D = det(A) = ad - be.

(A.78)

The two eigenvalues are given by the expressions

Al =

(0[T+v/T2_4D]'

A2 = (|)

[T - VT*-4D]

(A79a) .

(A.79b)

References 1. A. Erdelyi, Tables of Integral Transforms, 1954).

Vol. I (McGraw-Hill, New York,

2. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965). 3. E. Jaknke and F. Emde, Tables of Functions with Formulas and Curves (Dover, New York, 1943). 4. National Bureau of Standards, Handbook of Mathematical Functions (U.S. Gov­ ernment Printing Office; Washington, DC; 1964). 5. Chemical Rubber Company, Standard Mathematical Publishing Company, Cleveland, various editions).

Tables (Chemical Rubber

6. H. B. Dwight, Tables of Integrals and Other Mathematical New York, 1961).

Data (MacMillan,

Appendix B SERIES EXPANSIONS This appendix summarizes certain results concerning series expansions of func­ tions. Additional details and proofs of the stated theorems are given in the refer­ ences listed at the end of this appendix. B . l U n i f o r m Convergence Let {fn(x)}

be a sequence of real functions, each of which is denned on the

real interval a < x < b. For a particular i ] : such that a < x\ < b, consider the sequence of real numbers {/„(xi)}. Assume that this sequence converges for every T such that a < x < b, and let / ( x ) = Lim fn(x).

(B.l)

n—oo

This sequence of functions then converges pointwise on the interval and f(x)

is

called the limit function of the sequence. The sequence {fn(x)}

is said to converge uniformly to f(x) if, given any e > 0,

there exists a positive number N that depends only on e, such that !/„(*) - f(x)\ < e

(B.2)

for all n > A" for every x such that a < x < b. Consider the infinite series of real functions $>,(*),

(B.3)

1=1

each of which is defined on a real interval a < x < b. Consider the sequence

{fn(x)}

ai partial sums of this series defined as follows n

/„(*) = £«;(*). i=)

343i

(B.4)

244

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

The infinite series, Eq. (B.3), converges uniformly to / ( x ) , on a < x < 6, if the sequence of partial sums {/ n (x)} converges uniformly to / ( x ) on this interval. B.2 Weierstrass M Test for Uniform Convergence Let {Mn} be a sequence of positive constants such that oo

EM«

(B-5)

n=l

converges. Let {u„(x)} be a sequence of functions, each of which is defined on a < x < b, such that |«n(x)|<M„,

n = 1,2,3,...,

(B.6)

for all x on the interval. Then the series given by oo

n=l

converges uniformly on this interval. B.3 Properties of Uniformly Convergent Series Let {un(x)}

be a sequence of functions defined on a < x < b Assume that the

series (B.7)

oo

n=l

converges to a sum / ( x ) for a < x < 6, i.e.,

x

oo

f( ) = J2Un(x)n=l

The following theorems relate the properties of (ti„(i)} to / ( x ) .

APPENDICES

245

T h e o r e m 1. Let each u„(x) be continuous for a < x
I

/-6i

f(x)dx=

Jai

I

rbi

u\(x)dx

+ /

J a\

rbi

U2(x)dx + • ■ ■ + /

J a\

un{x)dx

+ • • •,

(B.9)

Ja-i

where a < aj < b\ < b. T h e o r e m 3 . A convergent series can be differentiated that the functions

term by term,

provided

of the series have continuous derivatives and the series of the

derivatives is uniformly

convergent.

T h e o r e m 4. If the series oo

oo

^un(x)

and

n=l

^

wn(x)

(B.10)

n=l

are uniformly convergent for a < x < 6, and h(x) is continuous for a < x < 6, then the following series are uniformly convergent for a < x < b oo

Y,[Aun(x)

+ Bwn(x)],

(B.ll)

oo

53fc(*)u„(i),

(B.12)

where A and S are arbitrary constants.

B.4 P o w e r Series A power series in x is a series of the form oo

£Cn*", n=0

(B.13)

246

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

where Co, C\,...,

C „ , . . . , axe constants. A power series in powers of (x — o) is a

series of the form oo

£C„(x-a)». n=0

(B.14)

The power series of Eq. (B.14) converges when x = a. This may be the onlyvalue of x for which the series converges. If there are other values of x for which the series converges, then the values of x lie in an interval, the convergence

interval,

having a midpoint at x — a. This interval can either be finite or infinite; see Figure B.l.

a—R

a

a+ R

Figure B.l. Regions of convergence and divergence of a power series that converges at x = a with a radius of convergence R. T h e o r e m 5. (1) Every power series oo

Y,Cn(x-a)n

(B.15)

n=0 has a radius of convergence R such that the series converges absolutely

when

\x — a\ < R and diverges when \x — a\ > R. (2) The number R can be zero (in which case the series converges only for x = a), a positive number, or infinite (unbounded,

in which case the series con­

verges for all x). (3) If R is not zero and Ri is such that 0 < Ri < R, then the series converges uniformly for \x — a\ < Ri. (4) The number R can be evaluated in the following two ways when the appro­ priate limit exists: R = Lim

cn

n—*oo C n +1

* = ££,ic^'

(B.16)

(B 17)

-

APPENDICES

247

(5) A power series represents a continuous function within the interval of con­ vergence. (6) A power series can be integrated term by term within the interval of con­ vergence. (7) A power series can be differentiated

term by term within the interval of

convergence.

B.5 Taylor Series of a Function of a Single Variable Let f(x) be a power series oo

/(*) = £

C B (* - a ) "

(B.18)

71 = 0

with convergence interval (a — R) < x < (a + R). This series is called a Taylor series of f(x) at x = a if the coefficients C„ are given by Cn = where f(n\a)

f("Val J —fA n'.

n = 0,l,2,...,

(B.19)

is the ra-th derivative of f(x) evaluated at x = a.

T h e o r e m 6. (1) Every power series with a non-zero radius of convergence is the Taylor series of its sum. (2) If two power series oo

oo

Y^Cn(x-a)n n=0

and

^

Z»„(x - a ) "

(B.20)

n=0

i a v e non-zero convergence radii and have equal sums whenever both series converge, then the series are identical, i.e., Cn = Dn,

n = 0,l,2,....

(B.21)

248

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS (3) If a power series has a non-zero convergence radius and has a sum that is

identically zero, then every coefficient of the series is zero, i.e., if oo

£C

B

( x - a ) " = 0,

(B.22)

n=0

then Cn=0,

n = 0,1,2,....

If a = 0, the Taylor series is called the Maclaurin series of f(x),

(B.23) i.e.,

n=0

T h e o r e m 7. Let oo

F(x) = ^ C „ ( x - a ) » ,

(B.25)

n=0 oo

D„(x - a ) B ,

G(;r) = £

(B.26)

n=0

be convergent power series with radii of convergence R\ and R2, respectively, 0 < R^ < R2.

where (B.27)

Then OO

F(x)G(x)

= Y,kn(x-a)n,

(B.28)

n=0

converges in the interval (a — R\) < x < (a + R\), and the coefficients kn are given by kn = C0Dn + C-i.D^-1 + C2Dn-2

+ ■■■ + CnD0.

(B.29)

B.6 Generalized Taylor's T h e o r e m [2] Let f(x)

be a real valued function denned for |x| < R, where R > 0, and

have all its derivatives exist. The kth. Taylor polynomial of f(x)

is defined by the

expression k

p^EE^a;^, i=0

(B.30)

APPENDICES

249

where /«>(0)

(B.31)

Note that if f(x) is only s times differentiable, then the Taylor polynomials are only denned for 0 < k < s. For this situation, f(x) can be written as f(x) = Pk(x) +

Rk(x),

(B.32)

where Rk(x) is the remainder. The following theorem gives the Taylor theorem for this case. T h e o r e m 8. Let f(x)

be defined for | i | < R and have s continuous

For each k < s — 1, f(x)

derivatives.

can be written as

/(*) = £ /

<0

(0)

x' +

Rk(x),

(B.33)

1=0

where the remainder Rk(x)

satisfies the relation

Rk(x) =

JoZf^(t){-^dt.

(B.34)

For each i?i with 0 < Rx < R and \x\ < Ru Mk{R,) \Rk(*)\ <

i*+l

L(fc+l)!j' '

(B.35) '

wiiere Mk(Ri)

= Max{|/ ( f c + 1 ) (x)|}

for \x\ < RL

(B.36)

B.7 Taylor Series of a Function of T w o Variables Let / ( i , t / ) b e a function of two variables x and y. Assume f(x, y) to be defined and continuous, and to have continuous partial derivatives of all orders at the point

250

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

(x,y) = (a,6). Then f(x,y)

can be expanded about the point (x,y)

= (a,b) in a

Taylor series that takes the form

d2f(a,b) 5i

d2f(a,b) .

,,,

nd

2

f(a,b),

,.

,,

+ •••.

2

(a; — aj

(B.37)

References 1. W. Kaplan, Advanced Calculus (Addison-Wesley; Reading, MA; 1952). See Chapter 6. 2. J. A. Murdock, Perturbations: Theory and Methods (Wiley-Interscience, New York, 1991). See Appendix A. 3. S. L. Ross, Differential Equations (Blaisdell; Waltham, MA; 1964). See Chap­ ter 10. 4. L. A. Pipes and L. R. Harvill, Applied Mathematics for Engineers and Physi­ cists (McGraw-Hill, New York, 1970, 3rd edition). See Appendix C. 5. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965). See Section 0.319.

Appendix C F O U R I E R SERIES C . l Definition of Fourier Series Let f(x)

be a function that is defined on the interval — L < x < L and is such

that the following integrals exist: JLf(x)mn(^dx,

j\x)cos(^fjdx, for n = 0,1,2,

(C.l)

The series

+

oo

f £

V L

+ bn sin

(n-nx V L I

(C.2)

n=l

where

an = f £ ) / % ( * ) cos ( ^ ) d z ,

K=

(i)fLfix)sin(^r)dx>

(C.3)

(c 4)

-

is called the Fourier series of f(x) on the interval — L < x < L. The numbers {an} and {&„} are called the Fourier coefficients of

f(x).

A function / i ( x ) such that / i ( * + p ) = /i(x),

P^O,

(C.5)

for all x is said to be periodic and to have period p. Since both sm(nnx/L)

and cos(mrx/L)

have period 2L/n, the only period

shared by all these expressions is 2L. Therefore, if the Fourier series of f(x)

con­

verges, then f(x) is periodic of period 2L, i.e., f(x + 2L) = f(x). If f(x)

(C.6)

is initially defined only in the interval — L < x < L, then Eq. (C.6) can be

used to define it for all values of x, i.e., —oo < x < oo. "251

252

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS In general, the Fourier series of f(x)

defined on an interval —L < x < L is a

strictly formal expansion. The next section gives the relevant theorem concerning convergence of Fourier series. C.2 Convergence of Fourier Series A function / ( x ) is said to be piecewise smooth on a finite interval a < x < b if this interval can be divided into a finite number of subintervals such that (1) / ( x ) has a continuous derivative in the interior of each of these subintervals, and (2) both / ( x ) and df/dx

approach finite limits as x approaches either end point of each of

these subintervals from its interior. C.2.1 Examples The function f(x) defined by

is piecewise smooth on the interval —7r < x < 7r. The two subintervals are [—7r,0) and (0,7r]. The function / ( x ) defined on the interval 0 < x < 5 by

/(*)

x2, 0<x-4)3/2, 4<x<5,

,_, . -

(C 8)

is piecewise smooth on this interval. Observe that in each subinterval both / ( x ) and df/dx are defined.

APPENDICES C.2.2 Convergence

253

Theorem

T h e o r e m 1. Let f(x),

(1) be periodic of period 2L, and (2) be piecewise

smooth

on the interval -L < x < L. Then the Fourier series of f(x) Y + 2^

anCosf-^-J+6nsinf—J

,

(C.9)

where

an = bn=

(i) [_ f{x)cos (rr)dx>

(l)fLf{x)sin(:T~)dx>

(C10) (C11)

converges at every point xQ to the value

f(xj) + f(*o)

(C.12)

2 wiere /(xjj~) is the right-hand limit of f(x) of f(x) to f(xo)

at xo- If f(xo)

at x0 and f(ig)

is the left-hand

hmit

is continuous at i 0 , the value given by Eq. (C.12) reduces

and the Fourier series of f(x)

converges to

f(x0).

C.3 B o u n d s on Fourier Coefficients [1, 2, 7] T h e o r e m 2. Let f(x) interval —L < x < L. of bounded variation. depend on f(x)

be periodic of period 2L and be piecewise smooth on the Let the Srst r derivatives of f(x)

exist and let f(x)

be

Then there exists a positive constant M (whose value may

and L) such that the Fourier coefficients satisfy the relation M K\ + \bn\ < — .

(C.13)

C o m m e n t s . A function / ( x ) , defined on — L < x < L, is of bounded variation if the arc-length of f(x) over this interval is bounded [2].

254

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

T h e o r e m 3. Let f(x)

be analytic in x and be periodic with period 2L.

exists a 9 and an A (which may depend on f(x)

There

and 2L) such that the Fourier

coefficients satisfy the relation |a„| + | 6 n | < A ^ .

(C.14)

C.4 Expansion of F ( a c o s x , — a s i n x ) in a Fourier Series At a number of places in the text, the Fourier series is needed for a function of two variables, F(u,v),

where it = acosx,

u = —asinx,

(C.15)

and in general F(u, v) is a polynomial function of u and v. To illustrate how this is done, consider the following particular form for F(u,v)

F(u,v):

= (l-u2)v.

(C.16)

Replacing u and v by the relations of Eq. (C.15), and using the trigonometric relations given in Appendix A, the following result is obtained: F(u,v)

= (1 — u2)v — (1 — a2 cos2 x)( —asinx) = —asinx + a 3 cos2 x sin x -a sin x + ( — J (sin x + sin 3x)

/a 2 -4\

.

fa3\ . „

= ( — 7 — ) a s i n x + (— J s m 3 x .

(C17)

This last expression is the required Fourier expansion of Eq. (C.16). For a second example, consider F(u, v) = u3 The following is obtained for this case: F(u, v) = u 3 = a3 cos 3 x = (-|-

J cos x + (^- j cos 3x.

(C.18)

APPENDICES

255

References 1. N. K. Bary, A Treatise on Trigonometric Series, Vol. I (MacMillan, New York, 1964). 2. R. C. Buck, Advanced Calculus (McGraw-Hill, New York, 1978). 3. H. S. Caxslaw, Theory of Fourier Series and Integrals (MacMillan, London, 1921). 4. R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill, New York, 1941). 5. W. Kaplan, Advanced Calculus (Addison-Wesley; Reading, MA; 1952). See Chapter 7. 6. W. Rogosinski, Fourier Series (Chelsea Publishing, New York, 1950). 7. E. C. Titchmarch, Eigenfunction 1946).

Expansions (Oxford University Press, Oxford,

8. A. Zygmund, Trigonometrical Series (Dover, New York, 1955).

Appendix D ASYMPTOTICS EXPANSIONS Essentially all of the power series expansions considered in the text are socalled asymptotic expansions. This appendix introduces the basic concepts relating to "asymptotics" and illustrates them by means of some examples. The references at the end of this appendix provide a fuller discussion of these matters and give the necessary proofs. In particular, the book by Murdock [3] discusses in detail the important issues of asymptotic analysis and the use of generalized asymptotic power series. D . l Gauge Functions and Order Symbols Let /(e) be a function of the real parameter e. If the limit of / ( e ) exists as e tends to zero, then there are three possibilities:

(o,

/(e) - I A, I oo,

(D.l)

with 0 < \A\ < oo. (The case where /(e) has an essential singularity at e = 0, such as sin(e_ 1 ) or exp(e_ 1 ), is excluded.) In the first and last cases, the rate at which /(e) - . 0 a n d / ( e ) -> oc oan nb expressed bb yomparing / / ( e with hertain known functions called gauge functions.

The simplest and often most useful gauge

functions are members of the set {e™} where n is an integer. Other gauge functions used are sine, sinhe, loge, etc. The behavior of a function / ( e ) , as e -» 0, can be compared with a gauge function g(e) by employing the symbols "O" and "o." D.l.l The Symbol O The symbol O is denned as follows: Let /(e) be a function of the parameter e and let g(e) be a gauge function. Let there exist a positive number A independent of e and an e0 > 0 such that |/(e)| < A\g(e)\,

for all |e| < e„, 256

(D.2)

APPENDICES

257

then /(e) = Ofa(e)]

as e -» 0.

(D.3)

The condition given in Eq. (D.3) can be replaced by Lim

m

£-►0

S(«)

(D.4)

CO.

<

Let / ( x , e) be a function of the variable x and the parameter e, and let
= 0[g(x,e)},

(D.5)

if there exists a positive number A independent of e and an eo > 0, such that | / ( x , e ) | < A\g(x,e)|,

for all |e| < e 0 .

(D.6)

If A and eo are independent of x, the relationship is said to hold uniformly. D.1.2 The Symbols o The symbol o is defined as follows: Let /(e) be a function of e and let g(e) be a gauge function. Let there exist an eo > 0, and let, for every positive number 6, independent of e, the following condition hold

1/(01 < %(0I,

for

M < eo,

(D.7)

then /(e) = o[g(e)]

as e -» 0.

(D.8)

The condition given by Eq. (D.8) can be replaced by Lim £-*0

»(0

= 0.

(D.9)

Let / ( x , e) be a function of x and e, and let g(x, e) be a gauge function. Then f(x,e)

= o[g{x,e)}

as e - 0,

(D.10)

258

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

if for every positive number 6, independent of «, there exists an eo such that |/(*,e)|<%(*,e)|,

for|e|<60.

(D.ll)

If 6 and e0 are independent of x, then Eq. ( D . l l ) is said to hold uniformly. D . 2 A s y m p t o t i c Expansion Let' {£„(e)} be a sequence of functions such that i , W = «[Ci(e)] Such a sequence is called an asymptotic

ase^O.

(D.12)

sequence.

Consider the series oo

m=0

where the am are independent of e and {6m(e)} is an asymptotic sequence. This is an asymptotic sequence, denoted by oo

y ~ X I a™6™(e)

as e -> 0,

(D.14)

m=0

if and only if n

y = J^ "m<5m(e) + 0[«„+i(e)]

as e -» 0.

(D.15)

m=0

The expansion given by Eq. (D.14) may diverge. However, if the series is an asymptotic expansion, then although Eq. (D.14) may diverge, for fixed n, the first n terms in the expansion can represent y with an error that can be made arbitrarily small by taking e sufficiently small. Thus, the error made in truncating the series after n terms is numerically less than the first neglected term, namely, the (n + l)th term. Note, however, that given a function y(«), the asymptotic expansion is not unique. In fact, y(e) can be represented by an unlimited number of asymptotic expansions, since there exists an unlimited number of possible asymptotic sequences

APPENDICES

259

that can be used. However, once a particular asymptotic sequence is selected, the representation of y(e) in terms of this sequence is unique. If y(e) has the asymptotic expansion oo

y(e) ~ Y, « m M 0

as t - 0,

(D.16)

m=0

for a particular sequence {Sm(e)}, then the coefficients an are given uniquely by a„ = L

i

m

^ b & ^ ) .

e-o

(D.17)

Sn(e)

D . 3 U n i f o r m Expansions Let y b e a function of x and e. The asymptotic expansion of y in terms of the asymptotic sequence {Sm(e)} is oo a

y(x, e)~J2

m(z)M<0

as t -> 0,

(D.18)

m=0

where the coefficients a m are functions of z only. This expansion is said to be uniformly valid if n

y(x,e)=

£am(x),Sm(€) +

fln-H(x,€),

(D.19)

m=0

where Rn(x,e)

= 0[Sn(e)],

(D.20)

uniformly for all x of interest. If these conditions do not hold, then the expansion is said to be nonuniformly valid. For the expansion to be uniformly valid, the term am(x)Sm(e)

must be small

compared with the preceding term a m _ i ( x ) £ m _ i ( e ) for each m. Since M « ) = °Pm-i(e)]

ase^O,

(D.21)

260

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

then am(x)

can be no more singular than am_-i(x) for all values of x of interest, if

the expansion is to be uniform. This means that am(x)/am-i

is bounded. Con­

sequently, each term in the expansion must be a small correction to the preceding term irrespective of the value of x. D . 4 Elementary Operations on A s y m p t o t i c Expansions

D.4.1 Addition and Subtraction In general, asymptotic expansions can be added and subtracted. For example, let {Sm(e)} be an asymptotic sequence and consider the asymptotic expansions of the two functions f(x, e) and g(x, e), where the expansions are defined for the same intervals of x and e; that is oo

/(*,e)~£am0r)M*).

(°-22)

m=0 oo

g(x,e)~

£ ] & m (aOMz).

P-23)

m=0

Then, for constants A and B oo

Af(x, e) + Bg(x, e) ~ ^

[Aam(x) + Bbm(x)]6m{e).

(D.24)

m=0

D.4.2 Integration If f(x, e) and am(x)

are integrable functions of x, then AX

°°

[X

I f(x,e)dx~J2S"'(e) •/xi

a

m=0

Jx

m{x)dx.

(D.25)

i

If f(x, e) and 5 m (e) are integrable functions of e, then ft ^0

°° f(x,e)de~Tam(x) m=

n

/e 6m(e)de. -/0

(D.26)

APPENDICES D.4.3

261

Multiplication

In general, two asymptotic expansions cannot be multiplied to form another asymptotic expansion. This is because in the formal product of f(x, e) and g(x, e), all products of the form Sn(e)Sm(x)

occur and it may not be possible to arrange them

so as to obtain an asymptotic sequence. Multiplication of asymptotic expansions is only justified if the asymptotic sequence {<5m(e)} is such that 6n(e)6m(x) form an asymptotic sequence or possess asymptotic expansions.

either

An important

n

asymptotic sequence that does have this property is {e }, the collection of (nonnegative) powers of e. If oo

/(z,e)~£am(z)em,

(D.27)

m=0 oo

g(x,e)~Y,bm{x)em,

(D.28)

m=0

then oo

f(x, e)g(x, *)~J2

c

™( a 0 em .

( D - 29 )

m=0

where m

cm(x) = J2 an{x)bm_n{x).

(D.30)

n=0

D.4.4

Differentiation

In general, it is not justified to differentiate asymptotic expansions with respect to either x or e. When differentiation is not justified, nonuniformities occur. The following theorem is useful:

262

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Theorem. Let f(x,e) respectively,

be an analytic function of x and e, where S and T are,

the domains of analyticity

valid asymptotic

of x and e. Let f(x, e) have a

uniformly

expansion oo

/(*,e)~5>m(*)em,

(D.31)

m=0

for all x in S. Under these conditions, the am[x) are analytic for x in S and 9f(x, e)

v ^ dam(x)

,

>

m=0

uniformiy in every compact proper subset T\

ofT.

D.5 Examples This section provides illustrations of some of the concepts introduced in earlier sections of this appendix.

For a fuller discussion and additional examples, the

references listed at the end of this appendix can be consulted. D.5.1 The Symbol 0 As e -> 0, sine = 0(e)

cose = 0(1)

sinhe = 0(e)

coshe = 0 ( l )

tanhe = 0(e) sine 2 = 0(e 2 )

and

sin(x + e) = 0(1) sin(ez) = 0(e)

cote = 0 ( e _ 1 ) 1 - cos2 e = 0(e 2 ) uniformly as e —> 0 nonuniformly as e —* 0.

D.5.2 The Symbol o As e -> 0, sine = o(l) sinh e = o(l) coth e = o ( e _ 3 / 2 )

cose =

o{e~1l2)

1 — cos e = o(e) sin 2 e = o(e)

APPENDICES

263

and sin(x + e) = o ( e - 1 / 3 )

uniformly as e —> 0

e~" — 1 = o(e1^2)

nonuniformly as e —> 0.

D.5.3 The Function sin(x + e) Consider the function sin(x + e) in more detail. For e —> 0 sin(x + e) = sin x cos e + cos x sin e

=

I1 _ h + i f + ••■) s i n * + ( e _ sT + HI + ■")cosx

/e2\ fe3\ = sin x + e cos x — I — J sin x — I — 1 cos x + • • •

(D.33)

For all values of x the coefficients of all powers of e are bounded. Consequently, the expansion is uniformly valid. D.5.4 The Function exp(—ex) For a nonuniformly valid expansion, consider the expansion of exp(—ex) for small e: °° (er)m exp(-5x)=£(-iri-i-.

(D.34)

m=0

This function can be accurately represented by a finite number of terms only if ex is small. Since e is assumed small, this means that x = 0(1). Note that if x is as large as 0 ( e _ 1 ) , then ex is not small, and a finite number of terms cannot give an accurate representation of exp(—ex). To obtain a satisfactory expansion for all x, all terms in Eq. (D.34) must be retained. D . 6 Generalized A s y m p t o t i c Power Series [3] Most of the calculational procedures given in the text are based on expansions that take the following form / ( * , e) = foHe)x]

+ h Ke)x]e + • ■ ■ + fn[v(e)x]en

+ 0(e"+1).

(D.35)

264

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Observe that t occurs not only in the powers of e m , but also appears in the coef­ ficients of these powers. A requirement is that this expansion be uniformly valid for all x. This will be the case if all the coefficients remain bounded as e —> 0, for fixed x. Such an expansion is called a generalized asymptotic power series. This is a special case of the generalized asymptotic expansion / ( * , e) = /„(*, e)60(e) + h(x, e)61(e) + • • ■ + / „ ( * , «)«„(«) + 0[6n+1(e)],

(D.36)

where {5„(e)} is an asymptotic sequence. References 1. N. G. de Bruijn, Asymptotic 1958). 2. A. Erdely, Asymptotic

Methods in Analysis (Interscience, New York,

Expansions (Dover, New York, 1956).

3. J. A. Murdock, Perturbations: Theory and Methods (Wiley-Interscience, New York, 1991). See Sections 1.8 and 4.2. 4. A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973). See Chapter 1. 5. J. G. Van der Corput, Asymptotic versity, 1962).

Expansions (Lecture Notes, Stanford Uni­

6. W. Wason, Asymptotic Expansion for Ordinary Differential Equations (Inter­ science, New York, 1965).

Appendix E B A S I C T H E O R E M S OF T H E T H E O R Y OF SECOND-ORDER DIFFERENTIAL EQUATIONS E . l Introduction The general second-order differential equation cPy dt2

dy

= f(y, dt , ) .

(E.l)

dy\ _ dt ~ ■!/2>

(E.2)

dy2 dt = f(yi ,V2 ,*),

(E.3)

can be written in the system form

by means of the transformation (y,dy/dt)

= (yi,y2)- A general system of coupled,

first-order differential equations is dy_ ■% = MvuV2,t), dt

(E.4)

^jT = /a(Vi,lfe,*)-

(E-5)

In this appendix, a number of theorems are given concerning the solutions of Eqs. (E.4) and (E.5). Proofs can be found in the references listed at the end of this appendix. The following assumptions and definitions apply to all the results of this ap­ pendix: (1) The functions /i(yi,y2,<) and f2{yi,yi,t)

are defined in a certain do­

main R of the three-dimensional (yi,y2,<) space, are continuous in this region, and have continuous partial derivatives with respect to yi, y2 and t. (2) A point having the coordinates (yi,j/2,*) will be denoted as P(y~i,y~2,t)-

265

266

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

E.2 Existence and Uniqueness of the Solution T h e o r e m 1. Let P(yi,V2,to) (ti
be any point in R.

There exists an interval

oft

t2) containing t0, and only one set of functions vi = M*\

vi = &(*).

(E-6)

defined in this interval, for which the following conditions are satisfied: (1) 4>i(t0) = y° and
values oft in the interval, ti < t < t2, the point

P[4>i(t), <j>2(t),t\ belongs to the domain R. (3) The system of functions given by Eq. (E.6) satisGes the system of differential equations (E.4) and (E.5). (4) The solutions, given by Eq. (E.6), can be continued up to the boundary of the domain R; that is, whatever closed domain Ri, contained entirely in R, there are values t' and t", where h
< t2,


such that the points P [ ^ ( t ' ) , ^ ( t ' ) . * ' ] and ^[^i(*")» fa(t"),t"] He outside

(E.7) Rt.

E.3 D e p e n d e n c e of the Solution on Initial Conditions The solutions of Eqs. (E.4) and (E.5) depend on the initial conditions (yJ,y2,i{iM,yl,yl),

y% = ^ ( M o . y ? , ^ ) ,

(E.8)

J/20 = ^(to,to,y?,2/2°)-

(E.9)

with y\ = Mto,t0,yly°2),

The following theorems give information concerning the dependence of the solutions on the initial conditions.

APPENDICES

267

T h e o r e m 2. Let

yi =

fc(*,**.vi,v5).

2/2 = &(*,**, yj.yj),

(E-iOa) (E.iob)

be a soiution to Eos. (E.4) and (E.5), defined for t in the interval, tx
y2(**) = y2*-

(E.n)

Let Ti and T2 be arbitrary numbers satisfying the condition tl
t2.

(E.12)

T i e n for an arbitrary positive e, there exists a positive number S = S(e, Ti, Tj) such that for the values oft0,

y° and y2 for which

\t0-t*\<6, the

\vl-vt\<6,

\y°2-y*2\<6,

(E.13)

solutions y\ = 4i(t,to,Vi,y°),

y2 = h(t,to,y°,y°),

are defined for all values oft in the interval T\
(E-14)

and satisfy the inequalities

l*i(*,to,yi 0 ,y2)-*i(M*.vr.V2)l<*»

(E.15)

|^(*,
(E-16)

T h e o r e m 3. If the functions fi(yi,yz,t)

and f2{y\,y2,t)

of Eqs. (E.4) and (E.5)

have continuous partial derivatives with respect to the variables j/i and y2 of order up to n > 1, then the solutions to this system have continuous partial

derivatives

with respect to y\ and y2 of the same order. T h e o r e m 4. If the functions fi{y\,y2,t)

and /2(yi,y2,*) are analytic functions of

the variables y\ and y2, then the solution, given by Eq. (E.8), is an analytic

function

268

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

of its arguments in a neighborhood fi{yi,y2,t)

and f2(yi,y2,t)

of every set of values for which the

functions

are defined.

E.4 D e p e n d e n c e of the Solution on a Parameter Let the functions f\ and f2 depend on a parameter A. For this case, Eqs. (E.4) and (E.5) become % - = /i(yi,y a ,*,A),

(E.17)

^=/a(Vi,tto,M).

(E-18)

T h e o r e m 5. If the functions f\(yi,

y2, t, A) and f2(yi,y2,t,

A) are continuous func­

tions of A, the solutions of Eqs. (E.17) and (E.18) yi=i(t,t0,y°,yl,X),

y2 = fa(t,ta,y~i,yl, A),

(E.19)

are aiso continuous functions of X. T h e o r e m 6. Let

fa(J/J,y2,t,

A) and f2(yi,y2,t,X),

and the first partial

derivatives

of fi and f2, with respect to yi and y2, be continuous functions of A. If yi and y2, given by Eq. (E.19), are soiutions of Eqs. (E.17) and (E.18), then the

di{t,to,yly°2,\) dy]

« = (1,2),

derivatives

i = (1,2),

(E.20)

are also continuous functions of X. T h e o r e m 7. If f\(y\,y2,t,X)

and f2(y-i,y2,t,X)

are analytic functions

arguments, then the solutions to Eqs. (E.17) and (E.18) are aiso analytic of all their arguments in a neighborhood

of their functions

of every set of values (t,to,yi,y2i^)

f°r

which they are defined.

References 1. A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of Oscillators (AddisonWesley; Reading, MA; 1966). See the Appendix, pp. 795-800.

APPENDICES 2. E. A. Coddington and N. Levinson, Theory of Ordinary Differential (McGraw-Hill, New York, 1995). See Chapter 2.

269

Equations

3. E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956). 4. N. Minorsky, Nonlinear 1962). See pp. 228-231.

Oscillations (Robert E. Krieger; Huntington, NY;

5. S. L. Ross, Differential Equations (Blaisdell; Waltham, MA; 1964). See Chap­ ters 10 and 11. 6. G. Sansone and R. Conti, Nonlinear Differential Equations (Pergamon, New York, 1964). Chapters VI and VII give excellent discussions of the topics pre­ sented in this appendix.

Appendix F LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS Essentially all of the approximation methods given in this book eventually lead to linear, second-order differential equations. This appendix gives the basic theorems and rules for solving this type of differential equation. Detailed proofs of the various theorems can be found in the references given at the end of this appendix. F . l Basic Existence T h e o r e m The general linear, second-order differential equation takes the form a0(t)^

+ a1(t)^

+ a2(t)y = F(t).

(F.l)

If F(t) = 0, then Eq. (F.l) is said to be homogeneous; if F(t) ^ 0, then Eq. (F.l) is said to be inhomogeneous. T h e o r e m 1. Let a0(t), a,\(t), a2(t) and F(t) be continuous on the interval a
Then there exists a unique solution y = (t)

ofEq. (F.l) sucii that rf(M = C „

^ T

= C2,

(F.2)

and the solution is defined over the entire interval a < t < b.

F.2 Homogeneous Linear Differential Equations The linear second-order homogeneous differential equation has the form u v a0(t)~

dv + a1{t)^+a2(t)y

= 0.

(F.3)

Again, it is assumed that a0(t), a-i(t), and a2(t) are continuous on the interval a
^70

APPENDICES

271

T h e o r e m 2. Let (t) be a solution of Eq. (F.3) such that

*(M = 0,

(F.4)

^ 2 = 0,

where a < t0 < b. Then <j>(t) = 0 for all t in this interval. To proceed the concepts of linear combination, linear dependence, and linear independence must be introduced and defined. F.2.1 Linear

Combination

If / i ( t ) , / 2 ( i ) , . . . , fn{i) are n functions and C\, C 2 , . . . , Cn are n arbitrary con­ stants, then the expression Cih(t)

+ C 2 / 2 (f) + • • • + Cnfn(i)

(F.5)

is called a linear combination of / i ( t ) , / 2 ( t ) , • • • > fn(t). F.2.2 Linear Dependent and Linear Independent

Functions

The n functions /i(<), fi{i), ■ ■ ■, fn{t) are called linearly dependent on a < t < b if and only if there exist constants C\, C2, ■ ■ ■, C„, not all zero, such that C i / i ( t ) + C 2 / 2 (t) + • • ■ + Cnfn(t)

=0

(F.6)

for all t such that a
= 0

(F.7)

for all t such that a < t < b implies that Cj = C2 = ■ • ■ = Cn = 0.

(F.8)

272

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

F.2.3 Theorems on Linear Second-Order Homogeneous Differential T h e o r e m 1. Let the functions / j ( t ) , f2(t),...,

Equations

/ „ ( t ) be any n solutions ofEq. (F.3)

on the interval a < t < b. Then the function C1f1{t) + C2f2(t) + where Ci,C2,...

,Cn

(F.9)

---+Cnfn(t)

are arbitrary constants, is also a solution of Eq. (F.3) on

a
on a
Definition. Let / i ( t ) and /2(t) be real functions, each of which has a derivative on a < t < 6. The determinant

hit)

/»(*)

dfi(t) it

dh(t) dt

(F.10)

is called the Wronskian of the two functions / i ( t ) and /2(t). W(fUf2,t)

Denote it by

= W(t).

T h e o r e m 3. Let / i ( t ) and fi(t)

be two solutions of Eq. (F.3) on a < t < b. Let

W(t) denote the Wronskian of f\(t) and f2(t).

Then either W(t) is zero for all t on

a
Z) f^ ° 1 ( ^firdz -

L Jto a o U )

(F.ll)

APPENDICES

273

for all t on a < t < b. T h e o r e m 5. Let / i ( t ) and fi(t)

be any two linearly independent

Eq. (F.3) on a < t < b. Every solution f(t) suitable linear combination of fi(t)

of Eq. (F.3) can be expressed as a

and /2(i), i.e.,

f(t) = C1f1(t) + C2f2(t), where C\ and C?. are arbitrary

solutions of

(F.12)

constants.

F . 3 I n h o m o g e n e o u s L i n e a r Differential E q u a t i o n s The general linear, second-order, inhomogeneous differential equation takes the form ao(<) - ^ + a,(*) ^

+ a0(t)y = F(t).

(F.13)

It is assumed that a&(t), aj(i), 02(1) and F(t) are continuous on a < t < b, with a0(t) ^ 0 on this interval. The Eq. (F.13) can be written as Ly = F(t),

(F.14)

where L is the linear operator J2

J

L = a0(t) — + a 1 ( i ) - + a 2 (t). T h e o r e m 6. Let v(t) be any solution of the inhomogeneous u(t) be any solution of the homogeneous

(F.15) Eq. (F.13), and Jet

equation

Ly = 0.

(F.16)

Then u(t) + v(t) is also a soJution of the inhomogeneous Eq. (F.13). The solution u(t) is called the homogeneous part of the solution to Eq. (F.13), and v(t) is called the particular solution to Eq. (F.13). The homogeneous solution

274

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

u(t) will contain two arbitrary constants. However, the particular solution v(t) will not contain any arbitrary constants. F.3.1 Principle of Superposition The principle of superposition for linear second-order inhomogeneous differen­ tial equations is given in the following theorem. T h e o r e m 7. Let Ly = Fi(t), be n different inhomogeneous

i = l,2,...,n,

(F.17)

second-order differential equations where the linear

operator L is defined by Eq. (F.15). Let /;(<) be a particular solution of Eq. (F.17) for i = 1 , 2 , . . . , n. Then n

!=1

is a particular solution of the equation n

Ly = YJFi{t).

(F.19)

1=1

F.3.2 Solutions of Linear Inhomogeneous Differential

Equations

Write Eq. (F.13) in "normal" form, i.e., or v

~d where p(t), q(t), and f(t)

+p{t)

dt

dv +q{t)v

=m

>

(F 20)

-

are continuous functions for a < t < b. Assume that

two linearly independent solutions, yi(t) and 2/2(2) are known for the corresponding homogeneous differential equation §+p(t)%

+ q(t)y

= 0.

(F.21)

APPENDICES

275

The general solution of Eq. (F.20) is y(t) = Cm(t)

+ C2y2(t)

+ ^ ~ where W(t0)

= W(y1,y2,t0)

J

f(x)eI^[y1(x)y2(t)

- yi(t)y2(x)]dx,

(F.22)

is the Wronskian of yi(t) and y2(t) evaluated at i = t0,

a
f p(z)dz.

(F.23)

JZQ

F.4 Linear Second-Order H o m o g e n e o u s Differential Equations w i t h Constant Coefficients For the special case of constant coefficients, the problem of obtaining two linear independent solutions of a homogeneous second-order differential equation can be completely solved. Consider the differential equation

a

cPv

dv

+ a i + a 2 j / =0

°^ i

'

,„

(F24)

where the coefficients ao, a\, and a2 are real constants. The equation a0m2 + aim + a2 = 0

(F.25)

is called the characteristic equation corresponding to Eq. (F.24). The two roots of Eq. (F.25), mi and m 2 are related to the general solution of Eq. (F.24) as follows: (1) Let mj and m 2 be real and distinct, i.e., mj =^ m 2 . The general solution of Eq. (F.24), in this case, is y(t) = Ciem>t + C2em*t, where Ci and C2 are arbitrary constants.

(F.26)

276

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS (2) Let mi and m 2 be complex conjugates of each other, i.e., mj = m 2 =

a + ib. For this case, the general solution of Eq. (F.24) is either one of the following equivalent forms:

v(t) = I Aeat yK>

co bt

<

+B ) '

a(

(F.27) <

\ C i e c o s & t + C 2 e "sinfc,

V

;

where A, B, Ci and C 2 are arbitrary constants. (3) Let m\ and m 2 be equal, i.e., mi = m 2 = m. The general solution for this case is y(t) = (Cl+C2t)emt,

(F.28)

where C\ and C 2 are arbitrary constants. F.5 Linear Second-Order Inhomogeneous Differential Equations with Constant Coefficients Consider the following inhomogeneous differential equation

a0^-

+a ^

+ a2y = Q(t),

(F.29)

where ao, a%, and o 2 are constants and Q(t) has first and second derivatives for an interval a < t < b. In general, if Q(t) takes the form of a sum of terms, each having the structure Qn,k(t) = f e * ' ,

(F.30)

then the general solution to Eq. (F.29) can be found, i.e., y(t) = C i e " " ( + C2em*< + v(t)

(F.31)

where the homogeneous solution is tt(<) = C 1 e m i ' + C 2 e m 2 t

(F.32)

and v(t) is a solution to the inhomogeneous Eq. (F.29). For the applications in this book, the following two rules will allow the deter­ mination of particular solution v(x) to Eqs. (F.29) and (F.30).

APPENDICES

277

Rule 1. Let no term of Q(t) be the same as a term in the homogeneous solution u(t).

In this case, a particular solution of Eq. (F.29) will be a linear combination

of the terms in Q(t) and all its linearly independent derivatives. The following example illustrates this rule. Consider the equation ^ f - 3 - ^ + 2y = 2 t e 3 ' + 3 s i n ( .

(F.33)

The characteristic equation is m 2 - 3m + 2 = 0,

(F.34)

and has solutions m.\ = 1 and m 2 = 2. Therefore, the solution to the homogeneous equation is u(t) = C1et+C2e2t,

(F.35)

where C\ and Ci are arbitrary constants. Observe that no term of Q(i) = 2te 3( + 3sini

(F.36)

is a member of the homogeneous solution. A particular solution of Eq. (F.33) will be a linear combination of t exp(3t) and sin t, and their linearly independent derivatives exp(3<) and cost. Consequently, the particular solution v(t) has the form v(t) = Ate3t + Be3t + C sin t + D cos t,

(F.37)

where A, B, C, and D are constants. These constants can be determined by sub­ stituting Eq. (F.37) into Eq. (F.33) and setting the coefficients of the linearly inde­ pendent terms, £exp(3i), exp(3<), sini, and cost, equal to zero. Doing this gives A = l,

* - ( § ) ,

C = l ,

D - l .

(F.38)

The particular solution v(t) is »(*) = *e 3 ' - ( J ) e 3 ' + (~)

* n * + ( ^ ) cost,

(F.39)

278

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

and the general solution to Eq. (F.33) is y(t) = C e ' + C2e2t + te3t - ( | ) e 3 < + ( ^ ) sin* + (fy

cost.

(F.40)

Rule 2. Let Q(t), in Eq. (F.29), contain a term that, ignoring constant coeffi­ cients, is tk times a term u-i(t) of u(t), where k is zero or a positive integer. In this case, a particular solution to Eq. (F.29) will be a linear combination of < i + 1 ui(t) and all its linearly independent derivatives that are not contained in u(t). As an illustration of this rule, consider the equation § - 3 $ + 2y = 2 < 2 + 3 e 2 ' . dtA at

(F.41)

The solution to the homogeneous equation is given by Eq. (F.35). Note that Q{t) = 2t2 + 3e2<

(F.42)

contains the term exp(2t), which, ignoring constant coefficients, is t° times the same term in the homogeneous solution, Eq. (F.35). Hence, v(t) must contain a linear combination of texp(2£) and all its linearly independent derivatives that are not contained in u(t). Consequently, v(t) has the form v(t) = At2 + Bt + C + Dte2t.

(F.43)

Note that exp(2i) is not included in Eq. (F.43) because it is already included in u(t). Substituting Eq. (F.43) into Eq. (F.41) and setting the coefficients of the various linearly independent terms equal to zero allows the determination of A, B, C and D. They are A = l,

5 = 3,

C=7-,

D = 3,

(F.44)

and the particular solution is v(t) = t2+3t

+ ^ + 3te2t.

(F.45)

APPENDICES

279

Thus, the general solution to Eq. (F.41) is y(t) = u(t) + v(t) = d e 1 + C 2 e 2 t + t2 + 3t + ^ + 3te 2 '.

(F.46)

References 1. G. Birkhoff and G. C. Rota, Ordinary Differential Equations (Ginn, Boston, 1962). 2. E. A. Coddington and N. Levinson, Theory of Ordinary Differential (McGraw-Hill, New York, 1955). See Chapters 1, 2, and 3.

Equations

3. W. Kaplan, Advanced Calculus (Addison-Wesley; Reading, MA; 1952). See Chapter 8. 4. E. A. Kraut, Fundamentals of Mathematical Physics (McGraw-Hill, New York, 1967). See Sections 6-18 and 6-21. 5. S. L. Ross, Differential Equations (Blaisdell; Waltham, MA; 1964). See Chap­ ters 10 and 11. 6. M. Tenenbaum and H. Pillard, Ordinary Differential Equations (Harper and Row, New York, 1963). See Chapters 4, 11, and 12.

Appendix G E X I S T E N C E OF P E R I O D I C S O L U T I O N S F O R CERTAIN SECOND-ORDER DIFFERENTIAL EQUATIONS This appendix gives, without proof, a number of important theorems dealing with the existence of periodic solutions for certain particular classes of secondorder nonlinear differential equations. Proofs and additional results are given in the references listed at the end of this appendix. The last four references [7, 8, 9, 10] present generalizations and recent new results on these topics. G . l Limit Cycles The second-order, nonlinear differential equation

can be written in the system form dy\ _ V2, dt dy% _ -yi dt

(G.2)

+ eF(Vi: . 1 / 2 ) ,

(G.3)

where yi = y. The (yi,y2) plane is called the phase plane. The trajectories in the phase plane, i.e., y% = yziyi), are solutions to the first-order differential equation dyi dy\

=

-yi+eF(yi,y2) 2/2

.

Solutions of Eq. (G.4) that are simple closed curves correspond to periodic solutions. Such isolated closed curves are called limit cycles. More precisely, a limit cycle is a closed trajectory in the (j/i, j/2) phase plane such that no trajectory sufficiently near it is also closed. A detailed discussion of the nature and properties of limit cycles is given in the books by Minorsky [6] and Farkas [10]. Every trajectory beginning sufficiently near a limit cycle approaches it either for t —> co or t —> —00. (See Figure G.l.) If all nearby trajectories approach a limit 280"

APPENDICES

281

cycle C as t —> oo, then the limit cycle is stable. If all nearby trajectories approach a limit cycle C as t —+ —oo, then the limit cycle is unstable. In the case where trajectories on one side of C approach it, while those on the other side depart from it, the limit cycle is semistable. FVom a practical point of view, semistable limit cycles are unstable. The problem of the determination of limit cycles is fundamental in the theory of oscillations of nonlinear nonconservative systems. These are the only kinds of systems in which limit cycles can exist. The matter of determining the presence of limit cycles for a given differential equation is a very difficult problem that can be solved by direct methods only in a few special cases. G.2 Lienard-Levinson-Smith T h e o r e m [1, 2] T h e o r e m 1. Consider the differential

g

+

equation

/ ( y ) | + ,(y) = o,

where f(x) and g(x) are differential functions.

(G.5)

(1) Let there exist j/i > 0 and y2 > 0

such that /(y) < 0 and f(y) > 0 otherwise.

for - yi < y < y 2 ,

(G.6)

for \y\ > 0.

(G.7)

(2) let yg(y) > 0

(3) Let /

g(y)dy = oo,

(G.8)

Jo ,-oo

/ Jo

g(y)dy = oo,

(G.9)

/ Jo

f{y)dy = oo.

(G.10)

282

Figure G.l. (a) Stable limit cycle; (b) unstable limit cycle; (c) semi-stable limit cycle.

APPENDICES

283

(4) Let G(y) = f g(t)dt Jo

(G.ll)

G(-yi)

(G.12)

and assume = G(y2).

Under these conditions Eq. (G.5) has an essentially unique periodic solution. Essentially unique in this context means that if y = (<) is a nontrivial periodic solution of Eq. (G.5), then all other nontrivial periodic solutions of Eq. (G.5) are of the form y = 4>{t-h)

(G.13)

where t\ is a real number. G.3 Levinson-Smith T h e o r e m [2] Levinson and Smith considered a generalized form of Eq. (G.5):

§ + >("*)§+*>-.

(G 14)

'

and established the existence of at least one limit cycle under certain conditions. T h e o r e m 2.

Assume yg(y) > 0

for \y\ > 0;

/(0,0)<0; /•OO

(G.16)

/* — OO

/ g(y)dy = / Jo Jo there exists some yo > 0 such that f(y,v)>0

(G.15)

g{y)dy = oo;

for\y\>y0;

(G.17)

(G.18)

there exists an M > 0 such that f(y,v)>-M

for\y\
(G.19)

284

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

there exists some y\ > y0 such that

I"1 f(y,v)dy > lOMyo,

(G.20)

Jyo

where v > 0 is an arbitrary decreasing positive function of y. Under these conditions, there exists at least one limit cycle for Eq. (G.14). Levinson and Smith have also established the uniqueness of the periodic solu­ tion when further restrictions on the functions f(y, v) and g(y) are satisfied. References 1. A. Lienard, Rev. Gen. Electricite 23, 901 (1928). 2. N. Levinson and O. K. Smith, Duke Math. J. 9, 382 (1942). 3. N. Levinson, Ann. Math. 45, 723 (1944). 4. S. Lefschetz, Differential Equations: Geometric York, 1962, 2nd edition). See Chapter XL

Theory (Interscience, New

5. G. Sansone and R. Conti, Nonlinear Differential Equations (Pergamon, Oxford, 1964). 6. N. Minorsky, Nonlinear 1974). See Chapter 3.

Oscillations (Robert E. Krieger; Huntington, NY;

7. G. Villari, J. Math. Anal. Appl. 86, 379 (1982). 8. P. J. Ponzo and N. Wax, J. Math. Anal. Appl. 104, 117 (1984). 9. Z.-H. Zheng, J. Math. Anal. Appl. 148, 1 (1990). 10. M. Farkas, Periodic Motion (Springer-Verlag, New York, 1994). See Chapter 3.

Appendix H S T A B I L I T Y OF LIMIT C Y C L E S H . l Introduction This book considers nonlinear differential equations of the form

£+■-"(-I)

<«•«

where e is a positive parameter, usually taken to be small, and F is a nonlinear function of its arguments. The function F does not depend on the independent variable t. Such differential equations are called autonomous. In a typical initial-value problem x and dx/dt are to be determined at time t when these two quantities are given at time t0. Equation (H.l) can be reduced to a system of two first-order differential equa­ tions by using the new variable , - | .

(H.2)

Thus, Eq. (H.l) becomes dx Tt=y, ^

(H.3a)

= -x + eF(xty).

(H.3b)

In the (x,y) plane, called the phase plane (see Appendix G and references therein), x is, in general, a measure of a "displacement" and y is a measure of the "velocity'' of the displacement. The Eqs. (H.3) can be generalized to the set g

= P(x,y),

^

= Q(*,V),

(H.4)

where P and Q are functions of x and y. The corresponding first-order differential equation for the phase plane trajectories are dy dx

Q{x,y) P(x,y) 285

(H.5)

286

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

In general, this equation has a solution of the form F1(x,y,C)

= 0,

(H.6)

where C is the integration constant. A plot of this equation for various values of C gives rise to a family of curves called integral curves. If the initial conditions are x(t0) = x0 and y(t0) = y0, it follows that the integral curve must pass through the point P(x0,y0).

For t > t 0 , the point P[x(t),y(t)]

curve Fi[x(t),y(t),Ci] F1(x0,y0,C1)

traces out a path denned by the

= 0, where C\ is the value of the constant C at t = t0, i.e.,

= 0.

The references listed at the end of this appendix give detailed discussions of the existence, uniqueness, and continuity properties of the first-order system of Eq. (H.4). A limit cycle is an isolated closed integral curve L that has the important property that all integral curves in its neighborhood either spiral toward or away from it. (See the discussion in Section 1 of Appendix G.) Limit cycles correspond to periodic solutions of Eq. (H.4) in the phase plane. This appendix is concerned with the question of the stability of the limit cycles that may exist for Eq. (H.4). The procedure given here follows closely the method of Davies and James [1, 2]. The notion of stability used here is the following: If a moving point in the (x, y) plane is subject to Eq. (H.4) and starts at time t = t0 from a point (xo,yo) that is sufficiently close to a limit cycle L, then as t —► oo, the moving point tends to the limit cycle. This type of stability is called asymptotic stability. The limit cycle is unstable if the moving point moves farther from the limit cycle as t —> oo. H.2 S t a b i l i t y C o n d i t i o n Assume that Eq. (H.4) has a periodic solution denoted by

x = (t),

V = i>(t),

(H.7a)

APPENDICES

287

with period 2ir, i.e., 4>(t) = 4>(t + 2TT),

V(<) = <M< + 2TT).

(H.7b)

Consider a small perturbation from this periodic motion; it can be written as x = flt) + ((t),

y = 4>{t) + »?(f),

(H.8)

where £(i) and 77(1) are assumed initially to be small as compared, respectively, with 4>(t) and xp(t). Substitution of Eq. (H.8) into Eq. (H.4) gives

^ + §=PW)

+ mMt) + ri(t)}

=m , , + ^

+

?3M, + ...,

+

^ l ,

( „. 9 )

and

' =W^]

+

«^ ox

+

-.

(H.10)

ay

Neglecting terms of second degree and higher in £ and r?, the following linear per­ turbation equations are obtained for £ and r/:

rf^

=

&

^+

dy

"'

dr, _ dQim,m] c l gwc^ffl] rft

dx

^

(H

-Ha)

(Hllb)

dy

Note that the coefficients in Eqs. ( H . l l ) are periodic functions of t with period 27r. Therefore, Eqs. ( H . l l ) are unchanged when t is replaced by t + 2n. The variable 77 can be eliminated from the Eqs. ( H . l l ) to obtain a single lin­ ear second-order differential equation for £. Denote the two linearly independent solutions of this equation by (i(t) and ^(<)- Similarly, let r?i(i) and T)2(t) be the

288

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

corresponding two linearly independent solutions of the second-order differential equation for 77. Since Eqs. ( H . l l ) are unchanged when t is replaced by t + 2TT, it follows that £i(£ + 2TT) and ^ ( ' + 27r) are also solutions and can be written as

Ut + 2TT) = A1(1(t) + BM*),

(H-12a)

(2(t + 2ir) = A2h(t) + BaUt),

(H.12b)

where Aj, B\, A2, and B 2 are arbitrary constants. This result follows from the fact that since £i(i) and ^2(t) are two linearly independent solutions, any other solutions can be written as a linear combination of them. Given £i(t) and £2(t), the solution 771(f) can be deduced from Eq. ( H . l l a ) in terms of £i(t), and 772(f) in terms of (,2(t). (Ai,Bi,A2,B2)

Consequently, the same constants

will appear in the equations for 771(7; + 27r) and 772(1 + 27r), i.e., jji(i + 2TT) = AlVl(t)

+ BlV2(t),

(H.13a)

r,2(t + 2TT) = ,4277! (t) + B2r,2(t).

(H.13b)

Let 0(f) be any solution for £(f) of Eqs. ( H . l l ) . Therefore, #(t) = a 1 { 1 ( t ) + / 9 1 f a ( t ) ,

(H.14)

where c*i and /?i are constants. Consider now the case where £ = 6{t) satisfies the condition 0{t + 2TT) = k$(t),

(H.15)

where fc is a constant. Using Eqs. (H.12) and (H.14) gives ai(i(t

+ 2TT) + &£,(< + 2TT) = k[ai^(t)

+

fte2(t)],

(H.16)

or

«i[Ai6(t) + Bj&(0] + A[i4 a 6(0 + #»&(*)] = fc[ai6(0 + fc&(t)]. (H.17)

APPENDICES Since d(t)

289

and £2(<) are linearly independent, it follows that (Aj - k)ai + A2Pi = 0,

(H.18a)

B i a ] + (B2 - k)p! = 0,

(H.18b)

and, consequently, k must be a solution of Ai — k A2 = k2 - (A1 + B2)k + (AiBt Bi B2 - k

- A a B i ) = 0.

(H.19)

When Ai, A2, B\, and B2 are known, Eq. (H.19) can be solved to determine k. Since Eq. (H.19) is of second degree, there will be two solutions for k, which can be written in the following form: k1=e2"'\

k2 = e2n^.

(H.20)

The fi\ and fi2 are called characteristic exponents. It follows that there exist two functions 6{i) such that 81(t + 2n) = e2"»i01(t),

(H.21a)

02(t + 2TT) = e2lr^92{t).

(H.21b)

Consider the Eq. (H.21a). It can be written as e -s**i 9l

(t + 2TT) = 6X (t),

(H.22a)

e-"i(«+2»)0j(* + 2TT) = e-^e^t).

(H.22b)

or

Thus, e x p ( - ^ i t ) ^ ( i ) is a periodic function in t of periodic 2n.

Similarly,

exp(—n 2 i)0 2 {t) is a periodic function in t of period 27r. Denote these periodic functions, respectively, by /n(<) and f2\(t),

so that

tfi(*) = e ' , f / i i ( * ) .

(H.23a)

290

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS (H-23b)

*»(*) = e « 7 a i ( t ) . where fu(t

+ 2TT) = fn(t)

and / 2 1 (f + 2TT) = / 2 1 (f).

We can take 6i(t) and 82(t) to be the two linearly independent solutions for £(f) of the original differential equations ( H . l l ) ; thus 6 W = e"I7iiW,

&(*) = e"*/2i(*)-

(H.24)

The two linearly independent solutions for Jj(f) will also be of the form m(t) = e " l 7 u ( * ) , where /i2(f + 27r) = /12(f) and fM(t

%(*) = e" 2 7 2 2 (f),

(H.25)

+ 2TT) = /22(f)- Note that, in general, £(f)

and 77(f) are not periodic functions in f.

If these functions are substituted into

Eq. (H.lla) and simplified, then the following two equations are obtained:

dfn ,

Solving for dP/dx

,

dP

^T

+ //l/ll =

^

+ ,2/21

^

dP

/ll +

= f/21

^/l2' +

f/2,

(H 26a)

-

(H.26b)

gives 9P _ /11/22 ~ /21/12 <9z

/ n / 2 2 - /12/21

Following a similar, dQ/dy

, „ „„..

/11/22 — /12/21

can be calculated; it is

9Q _ /22/11 - /21/12 9y

Mi/11/22 — M2/12/21

/ n / 2 2 — /12/21

M2/11/22 ~ Mi/12/21

,„

...

/ n / 2 2 - /12/21

In Eqs. (H.27) and (H.28), the following notation is used

Now adding Eqs. (H.27) and (H.28) gives !h+~dy~

=

7t [ l o s(/»/22 - /12/21)] + Mi + M2-

(H.30)

APPENDICES

291

Integrating this expression from t = 0 to t = 2n gives

r(£+f)^2^+^

(H 3I)

-

since the log term in Eq. (H.30) integrates to zero because the fij(t)

are periodic

functions of t with period 2-K. Hence, the sum of the two characteristic exponents

H\ + fi2

W Jo

dP[±(t),i>(t)} + dQ[(i)Mt)} dt. dx

dy

(H.32)

It will now be shown that if Eq. ( H . l l ) has a periodic solution, then one of the characteristic exponents is zero; that is, either fj,i = 0 or ft2 — 0. To do this, assume that £(t) and r)(t) are periodic solutions of Eq. ( H . l l ) . It then follows that t(t) = Cihit)

+ C2(2(t),

(H.33a)

r,(t) = Cim(t)+C2V2(t),

(H.33b)

where C] and C2 are two constants, not simultaneously zero, and i(t + 2w) = ((t),

rj(t + 2w) = r,(t).

(H.34)

Using Eq. (H.21) gives Cift(<) + C2(2(t) = Ci&(t + 2n) + C2(2(t + 2TT) = Cle2^Zl{t)

+ C2e2^(2{t),

(H.35)

and the result C,(l - e 2 "">)6(*) + C a (l - e 2 *" 2 )6(t) = 0.

(H.36)

A similar result also holds for rji(t) and rj2(t), namely, C,(l - e2^)m(t)

+ C 2 (l - e2^)r,2(t)

= 0.

(H.37)

292

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

Since C\ and C2 are not both zero, and since (£1,6)

an

^

( ' / l i ^ ) are linearly

independent, it follows from Eqs. (H.36) and (H.37) that either ^

or (J,2 must

vanish. It is easy to show that Eq. ( H . l l ) has at least one periodic solution, and as a consequence of the previous result one of its characteristic exponents is zero. To show this, note that

m = PW)M*)],

( H - 38 )

•?(*) = QW*)> 0(0]

are solutions, since if Eq. (H.38) is assumed to be correct d£ dP dx dP dy -dt=lx-M+-dy-li dr,

dQdx

=

=

dQdy

+

=

dP „ dP n DP , -dX-P+-^Q=-dX-li+-Qy-r1,

DP

dQ

dQ

^ ^^ W^ ^

P +

dQ

^

0 =

dQ

^

e+

Wr7'

(H.39a)

(H 39b)

-

Hence £ = P and r) = Q are periodic solutions. For this case, /^i can be set equal to zero and /u2 set equal to p. This gives, using Eq. (H.32)

From Eqs. (H.24) and (H.25), it follows that the criterion for stability or instability of the periodic solution x(t) = # t ) ,

y(t) = i>(t),

(H.41)

depends on whether \i < 0 or fj, > 0. In fact, (i < 0 =>■ asymptotic stability,

(H.42a)

H > 0 => asymptotic instability.

(H.42b)

Note that the integrand in Eq. (H.40) is evaluated at x = <j>(t) and y = V'(r)-

APPENDICES

293

For nonlinear differential equations of the form given by Eq. (H.l), the condition for stability is

■ fM^q

(H.43)

Jo oy If the period of oscillation is T rather than 2ir, then the condition for stability becomes Jo

oy

where y = x = 4>{t). Often a "first approximation" to the solution to Eq. (H.l) is x = 4>(t) ~ Acost,

y = ip(t) ~ — Asint.

(H.45)

For this situation, the stability condition becomes r2n

dF(Acost,

-Asint) dy

Jo

dt < 0.

(H.46)

To illustrate the procedure, consider the van der Pol differential equation fg

+

*= e(l-*2)|,

0<e«l,

(H.47)

where = (l-x2)y.

F(x,y)

(H.48)

A first-approximation to the solution is i = 2 cost,

y = -2sint.

(H.49)

Therefore, dF x

( 'y) dy

=i-

2 x

= l-4cos2i = -l-2cos2t,

(H.50)

and /■27T

r2ir

e I

Jo

ftp

lf-dt d

y

t"

= -e

(l + 2cos2t)di

Jo

= -2Tre<0.

(H.51)

294

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

The conclusion reached is that the above periodic function, Eq. (H.49), corresponds to an approximation to the stable limit cycle of the van der Pol equation. References 1. T. V. Davies and E. M. James, Nonlinear Differential Wesley; Reading, MA; 1966).

Equations

(Addison-

2. Reference 1, Section 3.7. 3. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qualitative Theory of Second-Order Dynamical Systems (Wiley, New York, 1973). 4. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983). 5. L. Perko, Differential Equations and Dynamical Systems (Springer-Verlag, New York, 1991). 6. Ye Yan-Qian, Theory of Limit Cycles (Translations of Mathematical Mono­ graphs, Vol. 66, American Mathematical Society, Rhode Island, 1986).

Appendix I Q U A L I T A T I V E T H E O R Y OF D I F F E R E N T I A L E Q U A T I O N S I.l Introduction Essentially all of the differential equations considered in this book can be rep­ resented as a system of two coupled first-order equations, i.e.,

£=*(*,»).

!«
a.!)

The assumption of continuity for P, Q and their first partial derivatives generally ensures the existence of a unique solution to Eq. (I.l) for a given set of initial values for x and y. Except for special cases of functional forms for P and Q, generally, sim­ ple explicit functions of t do not exist for the solutions. In addition to the analytic techniques that are presented in this book to obtain approximations to the solutions of Eq. (I.l), a powerful technique is the use of qualitative methods to investigate the behavior of the trajectories in the (x,y) phase space. These trajectories, y = y(x), are obtained as solutions to the first-order differential equation dy_ _

dx

Q{x,y)

p(x,yy

(

->

The general solution to Eq. (1.2) is often called a first-integral of the system given by Eq. (1.1). The simplest solutions to Eq. (I.l) are the constant solutions, x(t) = x and y(£) = y. They correspond to the fixed-points of this equation and are obtained by solving the equations P(x,y)

= 0,

Q(x,y)=0.

(1.3)

In applications pertaining to actual physical, biological, and engineering processes, only the real solutions have meaning. It is also of interest to note that the oscillating systems considered in this book, in general, have no more than one real fixed-point.

:

2S5 V

296

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

However, there does exist an abundance of other planar phase space models that have multi-fixed-points [1, 2, 3, 4, 5]. The purpose of this appendix is to give a brief introduction to the use of qualitative techniques to obtain information on the trajectories in the (x,y)

phase

space for the various types of differential equations considered in this book. More detailed procedures and their mathematical justification are presented in the various references listed at the end of this appendix. Appendix A.3 of Segel's book [5] provides one of the best introductions to this topic. 1.2 Basic Procedure (1) Determine the fixed-points of Eq. (1.1). These correspond to the solutions of Eq. (1.3). (2) Calculate the x nullcline, the curve in the phase plane along which dy/dx = oo, by solving P[x,Voo(x)]=0,

(1.4)

for yx = yoo(x)(3) Calculate the y nullcline, the curve in the phase plane along which dy/dx = 0, by solving Q[x,y0(x)]=0,

(1.5)

for y0 = yo(x)C o m m e n t . Note that the x and y nullclines intersect at the fixed-points. Also, they, in general, are not solutions to Eq. (1.2). (4) The x and y nullclines divide the phase plane into several open regions. In each region the sign of dy/dx is fixed, i.e., in a given region, dy/dx is bounded and either positive or negative. The next step is to determine the sign of dy/dx in each region. This can usually be done in a rather direct fashion.

APPENDICES

297

(5) Start at a carefully selected point in the phase plane and sketch the trajec­ tory passing through that point. Repeat this procedure for several other starting points. (6) It may be of value to examine the behavior of trajectories in the neighbor­ hood of a fixed-point. A linear approximation to Eq. (I.l) will usually provide this information [1, 2, 5]. C o m m e n t . Almost all of the systems considered in this book will have the relevant fixed-point being either a neutral center or a stable or unstable spiral. See Figure I.l. 1.3 Applications The first two examples are conservative oscillators [6]. They can be represented by the form g

+

/ W = 0.

(1.6)

The system equations are - = y ,

^

= -/(*),

(L7)

and the differential equation for the phase space trajectories is

dy

f(x)

dx

y

(1.8)

Observe that the x nullcline is the i-axis, while the y nullcline correspond to the isolated solutions of /(*)=0.

(1.9)

If these solutions are denoted by {x n },

n = l,2,...,7V,

(1.10)

then the fixed points all lie on the i-axis and are given by (*,^) = { ( * i , 0 ) , ( x a , 0 ) , . . . , ( * B > 0 ) } .

(1-11)

298

Figure I.l. Trajectories in the neighborhood of a fixed-point: (a) neutral center, (b) unstable spiral, (c) stable spiral.

APPENDICES

299

Equation (1.8) is separable and can be integrated to give y2 Y + V(x) = E = constant,

(1.12)

where V(x) is the potential function

V(x) = JZf{z)dz.

(1.13)

Equation (1.12) is called the first-integral of Eq. (1.6) and, generally, provides an implicit functional representation for the phase space trajectories. The constant E depends on the initial conditions, i.e., if x{t0) = x0 and y(t0) = y0, then

E=f

+ V(x0).

(1.14)

Example 1.3.1. Consider the following nonlinear, conservative oscillator ,

cPx

^+*

3

= 0.

(1.15)

The only fixed-point is at ( i , y) = (0,0) and its first integral is v2

y J

X* +

^

= E>0.

(1.16)

This latter equation defines a family of closed oval shaped curves in the phase plane. Larger values of E correspond to larger oval curves. (See Figure 1.2.) The conclusion is that for whatever set of initial conditions, the solutions to Eq. (1.15) are periodic. Example 1.3.2. The conservative, nonlinear oscillator

J-* + * 3 =0

(1.17)

300

Figure 1.2. Plots of Eq. (1.16) for various values of E. Note that E^ < E2 < E3.

APPENDICES

301

has the system equations dx

dv

,

Observe that for this oscillator there are three fixed-points located at (x,y) = { ( - 1 , 0 ) , (0,0), (1,1)}. The first integral is v2

x2

I4

Examination of this equation shows that while all the phase space curves are closed and thus all solutions are periodic, the general topology of the phase space is more complex than that for Eq. (1.15). Figure 1.3 gives a sampling of phase space trajectories for this system. The fixed-points are indicated by heavy dots. The figure-eight shaped trajectory sep­ arates the phase space into three regions. Inside each of the lobes, the trajec­ tories encircle a fixed-point. For example, the left lobe contains the fixed-point (x,y) = (—1,0) and the right lobe (x,y) start at the fixed-point (x,y)

= (1,0). Note that the two lobes both

= (0,0) and return there. This type of trajectory

is called a homoclinic trajectory. A simple calculation shows that his homoclinic trajectory corresponds to E = 0. For - - < E < 0,

(1.20)

the closed trajectories are confined to the interior of one of the lobes. For a given value of E in this interval, there exists two periodic motions. For E > 0, the closed trajectories enclose both fixed-points. Example 1.3.3. The van der Pol equation is g

+ . - e d - , - ) £ ,

,>0.

(1.21)

302

Figure 1.3. Plots of Eq. (1.19) for various values of E. The fixed-points are indi­ cated by the heavy dots.

APPENDICES

303

Written in system form it becomes dx = y,

(1.22a)

= -x + e(l-x2)y.

(I.22b)

Tt

^

There is a single fixed-point at (x,y) = (0,0). The trajectories in phase space are determined by the first-order differential equation dy _ -x + e(l - x2)y dx y

(1.23)

From Eqs. (1-22), it follows that the x nullcline is the x-axis, while the y nullcline is the curve yo{x) given by the expression

The x and y nullclines separate the phase plane into six open regions. These regions, as well as the sign of dy/dx in each region, is shown in Figure 1.4. Figure 1.5 shows the path of two typical trajectories, one starting far from the fixed-point and the other near the fixed-point. The path of the trajectory far from the fixed-point is drawn using information on the sign of the derivative dy/dx in each region and the fact that the trajectory has unbounded slope whenever it passes through the x-axis and zero slope whenever it passes through the y nullcline. The trajectory near the fixed-point is determined by examining the linearized form of Eq. (1.22), namely, dx

d

=y

Tt '

y

Tt=-x

,

+ ey

-

n OKA

(L25)

The solution to these equations are an unstable spiral for e > 0. To summarize, a typical trajectory far from the (fixed-point) origin spirals toward the origin, while a typical trajectory near the origin spirals away from it. Since such trajectories cannot cross each other, this result implies the existence of at least one closed curve in the phase space [7, 8]. Observing that a closed curve

304

aFigure 1.4. The curves are plots of the y nullcline yo(x) given by Eq. (1.24).

305

Figure 1.5. Typical trajectories for the van der Pol equation.

306

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS

implies a limit cycle, the general conclusion is that the van der Pol equation has at least one stable limit cycle. The total number of limit cycles cannot be determined from these qualitative geometric arguments. References 1. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qualita­ tive Theory of Second-Order Dynamic Systems (Wiley, New York, 1973, Israel Program for Scientific Translations). 2. L. Edelstein-Keshet, Mathematical Models in Biology (McGraw-Hill, New York, 1988). See Chapter 5. 3. M. Martin, Differential Equations and Their Applications New York, 1993, 4th edition). See Chapter 4.

(Springer-Verlag,

4. M. Humi and W. Miller, Second Course in Ordinary Differential Equations for Scientists and Engineers (Springer-Verlag, New York, 1988). See Chapter 8. 5. L. A. Segel, editor, Mathematical Models in Molecular and Cellular Biology (Cambridge University Press, Cambridge, 1980). See Appendix A.3: G. M. Odell, "Qualitative theory of systems of ordinary differential equations, including phase space analysis and the use of the Hopf bifurcation theorem." 6. E. T. Whittaker, Advanced Dynamics (Cambridge University Press, London, 1937). 7. N. Minorsky, Nonlinear Oscillations (Van Nostrand; Princeton, NJ; 1962). See Chapters 3 and 14. 8. V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equa­ tions (Princeton University Press; Princeton, NJ; 1960). See pp. 133-134.

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BIBLIOGRAPHY

313

20. "Perturbation Procedure for the van der Pol Oscillator Based on the Hopf Bifurcation Theorem," Journal of Sound and Vibration 127, 187 (1988). 21. "Construction of a Perturbation Solution for a Nonlinear, Singular Oscillator Equation," Journal of Sound and Vibration 130, 513 (1989). 22. "A Difference-Equation Model of the Duffing Equation," with 0 . Oyedeji and C. R. Mclntyre, Journal of Sound and Vibration 130, 509 (1989). 23. "Investigation of the Mathematical Properties of a New Nonlinear Negative Resistance Oscillator," Circuits, Systems and Signal Processing 8, 187 (1989). 24. "Calculation of the Transient Behavior for a Nonlinear, Singular Oscillator Equation," Journal of Sound and Vibration 134, 187 (1989). 25. "Symmetry Properties of van der Pol Type Differential Equations and Im­ plications," with W. E. Collins, Journal of Sound and Vibration 136, 352 (1990). 26. "Calculation of the Oscillatory Properties of the Solutions of Two Coupled, First-Order, Nonlinear Differential Equations," Journal of Sound and Vibra­ tion 137, 331 (1990). 27. "Investigation of Finite-Difference Models of the van der Pol Equation," in Differential Equations and Applications, editor A. R. Aftabizadeh (Ohio Uni­ versity Press; Columbus, OH; 1989); pp. 210-215. 28. "Investigations of the Conditions for Oscillations in Two-Variable Models of Biological Processes," BioSystems 24, 31 (1990). 29. "Failure of the Method of Slowly Varying Amplitude and Phase for Nonlinear, Singular Oscillators," with I. Ramadhani, Journal of Sound and Vibration 152, 180 (1992). 30. "Investigation of an Anti-Symmetric Quadratic Nonlinear Oscillator," with I. Ramadhani, Journal of Sound and Vibration 154, 190 (1992). 31. "Application of Generalized Harmonic Balance to an Anti-Symmetric Quad­ ratic Nonlinear Oscillator," with M. Mixon, Journal of Sound and Vibration 159, 546 (1992). 32. "Harmonic Balance: Comparison of Equation of Motion and Energy Meth­ ods," with S. Hiamang, Journal of Sound and Vibration 164, 179 (1993). 33. "Construction of a Perturbation Solution to a Mixed Parity System that Sat­ isfies the Correct Initial Conditions," Journal of Sound and Vibration 167, 564 (1993). 34. "Exact Solution to the Anti-Symmetric Constant Force Oscillator Equation," with T. Lipscomb, Journal of Sound and Vibration 169, 138 (1994).

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35. "Construction of a Finite-Difference Scheme that Exactly Conserves Energy for a Mixed Parity Oscillator," Journal of Sound and Vibration 172, 142 (1994). 36. "Reply" to the paper of B. N. Rao, Journal of Sound and Vibration 172, 698 (1994). 37. "A Finite Difference Scheme for Second Order Nonlinear Oscillator Differen­ tial Equations," with K. Oyedeji, Journal of Sound and Vibration 178, 285 (1994). 38. "Investigation of a Generalized van der Pol Oscillator Differential Equation," with W. Addo-Asah and H. C. Akpati, Journal of Sound and Vibration 179, 733 (1995).

INDEX

Advertising model, 24 Antisymmetric, constant force oscillator, 35-37 Asymptotic expansions definition of, 258 elementary operations on, 260-262 generalized, 263-264 uniform, 259-260 uniqueness of, 258-259 Asymptotic sequence, 258 Averaged quantities, 90, 222 Batch fermentation oscillations, 228-229 Brusselator model, 20, 22, 229-232 Characteristic equation, 241 roots, 242 Characteristic exponents, 289 Chlorine dioxide-iodine reaction, 23 Coulomb friction, 19-20, 102-103 Damping forces Coulomb, 19-20, 102-103 cubic, 188-189 linear, 27, 101 negative, 10 quadratic, 19, 64-65, 101-103 viscous, 17-19 Derivative expansion procedure, 191-192 Diatomic molecule, oscillations, 15-17 Differentiation of composite functions, 59, 108-111, 177-178, 191-192 of definite integrals, 241 Dimensionless forms of differential equations, 24—27 examples, 27-30 Dufnng's equation, 28-29 exact solution, 38-41 free oscillations treated by: harmonic balance, 153-154, 164-167; Krylov-Bogoliubov first approximation, 100-101; perturbation method, 65-66; two-time expansion, 186-188

315

316

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS Eardrum, vibrations of, 3, 6 Electrical circuits, nonlinear, 6-14 Elliptic integral of the first kind, 39 complete integral, 41, 43 Equivalent linearization, relation to Krylov-Bogoliubov first approximation, 95 worked examples, 97-99 Existence theorems linear equations, 270 nonlinear equations, 266 Fourier series bound on coefficients, 253-254 convergence of, 252-253 definition of, 251-252 expansion coefficients, 251 Gauge functions, 256 Generating solution, 71 Glycolysis, 22 Glycolytic oscillator linear stability analysis, 205, 207-209 nullclines, 202, 203 phase space diagram, 204 Harmonic balance comparison with perturbation methods, 163-164 first approximation, 150, 152 general method, 140-141, 147-149 rational approximation, 149-151 second approximation, 150-151 worked examples, 144-146, 153-162, 164-169 Harmonic oscillator, 30-31 Homogeneous linear differential equation, 270 Hopf bifurcation theorem, 210-215 subcritical, 212, 217 supercritical, 212, 216 Independent variable, transformation of, 59 Inhomogeneous linear differential equation, 273 Initial conditions dependence of solution on 266-268 Integral curves, 285-286

INDEX Jacobi elliptic function, 39, 41, 46 Fourier expansion, 41, 46 Kirchoff's laws, 6 Krylov-Bogoliubov first approximation, 87-91 two special cases, 91-92 worked examples, 99-106 Krylov-Bogoliubov-Mitropolsky method, 106-116 elimination of secular terms, 108 first approximation, 114 n-th approximation, 115 second approximation, 112-115, 115-116 worked examples, 117-123 Levinson-Smith theorem, 283-284 Lewis equation, 159-160 Lienard-Levinson-Smith theorem, 281-283 Limit cycles, 280-281 existence of, 72-79, 224 number of, 123-128 spurious, 123-128 stability of, 92-95, 224, 286-293 theorems on, 281-284 Lindstedt-Poincare method, 59-61 Linear damped oscillator, 27-28, 101, 117-118, 174, 192-195 Linear combination, 271 Linear dependent functions, 271 Linear independent functions, 271 Linear second order differential equations constant coefficients, 275-279 existence of solutions, 270 homogeneous equations, 270, 272-273 inhomogeneous equations, 273-275 solutions of, 275-279 Mass attached to a stretched wire, 3, 156-158 Mixed parity oscillator, 68-71 Multi-time expansions, 175-176 equations for the derivatives, 191-192 initial conditions, 192 worked examples, 192-197 Negative-resistance oscillator, 10, 14 see also Rayleigh equation, van der Pol equation

317

318

OSCILLATIONS IN PLANAR DYNAMIC SYSTEMS Nonlinear damping, see damping forces Nullclines, 296 Order symbols, 256-258 use of, 262-263 Parameter dependence of solution on, 268 differentiation of an integral with respect to, 241 Particle-in-a-box, 31-35 Pendulum, 2-3, 44-47 exact solution for period, 41-44 Periodic function, 251 Perturbation method existence of a periodic solution, 72-79 limitations of, 72 Lindstedt-Poincare method, 59-61 secular terms, 56-57, 60 worked examples, 61-71, 76-79 Phase plane, 295 Piecewise smooth function, 252 Power series definition, 245-246 radius of convergence, 246-247 Predator-prey model, 24 Rayleigh equation, 15, 29-30 free oscillations treated by: derivative expansion procedure, 196-197; two-time method, 190-191 Second order differential equations definition, 265 dependence of solution on initial conditions, 266-268 dependence of solution on parameter, 268 existence of solution, 266 uniqueness of solution, 266 see also linear second order differential equations Secular terms, 56-57 elimination of in the: derivative expansion procedure, 193, 196, 197; Krylov-Bogoliubov-Mitropolsky method, 108; perturbation method, 60; two-time expansion, 181, 183, 185, 186, 187, 189, 190

INDEX Self-sustained oscillations, see Rayleigh equation, van der Pol equation Series expansion uniform convergence, 243 uniformly convergent series, 244-245 see also Taylor series, power series Sgn function, 20 Shohat expansion, 79-80 worked examples, 79-82 Stellar oscillation model equation, 121-123 Taylor's series, 247-249 for function of two variables, 249-250 see also power series, series expansions Two-time expansion technique, 174-175, 177-179 equations for the third approximation, 178 initial conditions, 179 worked examples, 179-191 Two-variable systems, 218-225, 295-297 recipe for solving, 225-226 worked examples, 226-232 Uniform convergence, 243 Uniformly valid expansion, 259-260 Uniqueness of solutions, 266, 270 van der Pol equation, 14, 301, 303-306 free oscillations treated by: derivative expansion procedure, 195-196; harmonic balance, 154-156, 160-162; Krylov-Bogoliubov first approximation, 104-106; Krylov-Bogoliubov-Mitropolsky method, 119-121; perturbation method, 66-68; recipe, 226-227 Variables, dimensionless, 24-27 Viscous damping, see damping forces W C M oscillator, 227-228 Weierstrass M test, 244 Wronskian, 272

319