A PERTURBATIGN TECHNIQUE FOR '°

ORDINARY DIFFERENTIAL EQUATIONS
WITH PERIODIC COEFFICIENTS 7

Dissertation for the Segree of P21. D.

MICHIGAN STATE UNIVERSITY
ISRAEL LINDEIIFELD
1975

This is to certify that the

thesis entitled

A PERTURBATION TECHNIQUE FOR ORDINARY
DIFFERENTIAL EQUATIONS WITH
PERIODIC COEFFICIENTS

presented by

Israel Lindenfeld

 
 
 
     
    

  

LIBRAR Y
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Diva-53y

has been accepted towards fulfillment

of the requirements for

PhOD.

 

Date 2 y/‘Vé/f/é

0-7539

 

degree in Ma thema tics

 

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/ Major professor

 
  
 

 

 

 

 

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ABSTRACT

A PERTURBATION TECHNIQUE FOR ORDINARY
DIFFERENTIAL EQUATIONS WITH
PERIODIC COEFFICIENTS

BY

Israel Lindenfeld

We are interested in finding conditions under which
solutions, of a certain class of second order ordinary
differential equations with periodic coefficients, are

bounded. The class of equations under study is of the

form
.0 2 S n (S)
(1) X. + 03.x. + 6 Z Z a.k cossmtxk = O
3 3 3 5:1 k=l
j: l’ooo’n
where

dot denotes differentiation with respect to t
w. is the eigenfrequency of the unperturbed system

agi) are constants for all j,k and s

e > O is a small parameter

-

m is the frequency of the perturbed system

S and n are positive integers ‘2 l.

 

Cu

Israel Lindenfeld

We start by generalizing a problem solved by Wehrli
(Ingenieur-Archive 1963). The stability of the solutions
of a system of two coupled second order ordinary differen—
tial equations with periodic coefficients is investigated
by using classical Floquet theory. These differential
equations arise by considering the influence of a harmoni—
cally varying torsion moment on a rotating shaft carrying a
disc.

In Chapter III systems of the form (1) are treated by
generalizing a perturbation technique developed by Struble
and Fletcher (SIAM J. of Appl. Math., 1962). The perturba-
tion technique which we develOp gives rise to systems of
differential equations which can be solved exactly leading
to algebraic equations of degree 2_2 with complex co-
efficients. The perturbation technique developed by us is
applied to the Mathieu and van der Pol equations and yields
results which compare with the classical theory of these
equations.

The applicability of our technique is also clearly
demonstrated by the results which we obtain, for various
examples beyond those treated in Chapter III, and which
compare favorably with the results obtained either by
utilizing classical Floquet theory, or the Poincaré—
Lindstedt perturbation technique.

We also analyze a perturbation technique developed by

Hsu (J. of Appl. Mech. 1963). A careful analysis of Hsu's

Israel Lindenfeld

method enables us to explain its shortcomings, i.e., Hsu's
method leads to systems of first order differential equa-
tions which are solved by averaging (i.e., cannot be solved
exactly). This analysis enables us to modify it in the
specific case of the Mathieu equation by using two perturba—
tion parameters instead of one.

We wish to point out that the amount of computation
involved by using characteristic exponents (Floquet theory)
is huge in contrast to the amount of computation involved

by utilizing the perturbation method which we developed.

A PERTURBATION TECHNIQUE FOR ORDINARY
DIFFERENTIAL EQUATIONS WITH
PERIODIC COEFFICIENTS

BY

Israel Lindenfeld

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1976

ACKNOWLEDGMENTS

I would like to thank my major Professor Robert

Wasserman for his help and encouragement in the writing of
this dissertation. I wish also to thank Professor David
H.Y. Yen for his helpful suggestions and for introducing
me to the subject. My thanks are also extended to Profes-
sors J. Sutherland Frame and C.Y. Wang for their helpful
comments and to Professor Howard Teitelbaum (OMERAD) for
his help in proofreading. Last but not least I am grate-
ful to Mrs. Glendora Milligan for her careful and efficient

typing.

ii

TABLE OF CONTENTS

Chapter Page

I. INTRODUCTION AND PRELIMINARIES 1
1.0 Introduction . . . . . . . . . . . . 1
1.1 Floquet Theory . . . . . . . . . . . . l
1.2 Hill's Equation. . . . . . . . . . . . . . 6
1.3 Infinite Determinants . . . . . . . . . . 7
1.4 Some Perturbation Techniques (Regular

Perturbations, the Poincaré—Lindstedt

Method) . . . . . . . . . . . . . . . . . 9
1.5 The Method of Averaging . . . . . . . . . 12
II. A PROBLEM FROM DYNAMICS 15
2.1 Derivation of the Equation of Motion . . . 15
2.2 Necessary Conditions for the Existence of
Solutions for the System (2.1.9) . . . . . 18
2.3 Stability of Solutions . . . . . . . . . . 23
2.4 Influence of a Damping Factor . . . . . . 28
2.5 The Case a = g-(s any positive integer) . 28
III. GENERALIZATION OF STRUBLE'S METHOD FOR SYSTEMS
OF SECOND ORDER O.D.E.'S 31
3.1 Theory of the First Order Approximation. . 31

3.2 Theory of the Second Order Approximation . 43
3.3 The Mathieu Equation . . . . . . . . . . . 49

3.4 The van der Pol Equation . . . . . . . . . 54

iii

.."" .1
1" v~

 

VI .

Chapter Page

IV. ADDITIONAL APPLICATIONS OF THE GENERALIZED
STRUBLE METHOD 58
4.1 A First Order Approximation for the
Generalized Wehrli System . . . . . . . . 58
4.2 A Second Order Result for Wehrli's
Original System . . . . . . . . . . . . . 61
4.3 The Differential Equation
I + (6 + Ecost)-mx = O . . . . . . . . 64
4.4 The Differential Equation
I + (a - ecoszt)(l — Ecoszt)_lx = o . . 69
V. APPLICATION OF PERTURBATION TECHNIQUES DEVELOPED
BY PORTER AND RAND 73
5.1 A Third Order Result for Wehrli's Original
System . . . . . . . . . . . . . . . . . 73
5.2 An Application of Rand's Technique . . . 77
VI. HSU'S METHOD AND ITS MODIFICATION 82
6.1 Hsu's Method . . . . . . . . . . . . . . 82

6.2 Modification of Hsu's Technique as

Applied to the Mathieu Equation . . . . . 88
CONCLUSIONS 96

APPENDIX A: THE INFINITE DETERMINANT A(O) . . 99

APPENDIX B: EXPANSION OF A(O) UP TO TERMS OF

C(64) . . . . . . . . . . . . . . 101
APPENDIX c: THE COEFFICIENTS K1 AND K2 . . . 103
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . 104

iv

Figure 1

Figure 2

LIST OF FIGURES

26

27

 

 

! Isl I II 4

PI

er

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”he

CHAPTER I

INTRODUCTION AND PRELIMINARIES

1.0 Introduction.

 

The main purpose of our present work is to develop a
perturbation technique for certain classes of ordinary
differential equations with periodic coefficients. It is
necessary to develop this kind of approximation technique

since the computation of the characteristic exponents is a

 

difficult task (see Chapter I, Section 1 and Chapter II).
We shall also apply the perturbation technique, developed
by us in Chapter III to a number of examples (see Chapters
III and IV) and Obtain results which compare favorably with

the ones obtained by using rigorous methods.

1.1. Floquet Theory.

In this section we will outline some of the basic
prOperties and state some of the fundamental theorems gov-
erning the behavior of systems of first order linear ordi-

nary differential equations of the form
(1.1.1) )2 = A(t)X

where X is a column vector in the variables x1,~o,xn .

 

’_n74

far i

Dot denotes differentiation with respect to t (unless
otherwise indicated) and A(t) is a n x n matrix with
continuous and periodic entries,i.e., A(t + T) = A(t) for
some T #'O. We note that the periodicity of the entries
does not entail the periodicity of the nontrivial solutions,
for if k = (tanzt)x (x = x(t)), we obviously have

[tan (Ir-I-t)]2 = (tantflz, but any nontrivial solution of the

 

above equation is of the form x = x0 exp((tantfl-¢fl, and is
not periodic.

A fundamental theorem by Floquet describing the prop-

erties of the solutions of a system like (1.1.1) states

Theorem (F). Given the system (1.1.1) with A(t) a n by

 

n matrix with continuous periodic entries of period T #’0,
then there exists a nonzero constant a (real or complex)
and at least one nontrivial solution x(t) of the system

(1.1.1) having the property
(1dum xu+T)=cudU.

It can be shown that a is an eigenvalue of a certain con-
stant nonsingular matrix hence a # O; conversely it can be
shown that any solution having the property x(t + T) =

a)((t) for some a fi 0 then a must be an eigenvalue of

the above mentioned matrix.

Definition. The distinct characteristic roots

a1,---,am (l g.m g'n) of this matrix are called the

characteristic numbers or nultipliers. The numbers

0 -°-,om defined by the relation

1!
io.T __
aj = e 3 (i =\/Ll, j = l,°°°,m) are called the charac-

teristic exponents. From this definition it follows that

 

the characteristic exponents are determined up to multiples
of 2N. It can also be shown that if 01,'-°,om are the
characteristic exponents of the system (1.1.1) then there
are at least m solutions of the form
io.t
x.(t) =p.(t)e 3 , j = l,°°°,m, with p.(t+T) =p.(t).
J J J 3

We wish to point out that the main feature of Floquet theory

is the way the solutions are expressed, i.e.,an exponential

times a periodic function.

Definition. We say that a system of first order O.D.E's is

 

stable if all its solutions are bounded for t > O and

 

unstable if unbounded solutions exist as t a a. It can be

 

shown that if the characteristic multipliers aj. satisfy

Iajl < 1 then the solutions of the system (1.1.1) are stable.

Application. Consider the second order linear O.D.E.

 

(1.1.3) 3': + p(t)x = o

where p(t) is a continuous periodic function of period N.
It is well known that equation (1.1.3) has two continuously
differentiable solutions x1(t), x2(t) which are uniquely

determined by the conditions

' u
-r-. '1'" u .. 15h

‘. .“0.,.‘, . g...

 

 

F1
9C.

SC

Wit
We

for

The:

l ?
the

Il.1.

II
0

xl(O) = l X1(O)

ll
H

x2(O) = O x2(O)

It can be shown from the preceding considerations and

Floquet theory that the characteristic equation for (1.1.3)

 

is

(1.1.4) (12 -[xl(1r) + iZITrHa + 1 = O

From equation (1.1.4) we can conclude that if a1 # a2 then
equation (1.1.3) possesses a pair of linearly independent

solutions which can be written in the form
1

w1(t) = e Gtp1(t)

w2(t) = e-iOtPZIt)

1N0 -ino

 

 

w1th pj(t4-N) = pj(t) j = 1,2 and e = a1, e = a2.
We also observe that since alaz = 1 it would be necessary
for stability to require

(1.1.5) In.” = [dzl = l .

These are also sufficient conditions for stability if
a1 #’a2. If d1 = 02 then (1.1.5) is not sufficient for
the stability of the solutions of equation (1.1.3) and addi-
tional conditions have to be imposed, see Stoker [24].

The case where a = a = 1 or a = a ==-1 is of

1 2 l 2
ggeat importance for in this case the differential equation

 

(1.1.3) possesses a nontrivial periodic solution of period

N (2F): respectively. Let xl(N) + 12(n) = A then it

follows from (1.1.4) that

a) [A] > 2 leads to unstable solutions of the

D.E. (1.1.3).

b) IAI < 2 leads to stable solutions since this

implies a1 # a2 and [a

Since A is real we conclude that the transition from sta—

 

bility to instability occurs for [AI = 2 ,i.e., A = 2 or
1 = d2 = l or a1 = a2 = -l

and thus to periodic solutions of period N or 2N

A = -2 which corresponds to a

respectively. Thus the transition from stability to insta-

 

bility is characterized by periodic solutions of period N
or 2H. We wish to emphasize that the principal difficulty

in the computations of the characteristic exponents (or

 

equivalently the multipliers) is the fact that the charac—

teristic equations for D.E's with periodic coefficients

 

depend on the solutions which are unknown, and the solutions
depend on these exponents (see equation (1.1.4)). This
results in a vicious circle and thus different methods of

attack have to be used (e.g., approximation techniques).

 

Some of these techniques will be outlined in this introducé
tory chapter and some will be developed and utilized in the
subsequent chapters. Further details on Floquet theory can
be found in Minorsky [16], pages 127-130 and Stoker [24],
pages 193—198.

1.2. Hill's Equation.

 

In this section we present some results for the second

order O.D.E.
(1.2.1) 32+ [1+Q(t)]x=0

where 1 is a parameter and Q(t) is a real periodic
function of t with period N and is of bounded variation.
The behavior of the solutions of an equation like (1.2.1)
was investigated by Haupt [6] and is stated in the following
theorem (which will be used by us implicitly in the follOW-

ing chapters).

Theorem (H). To every differential equation of type (1.2.1)

 

there correspond two monotonically increasing infinite

sequences of real numbers:

(1.2.2) )‘O'xl’XZ'

l I I I
(1.2.3) A1,A29A3px4,
such that

a) Equation (1.2.1) has a solution of period F if and
only if I = 1n, n = 0,1,2, --- and a solution of

period 2F if and only if 1 = 1 , n = 1,2,3, ---

I
n
b) The ln's and 15's; satisfy the following in—

equalities:

(1.2.4) 1 < l

I I I I
0 1.3 X2 < xl-S X2 < A3.S X4 < 13 S_A4 <
and the relations:
(1.2.5) lim (1n)-1 = 0: lim (IQ’1 = o:

11an naco
c) The solutions of (1.2.1) are stable in the intervals:
I I I I
(1'2'6) ()OIX1)I(X2IX1)I(Xz'x3)l(x4lx3)o

d) At the end-points of these intervals the solutions of

equation (1.2.1) are in general unstable.

e) The solutions of (1.2.1) are stable for A = X2n+1

_ _ I
or I — X2n+2 and they are stable for I - 12n+1 or

I

— I .-
— x2n+2 (x2n+1 ' x2n+2)

_ I o o
l - 12n+2 if and only If 12n+1

respectively.

f) If 1 is complex, (1.2.1) has always unstable

solutions.

For an exhaustive treatment of Hill's equation see

Magnus and Winkler [l2] and the references given therein.

1.3. Infinite Determinants.

We have already pointed out in Section 1.1 that the
computation of the characteristic exponents is a difficult
task. One of the techniques used by us to facilitate this
task (see Chapter 2) is infinite determinants and we shall

define this concept and give a very brief outline in this

section of some of their prOperties. We shall write a

L

determinant in the form, Ua k , 'where m,n vary over

m,nII
all the integers from k to L and m denotes the rows;

n denotes the columns of the determinant. We will consider
the case k = -¢>: I = m or k = 0,.L = m . These we shall

call two sided and one sided infinite determinants, respec—

tively. If

I l
' II a: ' I on
we shall say that
on +00
I
‘am,nH0 ' Ham,n”-w

exist or converge. We shall say that a determinant is of

Hill's type if it satisfies:

 

111;:1’1 ‘amon Om'nl < 0° '
where 6m,n = 0 for m #'n.: 6m,n = 1 for m = n and
the sum extends over all the values of m and n. It is

clear that any finite determinant is of Hill's type. It
can also be shown, see [12] that an infinite determinant

of Hill's type converges. For more information on

 

infinite determinants, see Magnus [11] and Magnus and
Winkler [12]. For a special class of infinite determinants
which are closely related to Chapter II, see Fleckenstein

[4] and Lemaitre and Godart [10].

1.4. Some Perturbation Techniques (Regular Perturbations,
the Poincare—Lindstedt Method).
Since we are mainly concerned in our present work

with systems of differential equations of the form

°° 2 S n (s) .

x. +w.x.+ 6 Z Z a. cos swt =0, J=1."'.n

3 3 5:1 k=1 3k xk

(see Chapter III, Section 1 for notation), we will describe
some techniques which will enable us to make statements
about the stability of their solutions for small values of
the parameter 6. These techniques enable us to make state-
ments about the stability of the solutions without resort-

ing to the difficult task of computing the characteristic

 

 

exponents directly, (see Chapter I, Section 1).

 

The basic technique which was applied to a second order

O.D.E. of the type
.0 2 0
(1.4.1) x+ wx= €f(x,x)

where w is a constant and f is an analytic function of
x and x, dates back to Poisson. He tries a solution of
the form

(1.4.2) x= 23 equ
q o

and recursively improves upon the zeroth order solution
x = x(0) (i.e., the solution of equation (1.4.1) for €==0).
The shortcomings of this technique become apparent if we

examine the equation:

10

(1.4.3) 32 + x = — €x(€ > o . 6 small).
The solution of this equation is found to be
(1.4.4) x = A cos V’l + E t + B sin.¢’l + E t

where, A and B are constants. Thus the solution of
equation (1.4.3) is bounded for all t > 0. If we utilize

the expression (1.4.2) and substitute into (1.4.3) we

(I)

obtain already for x terms of the form tsint, tcost

called secular terms and thus the full first order solution
(0) + EKm

 

which is x = x becomes unbounded for t >'%,

i.e. the expansion (1.4.2) is not uniformly valid for all

 

t. Moreover, Poisson's technique (regular perturbation

 

technique) does not enable us to eliminate terms of this

 

kind (i.e., secular terms). It becomes quite clear why an

 

expansion like (1.4.2) fails if we examine the function
sin(14-E)t with period 2N/14-E, and E a small positive
parameter. If we expand this function in a Taylor type
series we see that

2 2 €3t3
(1.4.5) sin(l+-E)t==sint + Etcost -

 

2 3!

It is hard to establish the periodicity of the right hand
side of equation (1.4.5) because of the secular terms (i.e.,
terms of the form tsint, tcost, which become large for
large t). To overcome this difficulty, i.e., in order to

Obtain uniformly valid expansion we shall outline briefly a

 

technique develOped by Poincaré and Lindstedt.

6 E sint-—-———— cost+-.--

11

We treat again an equation of the form
(1.4.6) 32 + wzx = €f(x,x),

where E > 0 is a small parameter, w constant and f
analytic in x and x . To account for the frequency

change (i.e., the periods dependence upon 6) we set

(1-4-7) t = 8(1 + 6001 +€2w2 + ---)

and substitute (1.4.7) into equation (1.4.6).
After some computations we Obtain

2

 

(1.4.8) (1+Eu) +---)’2 d X + wzx = €f[x, (l+€w+--o)-193‘. .
l dSZ 1 (33
We set
(1.4.9) x = Z: qu
q=O q
substitute (1.4.9) into equation (1.4.8) and equate co—
efficients of like powers of E . By doing so, we Obtain

equations which determine the xq's successively. The solu-
tions to the various D.E.‘s contain Secular terms, but the
presence of the yet unspecified mq's enables us to elimi—
nate the secular terms for certain choices of the wq's.
The power of the Poincare-Lindstedt technique lies in

the double expansion procedure i.e., first introducing a new

time scale, equation (1.4.7) and then expanding the solution

 

in the form (1.4.9). For more information on this technique

and its application we refer to Minorski [l6] and Stoker [24].

12

1.5. The Method of Averaging.

In this section we shall give a brief outline of a
technique developed by Krylov and Bogoliubov (in short the
K.B. method). For a detailed account on the theoretical
foundation of the K.B. method we refer to [2]. We consider

again the D.E.
" 2 .
(1.5.1) x + w x = €f1x,)d,

(see Section 1.4 for notation).
It is clear that if no perturbing force is present
i.e., E = 0 the vibrations are purely harmonic and thus

x = Acosw, I = wt + 9 with
(1.5.2) A = 0; I = w o

i.e., the amplitude is constant and the phase angle is

uniformly rotating. For 6 ¢ 0 we assume a solution in the

form

(1.5.3) x = A(t)cos[wt + Q(t)] s A(t)cosw(tL
Thus

(1.5.4) 5: = Acosw — (u) + emsin) . I 5 Mt).

We now impose the condition
(1.5.5) AcosI — Aesinw = 0.

Thus

13

(1.5.6) 3} = — AwsinI).

Note: This is equivalent to the velocity in the case

6 = 0; ‘we could have thus required instead of equation

 

(1.5.5), that equation (1.5.6) be valid.

 

Differentiating equation (1.5.6) w.r.t.t and sub—

stituting into equation (1.5.1) yields
(1.5.7) —Awsin¢ - Awécosw = €:f

where f E f(Acos§, -Awsin¢).
From equation (1.5.5) and (1.5.7) we obtain by solving

for A and 9

(a) is. = - f)- fsinI)
(1.5.8)
° _ i
an 6- Au)fcosIr.

The system of first order O.D.E's is nonlinear and coupled.
We observe that A and 8 vary slowly with time (since 6
is small). Assume now that we integrate equation (1.5.8a)

over a certain period T, *where I goes from 0 to Zn.

We obtain

t+T 2N
(1.5.9) [ Adt = A(t+ T) -A(t) = -5] £51m) 9.31.
t I1) 0 I

Since 8 is slowly varying we have

+ I): 27r/T + 0(6)

{-
II
8

thus we obtain

14

2V
A(t+T)—A(t) _ 3A;_ 6 .
T > dt — - 21M) JO feinwdw,

 

(1.5.10)

where 3% represents the change of amplitude averaged over

a period. Similarly we obtain for the averaged phase change

2N
d9 _ E
(1.5.11) EE--—2—TT_A-u-I.I;) fCOSW d1).

We note that equations (1.5.10) and (1.5.11) are decoupled
since we can now solve for A and 6 successively. For
more information on this technique and its applications we

refer to [2] and [28]. Bogoliubov and Mitropolsky'[2] and
Wang [28].

CHAPTER II
A PROBLEM FROM DYNAMICS

2.1. Derivation of the Equations of Motion.

We consider a weightless circular shaft rotating around
its axis with frequency 0. .At some point along the shaft a
disk of mass m is attached whose center of mass is not on
the axis of the shaft. As the shaft rotates this eccentric
mass causes the shaft to twist about its axis and the axis
to bend. The center of mass moves in the x1,x2 plane per-
pendicular to the axis of the shaft. The equations of

motion of the center of mass are

_ d
(2.1.1) ,-a_?

B
:3
X
+
"U
ll
0

where the independent variable is the dimensionless time
T = fit and (P , Px ) is the internal elastic force of

x
l 2
the shaft. From elasticity theory the displacement of the

center of the shaft in the xl,x2 plane is given by

(2.1.2)

15

16

where the "influence numbers" aij(i,j = 1,2) are functions
of the constant material properties of the shaft and of its

motion. Wehrli shows that in this case all = a22 = con-

stant and a = a21 is a constant multiple of the magni—

12
tude w of the torsion moment of the shaft (see Wehrli [29],
equation 3.10). Solving (2.1.2) for P and P ,
X1 X2
substituting into (2.1.1) and drOpping the non—homogeneous

terms which arise by replacing the coordinates of the center

of the shaft by those of the center of mass, we Obtain

“‘0 X1 + a‘11"1“ a12x2 '0
(2.1.3)
2" _
nul x2 + a21xl + a22x? 0
where
2

denotes a positive constant, mo denotes the critical

angular velocity of the shaft without the torsional moment

and

 

(2.1.5) 2 = -——-= a constant multiple of the torsion
moment W.

Wehrli [29] assumes W to be of the form:

(2.1.6) W = WO + W1 cosflt.

17

We generalize Wehrli's result by taking W to be of the

 

more general form

n
(2.1.7) W = Z: W.cosz.
i=0 3
W.
We assume -—£ = u. = constant for i = l,°'°n, u. # u. for
WO 1 1 j

i % j and make the additional assumption

€<<11

where EI denotes the flexural rigidity of the shaft and L

denotes its length. Thus using equation (2.1.7) we can write

 

 

a12 a21 n
(2.1.8) 2 = 2 = e Z akcosk'r
NM) m0 k=0
with
a. = azk u (l < j < n) u E l
j—l 1 j—l —- - ' 0

and k1 a numerical constant. Thus substituting (2.1.4)

and (2.1.8) into (2.1.3) yields the system of second order

ordinary differential equations

n
x1 + ale + 6[ Z akcosk'r] x2 = 0
k=0
(2.1.9)
2 n
x2 + a x2 + €[k§o akcosk'r] x1 = 0

18

2.2. Necessary Conditions for the Existence of Solutions

for the System (2.1.9).
The system of equations (2.1.9) is a system of 0.D.E.'s
with periodic coefficients. By Floquet theory (see Chapter

1, Section 1) we can write the solutions in the form:

I +m O
101' IVT .

where o is the characteristic exponent and the c 's are

jv
constants to be determined. By substituting (2.2.1) into
the system of equations (2.1.9) we obtain a doubly infinite

system of homogeneous linear equations

r 2 2
[a - (o + v) ]c1V + anc2V +
E n
+ 2' Z‘ as[C2V—s + C2v+s] = 0
3:1
2 2
2.2.2 -
( I I [a (o + v) ]C2v + anc1v +
E n
+ 2' Z? as[ClV-s + lv+s] = 0
8:1
I V = 0, :_l, i.2v .

 

The system of homogeneous equations (2.2.2) has a nontrivial

solution for the 's if and only if the determinant of

cjv
the system is equal to zero. Let us denote the infinite de-
terminant obtained after dividing each row by a2 - (o + v)2
by A(O), (see Appendix A). We then can prove that A(o)

has the following prOperties:

19

1) 8(0) converges uniformly and absolutely except in

the neighborhood of points of the form 0 = - v + a .

ii) A(O) is a periodic function of o with period 1.
iii) A(O) is an even function of O.

nggf: That A(o) converges uniformly and absolutely
except in the neighborhood of points of the form 0 =-v+ a
is a direct outcome of the uniform and absolute convergence
of the series

+m

2: l

v=—co az-(O+v)

 

2

which converges uniformly and absolutely except in the neigh-
borhood of points of the form 0 =-Vj;a. Thus by enclosing
these singular points in small enough disks and excluding
those disks from the domain of convergence it follows that
the above series is uniformly and absolutely convergent, see
[31]. From the convergence of this series it follows that
A(o) is a determinant of the Hill's type and thus converges,
(see Chapter I, Section 3 and [12]). The periodicity of

A(O) follows from the remark that A(o) remains unchanged
if we replace 0 by 04-1 and at the same time replace v
by v-—1 (since v runs from -w to m, the same is also true
for v-l). Thus A(o4-1) = A(o). Similarly replacing o

by -<I is equivalent to replacing v by -‘v. Hence

A(—o) = A(o). It thus follows that A(o) is an even

20

periodic and meromorphic function of o with poles of

order two at most at the points a = - v i a (a 7! -:—, 3 any
positive integer). If a ='% (5 any positive integer) then
A(o) possesses poles of order 4 at most and this case

will be treated separately.

Assume now a #-% (5 any positive integer) and

consider the meromorphic function:
(2.2.3) F(o) = Kl[cotn(o-a) — cotn(o+a)] +
2 2
+ K2[csc N(o—a) + csc n(o+a)] .

We observe that F(O) possesses the same poles of order two
K

as A(O). The residues of F(o) at a pole are :_7%- and
K
._%. If we choose the constants K1 and K2 in such a way
n
that for the principal parts c_1 and c_2 in the expansion
of A(o), we have

C +K_1 c -52

—l =-— U’ ' -2 _ 2 '

F

then the function G(o) defined as
(2.2.4) 0(0) = 8(0) — F(o)

represents an entire transcendental function.
Since A(O+JJ = A(O) it is sufficient to investigate
the behavior of 6(0) in the strip 0.3 Rec5<].. From the

uniform and absolute convergence of A(o) and the fact that

21

 

 

 

lim 1 2 = 0
Im04¢>a - (o+v)

where II denotes absolute value, it follows that
(2.2.5) lim A(o) = l.

Im04m
We also have
(2.2.6) lim FIG) = 0 -

Imo-m

From equations (2.2.5) and (2.2.6) it follows that G(o)
(defined by equation (2.2.4)) is bounded in the entire com-
plex o-plane and thus by Liouville's theorem reduces to a

constant, i.e., 0(0) I l . Thus by using formulas (2.2.3),

(2.2.4) and the fact that G(o) E 1 yields after some

 

 

manipulations:
2Klsin2na 4K2(l-c082nac082vo)
(2.2.7) A(o) = 1 + +
p 2
P
where p = cosZna — cosZNo.

We observed that in order to obtain a nontrivial solu-
tion to the system of linear equations (2.2.2), A(O) has

to vanish. Thus if K1 and K2 are known, we obtain the

following relationship by setting A(o) = 0 in (2.2.7)

(2.2.8) cosZNo = KlsinZNa + (2K2+l)c052na :

l

221ra]2

cos2ra)2-4K sin

:_[(Kls1n21ra+2K2 2

22

We shall now exhibit some of the terms of the expansion

of A(O) in powers of 6 which will enable us to evaluate

 

 

K1 and K2.
2 2
2 +°° 30 1 n a"
(2-2-9) A(O) = l - E Z‘ 7 + '5' Z x+ +
V="°° Av j=1 v \H—j
4 +m n (4ag-a§)2
+ e 23 _Zf 2 2 +
v=-<P j—l 16AvAv+j
a4 m 6
+ —%- Z: 2 + + 0(6 )
'= +1, 21 A .
Av j n n v+3
where AV = a2 - (o+-v)2. A detailed computation of A(o)

up to and including terms of 0(64) will be given in

Appendix B.

Remarks:

i) From formula (2.2.8) it follows that in order to

evaluate K1 we have to consider only the coefficients of

62 in the expansion of A(o), but for K2 we have to

consider also the coefficients of 64 (see also Appendix

C).

ii) In order to evaluate K2 we do not need all the

4
coefficients of 6 (see Appendix C).

We thus find:

23

 

 

 

 

2 2
Fa n a.
fKfi—QGAIE 373—2: 62"0‘64’
4a j=l 4a - j
2.
Nzaz 2 n (4a2-a?) (4a2+-j2)
_ 0 2 F 0 1
K2" 2€+—2.Z .2 2.22
(2.2.10) ( 4a a J=1 32: (4a -3 )
w 2 -2
.. 2:
j=n+l,n21 2] (4a -j )
K
+O(€6)

2.3. Stability of Solutions.

From Floquet theory (Chapter 1, Section 1) it follows
that the solutions (2.2.1), of the system of ordinary
differential equations (2.1.9) will be bounded if we require
the 0's to be Eggl_and distinct. Using equations (2.2.8)
and (2.2.10) we observe that the 0's are continuous
functions of 0 and 6. Thus the (0,6) plane can be
divided into regions of stability and instability whose
boundaries (boundary curves) have the proPerty, that along
them there are double 0's and in some cases quadruple

'8 (since 0 can appear either as an integral multiple of

th o

or 0 can appear having opposite signs and counted twice
each time). In the first case we speak about instability
regions of the first kind and denote them by I1 and in the
second case we call them instability regions of the second

kind and denote them by I An investigation of the sta—

2.

bility of the solutions is meaningful only for small E.

 

23

 

 

 

 

2 2
"'3 n a.
I—62+- 2 73—: e2+o<e4>
4a j=l 4a - j
2.
Irzaz 2 n (4a2—a?) (4a2+j2)
_ O 2 W z: O 1
K2 " ' 2 E I —2' . .2 2 .2 2
(2.2.10) ( 4a a 3=1 323 (4a -3 )
a. 2 .2
4 (4a 4—3 ) ... 4
+ a , Z .2 2 .2 2 + e +
j=n+l,n21 2] (4a ‘-3 )
K
+ 0(66)

2.3. Stability of Solutions.

From Floquet theory (Chapter I, Section 1) it follows
that the solutions (2.2.1), of the system of ordinary
differential equations (2.1.9) will be bounded if we require

the 0's to be real and distinct. Using equations (2.2.8)

 

and (2.2.10) we observe that the 0's are continuous
functions of Q and 6. Thus the (0,6) Plane can be
divided into regions of stability and instability whose
boundaries (boundary curves) have the pr0perty, that along
them there are double 0's and in some cases quadruple

0's (since 0 can appear either as an integral multiple of

%- or 0 can appear having opposite signs and counted twice
each time). In the first case we speak about instability

regions of the first kind and denote them by I1 and in the
second case we call them instability regions of the second

kind and denote them by I An investigation of the sta-

2.

bility of the solutions is meaningful only for small 6.

Wed

EXCE

* .
Inst

 

tang

 

 

and

Rest

 

the C<

Obtai:

 

 

 

I I'

 

24

We Observe that the neighborhood of the 0 axis is stable
except at points where:

3’2.
0

(2.3.1) =—:-. 6:0 (s=1,2,---).

In order to show that equation (2.3.1) gives rise to
instability regions and in order to find the slopes of the
tangents to the boundary curves corresponding to I1 and

12, we set

2
(2.3.2) Q—‘gw

2 _
O + 016 + 0(6 ), s — 1,2,

and substitute (2.3.2) and (2.2.10) into equation (2.2.8).

Restricting ourselves to terms of order 62 we obtain:

 

 

 

s+l 2 2
((a) cosZno = (-1)S+-L-1) 2” s [4szflij;(8klw s)01+-
32w 0
O
2 2 2 2 3
+ (4—ps)k1wo]€ + 0(6)
(2.3.3) <
s = 1, , n
8+1 2 2
- 3
(b) cosZTrO = (-l)8 + ( 1) gs [sflliklwo]2€2+0(€ )
8w
k 0
s = n + l, .

For boundary curves of the first kind (11) we require
the content of the square brackets in (2.3.3) to vanish and

Obtain:

 

 

Not

 

lie)

the:

The
of
pos

3184

the

CUM

uti}

has

 

Varj

 

25

 

 

k w
r _ l O
(a) a]. "' i 28 (2 i “5)
s = l, , n
(2.3.4) ( k
w
(b) 01 = i. 150
x s = n + 1,

 

Note that in equation (2.3.4b) 01 has to be counted twice.
From equation (2.3.3a) we can conclude that if 01
lies between - é%'(2 :_us)k1wo and €%'(2 i u8)k1wo,
then purely imaginary o's will occur which indicates in-
stability (for if 0 is a characteristic exponent so is -o).
The 01 values in equation (2.3.4a) give us the direction
of the boundary curves of two instability regions whose
position and width depend upon kl and us and which can
also coincide.
From equation (2.3.4b) we conclude that for s > n
the instability regions are very narrow since the boundary
curves touch each other.

For instability regions of the second kind (I ) we

2
utilize again equation (2.3.3a,b) and conclude that fll==0
has to be counted twice and is independent of s.

The following two diagrams give us an idea of the

various situations (see explanations below diagrams).

26

 

 

 

 

0
q
\:\\\;T
I \\\*
==O
ZQO O1
/ / I/
i \
I / ’
mo ' , //
I .
1
2
2w0 l
3 i
) shaded areas indicate instability
ZwO
s ) thick lines: instability

 

 

Figure 1.

Note: If us 74> O for s = l,---,n, we will have for

Zwb

8

 

 

each (s = 1,---,11) instability regions of the type

indicated by the shaded areas in Figure 1.

27

2w

 

s==n+-l.--- . \\\\\\\‘

 

 

Figure 2.
2w0
If s > n, ‘we will have for each —§—- instability

regions of the kind indicated in Figure 2, all of them very

narrow.

 

28

2.4. Influence of a Damping Factor.

If we assume only the existence of external damping

and its pr0pertiona1ity to the velocity, we have to add

to the system of 0.D.E.'s (2.1.3) terms of the form:

Zmnyl, Znyflxzo

= d/dT
(y-damping constant) respectively.
Thus using equations (2.1.4), (2.1.5), (2.1.7) and
setting
-35.
(2.4.1) Xj = yje , j = 1,2

 

r 13 [ £23 .1
Y + (a - )y + 6 a COSkT y = O
1 Q2 1 .k=0 k J
(2.4.2) (
§ + (a2 - 2)y + €[: 23 a cosk¢]y = 0

Using equations (2.2.1) and (2.4.1) we conclude that

the solutions of the system of 0.D.E.'s (2.4.2) are stable

if the characteristic exponents satisfy the relation
.1
IImOI < 0.

2.5. The Case a 2 (3 any positive integer).

If a = g- (5 any positive integer) the determinant
A(o) (see Appendix A) represents a meromorphic function
having poles of order 4 at

o = v + %'(V = O .:_l, + 2,--').

29

Define
(2.5.1) H(o) = A(o) + Qlcscn(o + %) -
Q Q 2
- —2~g% cotn(0 + g) - —¥%-ii§-cotn(o +-%)
F 2F do
0 3
— -—4-3--g-—3-cotwr(o + :3).
6V do
We are now able to choose Ql.°--,Q4 in such a way that
H(o) remains finite at the poles of A(o). Since
A(o + 1) = A(O), (see ChapterIEL Section 2) we have to in-

vestigate the behavior of H(G) only in the strip
0 S_Re0 < 1 for Imo ——> w . We thus find

(2.5.2) lim H(O) = 1
O‘Re0<l

Imo ——> w

and thus by Liouville's theorem H(O) I1. We also have
A(-o) = A(O) (Chapter II,Section 2), thus we conclude
01 = Q3 = O .

We recall that the doubly infinite system of homo—
geneous linear equations (2.2.2) has a nontrivial solution
if and only if A(O) = O . Thus by setting A(U) = O we

Obtain

 

2 r
1 L

_ .1. 2
”6 (302-204): /(302-2o4) 4604].

(2.5.3) [sin(n+~%)

In order to compute 02 and Q4 we utilize again the ex-

pansion of A(O) given by equation (2.2.9) and choose 02

 

_r , L7...)

 

30

and Q4 in such a way that for the principal parts of A(o)

at a pole we have

 

 

 

___3 __4_
(2.5.4) c_2 — 2 , c__4 4.
n F
We thus obtain
( v2 2 2 2 4
Q = - (4a - a )e + o(€ )
2 2 O s
25
4 2
F 2 2 4
(2.5.5) 0 =-—- (4a -a) E +
4 16s4 0 S
k for s = l, , n
and
F 2V2 2 2 4
02 =‘- 2 a06 +—O(E )
s
/ a4
_ 4 _Q_ 4
(2.5.6) ) Q4 ~ W 4 6 +
s
for s = n + 1,
\

 

If we substitute (2.5.5) or (2.5.6) into (2.5.3) and use the

trigonometric identity

2

(2.5.7) [sinn(0 + §)] = %{1 + (-1)S+1coszno],

*we obtain 01 = O which is in agreement with equations
(2.3.3a,b), i.e., verifies our statement about quadruple

:points in the case 01 = O.

CHAPTER III

GENERALIZATION OF STRUBLE'S METHOD FOR SYSTEMS
OF SECOND ORDER 0.D.E.'s

3.1. Theory of the First Order Approximation.
In a paper by Struble and Fletcher [27] a technique is

develOped in order to find a perturbational solution of the

Mathieu equation
(M) x + nzx = (Ecost)x, ' = d/dt

where n2 and E are constants.
In this chapter we will generalize Struble's technique

and apply it to systems of second order ordinary differential

equations of the form

n
00 (S) _
(3.1.1) X). + wjx. + e _ 23 (ajk cos swt) xk — o

s l k=l

M
MU)

where

wj is the eigenfrequency of the unperturbed system

6 > O is a small parameter

w is the excitation frequency of the perturbed system
agi) are constants for all j,k and s

S and n are positive integers.

31

32

 

Let
N ( )
(3.1.2) x. = C.sinm.t + D.cosw.t + Z? qu.q
3 3 3 J J q=l
j = 1, . n
where
(i) (i)
C. = C. t,E , D. = D. t,€ , x. = x. t
J J( ) 3 3( ) J J ( )
i = 1,"°, N and N is a positive integer.
Substituting (3.1.2) into (3.1.1) yields
r u u .
C.sinw.t+D.cosw.t+ 2w.(C.cosw.t -
J 3 J J 3 3 3
° N "( ) 2 < )
— D.sinw.t) + Z Eq(x.q + w.x.q ) +
J J q=1 J J 3
(3.1.3) {
S n (s)
+ E E‘ Z: ajk cos swt[Ck51nmkt+chosu1kt +
s—l k—l
N
+ Z‘ quéq)} = O.
k q=l

Restricting ourselves to terms of zero order and first order

in E we obtain by using (3.1.3)
.é.sinw.t +D.cosw.t+ 2w. (é.cosw.t -
J J 3 3 J 3 J

~(1) 2X(1)

(3.1.4) - Djsinwjt)+ e('j + w] ) +
S n (s)
+ E 3:21 13:31 ajk cos swt[Ck51nmkt+choswkt} = 0.

Examination of (3.1.4) suggests the following distribution

of terms:

33

 

 

f .
C — 2w.D = O
J 3
(3.1.5) ( .
D + 2w.C. = O
3 J
K
and
(’ e(§(1) + w?x)1)) =
J J
(3 l 6) <--6 ‘82: g: a(s)cosswt[C sin t+
" ‘ _ _ jk k “’k
s—l k—l
t + choswkt}.

Remarks:

i) Equations like (3.1.5) will be called variational

 

equations while an equation like (3.1.6) will be called a

perturbational equation.

 

ii) At any step of the process the variational equations
are associated with sinwjt and coswjt and the perturba-

tional equation with the remaining (nonresonant) terms.

It is quite clear that the system (3.1.5) has the

solution C. E C . and D. E D . where C . and D . are
3 03 J 03 OJ 0]

constants for j = 1,°--, n.

If swli.mk is appreciably different from i_wj for

all j,k and s (1.3 j, k.g n: 1.3 5‘3 S), then equation

(3.1.6) has the solution:

 

34

 

C sin(su)+ )t+D cos(sm+ )t
(X(1)=_% g Za(s){ k 2‘“). k wk
3 5:1 k: l mj - (sw+wk)2

chos (sw - wk)t — Cksin(sw - wk) t }

2

(3.1.7) < + 2
wj - (sw—mk)

 

j = l,...,n.

 

Hence, if sw‘: wk is appreciably different from :_wj

for all j,k and s the solution of equation (3.1.1),

correct up to and including terms of first order in E is

 

given by
x. = C. sinw. t + D. cosw. t + €x(l)
J j j j j 3
where C. and D3. are constants for j = 1,'°', n and
x(l) is given by (3.1.7).

3'
Case(a): Single Resonance. Assume now that sw4-mk==wj+-n
(n—small real number) for a certain set of values of
j,k (j # k) and s. For simplicity we denote these values
by’ L,,m and s1 . In this case the solution (3.1.7) will
not be valid since we have resonance or terms with small
divisors. We remove these resonance creating terms from

(3.1.6) and write:

35

C251nwzt + chosmzt +

(3.1.8)< + 2w£(C£coswzt — DL51nw2t) =

e (51) .
_ - E-azm [Cm51n Unirn)t + Dmcos(wz+-n)t]

 

In a similar fashion by interchanging the indices 2
and m and removing the resulting "offending terms" from

(3.1.6), we obtain

(3.1.9) C sinu)tn+D cosw t+-2w (C cosw t-—D sinw t) =
m m m m m m m m m
e (51) .
_ - 2 amt [-CLSID(-Wmi'n)t4'D£COS (—Qm+-n)t].

Equating the coefficients of sinwlt, coswlt on the
left hand side (L.H.S) and right hand side (R.H.S.) of
(3.1.8) and similarly equating the coefficients of sinmmt,
cosgmt on the L.H.S. and R.H.S. of (3.1.9), we obtain,

after some computations, the following system of second

order ordinary differential equations:

 

r .. . E (81) .
(a) Cl. - 2WD; = — -2— atm [Cmcosnt -Dmsinnt]
" . E (31)
J =_- _ _ '
(b) D‘+-2wLC‘ 2 azm [Cm51nnt+-Dmcosnt]
(3.1.1o)< (S )
oo 0 — g— l t D . t]
(c) Cm--2mem — - 2 am! [Czcosn +- £51nn
-- - e (51) .
(d) Dm + 2thm==- E'amz [-CL51nnt4-D£cosnt].

K

36

Since we are concerned here with solutions of the system of
equations(3.l.lo) which are correct to first order in 6,
we assume 1') = 0(6) and drop the terms .CL , DL '22!“ and
Em in equation (3.1.10). From the resulting equations

obtained by merely dropping these terms, we can conclude

that the omitted terms are of second order in 6.

 

In order to solve the reduced system of equations
(which we call the reduced variational system) obtained by

,D

dropp1ng CL £"Cm and Dm' ‘we set
C1 + IDL = Yl
CL ' 1D2 = Y2
(3.1.11)
C + iD = Y
m m 3
Cm - le = Y4 i =\/ —1 ,

then the reduced variational system transforms into

 

 

 

 

o ' (S ) '
f _ 16 1 int
(a) Y1 ‘ 4w! aim Y3e
. (s ) .
_ 16 1 -1nt
(b) Y2 ‘ ‘ 4m! azni'y4e
(3.1.12) <
(c) T -’ 16 a(sl)Y e—int
3 — 4w m1 1
m
. (s )
__—16 1 1nt
. (d) Y4 ' 4mm amt Yze

 

Looking closely upon the system (3.1.12) we see that we have

t1) solve either equations (3.1.12a) and (3.1.12c), 93

37

equations (3.1.l2b) and (3.1.12d). Choosing equations

(3.1.12a,c), we solve them by setting

1 = Aept+filnt ' y3 = Bept-é1nt

where A and B are constants. Substituting, we obtain

a system of linear equations

 

. (s )
1 . 16 l _
(p + §fl1)A — 4mg a)"m B — O
(L) if“... ins—o
- 4uh mL P-211 _

Requiring A and B not to vanish simultaneously implies

that the determinant of the system (L) has to be equal to

zero, which yields

 

2 1 2
(3.1.13) p +311 + 1:1:me :0.
L m

Definition: An equation like (3.1.13) is called an indicial

 

 

 

eguation.
NOte: The transition curves, i.e.,the boundary curves be-

tween the regions of stability and instability are obtained

iby'setting p = O in (3.1.13).

Without loss of generality we assume now that wt and

. . . (S1) (51) .
(fin are both p031t1ve. Hence, 1f azm amt > O i.e.,
(s1) (s1) .
galm and amt have the same s1gn, then the system

(3.1.12) will be stable i.e., all its solutions will be

bounded .

38

Remark:

We observe that the relations

sw :_(w - w )

SL1)

+ (wt + mm)

give us all the mathematically possible critical frequencies.

The only physically admissible ones are those for which

i (ml — wk) 2.0 or wt + mm > 0 (Since we have chosen wt and
mm to be both positive).

The preceding analysis of the single resonance (Case a)
was restricted to the case j #‘k. If we assume j = k,

we obtain sw = n (n—small real number) for some 5 and

the variational equations reduce to

(s

 

 

 

. . ) .
f _ 16 1 int
Y1 — 4w; all Yle
(3.1.14) < ( )
. s
= _ 16 1 1nt
L Y2 4% a“. Yze

and it is clear that the solutions of (3.1.14) are bounded,

i.e., the system is stable.

 

Case (b): Assume now that sw - wk = wj + n (j # k,
n-small real number) for certain values of j,k and 5,
say L..m and 51 respectively. Following the same proce—
dure as outlined in Case (a), we Obtain the following

results:

39

i) The indicial equation is

62a(sl)a(sl)
2 _ 1_ 2 +_ mu m2
p _ - 471 16m w
L m

 

ii) Assuming that both wt and mm are positive, we

can conclude from i) that if

(81) (31) . ($1) (51) .
azm amt < O, i.e., atm and amt are of opp051te

signs, the reduced system of variational equations is stable.

Remark:
In each of the Cases (a) and (b) treated above, we

dealt for simplicity's sake only, with the reduced systems of

variational equations.,i.e.,we omitted the terms C£,1D£,

Em and Bm" since they were all of 0(62).

For the sake of completeness let us now treat the Egg-
reduced system (3.1.10). Adding equations (3.1.10a),
(3.1.lOb), (3.1.10c), and (3.1.10d) respectively, using
(3.1.11), setting

Y1 = Aept+§1nt . Y3 = Bept-fiint

and finally requiring that A and B do not vanish simul—

taneously yields the indicial equation
2 . 1 2
(3.1.15) [p + 1p(2wL + n) - win — 1T1] x
x [p2 + ip(2w - n)'+ w n —-£ 2] =
m m 47]

—-—-

62 a(81)a(81)
4 £m mL '

4O

Letting p = O and neglecting terms of order n3 and n4

in (3.1.15) yields

2 ($1) ($1)

2 6 32m amz

(3.1.16) n = — 4w w
L m

which is the same result as the one obtained by setting
p = O in equation (3.1.13). Thus by letting p = O in
(3.1.15) and neglecting the terms of higher order than
0(n2), we obtain the transition curves between the regions
of stability and instability.

A more complicated problem is to investigate equation
(3.1.15) and find out the conditions for which Reps O .
We shall not investigate this question in detail, but will
only note that the question posed above is equivalent to

the question of investigating the behavior of the polynomial

Q(z) = [(z+i)2 + E3i][(z-i)2 + 6%]
z = x + yi, i =./ -1

‘with 61 — 51 = 2 . We also refer the reader to [13],
‘pages 179-186 for further information on the zeros of poly-

:nomials with complex coefficients.

(:ase (c): Multiple resonance. It is possible that sw

 

\flill be nearly equal or equal to i-(wj"mk) for more than
cnne set of values of j,k and s . This case will be re—
ferred to as multiple resonance. There are many ways for

nuiltiple resonance to occur: for example all of the wj's

41

may not be distinct. Since there are so many possibilities

we will consider only one particular case which will help

us exhibit the general method of analysis. Assume that

slw + wr 2 mp + n1 p # r
(3.1.17)

32w + wp = wq + n2 q # P

where n1 and n2 are small real numbers. We omit again

terms of order 62 (i.e., Cj and Dj j = p,q,r) and

thus can write (using 3.1.4):

(3.1.18) 2w (C cosw t — D sinw t) =
P P P P P
-E Lapq [-Cqsin(szw-wq)t4-chos(s2w-wq)t]+-
(81)

. 1
+ apr [Cr51n(slw+-wr)t+-Drcos(slw+-wr)t]J.

(3.1.19) 2wq(choswqt - Dqsiant) =

(S)
E 2 .
= —-— a C + t4-D co + ‘t .
2 qp [ p31n(52w mp) p 3(52w mp) ]

(3.1.20) 2wr(Crcoswrt - Dr51nwrt) =

(81)

__ea
_er

-C sin w- t+-D cos m-w t .
[ p (81 mp) p (81 p) ]

IJsing (3.1.17), equating the coefficients of sinwjt,
cosuBt (j = p,q,r) on the right and left hand side of
equations (3 . l . 18) thru (3 . 1 . 20) respectively and finally

setting:

42

 

C + iD = Y
( + 'D = Y
\ Cq _ l q 34
L.Cr :IDr = Y5

we obtain

 

 

 

, . (s ) -in t (s ) in t
Y=16(a 2Ye 2+a 1Ye1\
l 4gp ..pq 3 pr 5
. . (s ) in t
__ 16 2 2
(3.1.21) Y3 — 4w aqp Yle
q
. - (s ) —in t
Y5=416 a lYle 1
k wr rp

 

and a completely similar set of equations for the

Yk s (k = 2,4,6).

(3.1.21) are obtained by inspection.

I = pt
Yl Ae
pt+in2t
(3.1.22) ( Y3 = Be
pt—inlt
KY5 = C8

 

Stflostituting (3.1.22) into (3.1.21) and requiring A,B

The solutions of the system of equations

We set

and

c: not to vanish simultaneously gives us (as before) the

ind icial equation .

 

The indicial equation in this case turns

CNJt to be a third degree polynomial with complex coefficients

and thus determining the condition for Rep: 0

is a fairly

43

difficult task. We refer again to Marden [13], pages 179—

186 for further information.

3.2. Theory of the Second Order Approximation.

In this section we will investigate the question
whether there are any other stability (instability) regions
besides those discussed in section 1 of this chapter. In
order to answer this question it is necessary to retain
higher order terms in 6 in the analysis (terms of at least
second order in 6 ). This is also necessary in order to
refine the details for the first critical (resonance) region.
We shall also restrict the analysis to the determination of
second order critical (resonance) regions.

The procedure to obtain an improved approxflmation for
the first critical region will only be indicated for the
general case and carried out in detail for the Mathieu equa-
tion (see section 3.3). Before proceeding we require that
su): “k be appreciably different from i.wj for any j,k
and s . In this case by restricting ourselves to second

order terms in 6 in (3.1.3) we can write:
(.é.sinw.t+.D.cosw.t+2w. C.cosw.t-D.sin .t +
J 3 3 J 3( J 3 3 w) )

+ 62(§§2) 4-w7xiz)) =

) 3 3
(3.2.1) (
s n
= - 62 Z Z a)? cos swt xlil)
s=1 k=1 3

 

44

(1)

Here we used the fact that xj satisfies equation

(3.1.6). Substituting for xél) from (3.1.7) yields the

perturbational equation:

 

 

 

2 " (2) 2 (2)
6 . + w.x. =
(x3 3 J )
2 S
:5. 2: 9:. 2 3‘: a(s)a(P)[A1+A2+A3+A4+
4 3:1 k=l p=1 m=l jk km wi-(pw-w )2
(3.2.2) ( m
Bl+BZ+B3+B4]
+ 2 2
um-(mmsg
\ j=lo°'°ono
where:
A1 = Dmcos[(s+p)w-wm)]t: A2 = Dmcos[(s-p)w+-gm]t
A3 = - Cmsin[(s+p)w-mm]t: A4 = Cmsin[(s—p)w+-mm]t
B = -

1 Cm51n[(s+p)w+-mm]t; B =

2 Cm51n[(s-p)w-wh]t

B3 = Dmcos[(s+p)w+-wm]t; B4 = Dmcos[(s-p)w-mm]t.

By examining (3.2.2) we observe that the possible critical

frequencies are:

(Cl) pm = :_(mm + wk)

(c2) (p+8)w = 3; (mm : wj)
(c3) (s—p)w = 1 (mm 1 wj)

45

We emphasize again that the physically admissible cases

 

of critical frequencies are those for which m is positive.

We also note that pm = :_(mm‘:_mk) leads to first order

 

single resonance which we excluded from our considerations.

 

If (s+p)m is appreciably different from :(mm :_mj)

equation (3.2.2) has the solution:

8 n S n A + A
r x$2)=%- z 2: 23 z a¥:>ag){—__1AB 3+
3 5:1 k=l p=l m=l 3
(3°2'3) < A2 + A4 B1 + B3 B2+ B4
+_____+__+___}
L AC DE DF

 

where the Aj's and Bj's (j = 1,°°',4) have the same

meaning as those in equation (3.2.2) and

A = mi - (pm - mm)2
B = mi — [(s+p)m - mm]2
C = m; — [(s-p)m + mm]2
D = mi - (pm + mm)2
B = mg - [(s+p)m + mm]2
P = m; - [(S—p)m - mm]2

Case (a): Second Order Single Resonance. We assume

(p+s)m - uh’= mj + n (n—small real number) for a certain
set of values of p + s = u,j and m.. We also assume
mj fi'uh‘ for j #’m . Before analyzing this case we make

the following remarks:

46

i) We observe that p4—s = u can be expressed as a
sum of two integers p and s, in different ways, i.e.,
(p+s)a ? pa + Sa' where a denotes the number of
different ways u (which is given) can be partitioned as

a sum of two positive integers.

ii) Since (5+p)a can be equal to (51"pl)6 for some
6(B's), we also have to take into account those terms for
which (s+p)a = (sl--pl)B . Here we assume (sl-p1) > 0.
otherwise choose pl-sl.

Back to the analysis of second order single resonance,
i.e., (p+s)m-—mm = mj4-n (for certain values of p4—s==u,
j and m) . We use here the technique developed by us in
Section 1 of this chapter, i.e., at any step of the process
we associate the variational equations with the coefficients
of sinmrt and cosmrt (r = j,m) respectively and the
perturbational equation with the remaining nonresonant terms.
We note that in the case of the second order (single) re-
sonance, we have to solve the 3211 set of variational
equations, i.e., we are in general not allowed to omit the
terms Ej" I3 , Em and Dm since they are all terms of
C(62) (see the observations made following equation (3.1.10).
Since the arguments in this case are a word by word repeti-
.Eigp_of those in Section 3.1, we do not exhibit the details
'but write down only the variational equations and solve
them. Setting Cj iiDj = Y and Cm :JU) = Y3 we

12 m 4
Obta in:

47

, .. . 62 int
(a) Y1 + 21ij1 = - 7r-Q Y4e
" . 62 —int
(b) Y2 - 21ij2 = - TT'Q Y3e
(3.2.4) ( .. . 62 int
(c) Y3 + 21mmY3 = - ——-w Y2e
.o o 62 -int
p(d) Y4 - 21me4 = - T W Yle ,

 

where

n
_ (S) (p) 2 2
Q — Xi kg: ajk a1 /mk - (pm-mm)

n
(S) (p) 2 2
2:11:51 amk a'kj /mk- (pm-m3.)

and 2i means that the sum extends over (s+p)a and
(sl-pl)B (see Remarks 1) and ii) preceding this discussion).
Observe that m and j are fixed, It is quite clear that
we need solve either equations (3.2.4a,d) or equations

(3.2.4b,c). We choose (3.2.4a,d) and write

Y = Kept-Filnt

Y4 = Bept—tnlt.

Substituting into (3.2.4a,d) and requiring A and B not
to vanish simultaneously gives rise to the indicial
equation.

The transition (boundary) curves between the regions

of stability and instability are obtained by setting p==0

in the indicial equation.

48

A more complicated question is to determine the condi-
tions for the stability of the solutions Yj(j = 1,---, 4)
of the system (3.2.4). In order that the Yj's be bounded
with time, we have to require Repng . Since in our case
the indicial equation is a fourth degree equation with
complex coefficients, determining the conditions for Repfgo
is not an easy task. We refer again to Marden [13], pages
179-186. The preceding discussion also applies to the
cases (s+p)m + mm = i'mj + n and to the cases

(s-p)m:mm = :_mj + n and will not be carried out.

Case (b): Second Order Multiple Resonance. For the second

 

order analysis there is an even larger number of ways in

which multiple resonance may occur than for the first order

 

analysis (see the paper by Hsu [8] for a more detailed dis-
cussion of multiple resonance). These cases are analyzed in
a similar manner as the one in section 3.1, (equations
3.1.17 - 3.1.22).

We already pointed out at the beginning of this section
that in order to refine the details for the first critical
(resonance) region, we have to retain higher order terms in
6 in the analysis. We indicate here only the essentials
of this procedure and carry it out in the next section for
the Mathieu equation. To obtain an improved approximation
to a first order instability region which corresponds to a
critical frequency of the type su):_uk_= :_mj + n (n-small)

it is necessary to modify the preceding analysis. The terms

49

in equation (3.1.6) and hence in eguation (3.2.1) which

 

lead to the first order variational system(s) must be re-
moved. These terms are then added to the first order
resonance terms appearing in the modified equation (3.2.1)

to obtain an improved (second order) variational system.

3.3. The Mathieu Eqpation.

 

In this section we shall apply the perturbation tech-
nique, which we developed in Section 1 of this chapter and
which we shall call the Generalized Struble Method (in short

GSM), to the Mathieu equation
o. 2
(3.3.1) x + n x = (6cost)x

where 0 < 6 <<Z]. and n is a constant.

Equation (3.3.1) is an equation of the form (3.1.1)
. _ - ,-_ , _ . _ (S)_(S)_
with mj — mk — n, j — k, m — 1, s — l and ajk --akj -1.

Applying GSM to (3.3.1) we set

N q (q)
x = Csinnt + Dcosnt + Z: 6 x
q=1

substitute into equation (3.3.1) and restrict ourselves at
first to terms of 0(6). We observe that in this case, the

first order perturbational equation will have an unbounded

solution for n near %- or n near --%. Assume now
that n is near -%, we write l-2n==n (n-small real

number) and utilize the results obtained in Section 3.1 for
the case of single resonance (Case (b)). The reduced first

order variational system turns out to be

50

. 6 .
C — ZH-(- Cs1nnt + Dcosnt)

(3.3.2) . E
D = m (CCOSTH: + Dsinnt)

and the first order perturbational equation becomes

(3.3.3) 6(x(1)4-n2x(1)) = §[Csin(n+l)t + Dcos(n+1)t].
In order to solve equation (3.3.2) we set C + iD = Y1,
C - iD = Y2 and obtain
{(1 = 27% eint Y2

(3.3.4) {(2 = — l—E— e’int Y1

n = l-—2n .
To solve equation (3.3.4) we set

Y1 = Aept+§int' 'Y2 = Bept-fiint.

Substituting into (3.3.4) and requiring that A and B do

not vanish simultaneously yields the indicial equation

 

Letting p = O and remembering that n = l — 2n yields
(3.3.5) |1 — 2n |= 6/2n.

From (3.3.5) we obtain

51

or in a more standard form

2 _.1 .g 2
n — 4 i.2 + 0(6 ).

Thus the transition curves for the first critical region

(n near %) are given in the case of the Mathieu equation by

2 _ 1_ g. 2
n — 4 i.2 + 0(6 ).

In order to refine the details for the first critical

region, i.e., refine the details for n near ‘%, we
solve at first equation (3.3.3). The approximate integral
(3 3 6) x(1) = _-—l;— [Csin(n+l)t + Dcos(n+1)t]

' ' 2n+l

satisfies equation (3.3.3) to zero order in 6, since C
and D are of first order in 6 (see(3.3.2)). Utilizing
now equations (3.3.2), (3.3.6) and retaining terms up to
and including 0(62) in the expression obtained by substi-

N
tuting x = Csinnt + Dcosnt + Z3 qu(Q) in equation
q=l

(3.3.1). Yields the following system of variational

equations
r . . 2
O . . _ é 1nt 6
Y1 + ZlnYl ‘ ’ 2 Yze ‘ 4(2n+1) Y1
3. - 21...} = .5... e-int- __i...
(3.3.7) ( 2 2 2 1 4(2n+1) 2

(Y1 =Cj-_iD, i=f1T)

 

and the perturbational equation:

52

n 2
62(x(2)+-n2x(2))==gé—{Csin(n—2)t+-Dcos(n-2)t] -

(3.3.8)
2

- Wféfi)‘ [Csin(n+2)t+ Dcos(n+2)t] -

The variational system (3.3.7) is solved in exactly
the same way as the system of equations (3.3.4). This

leads to the indicial equation

 

2 2
r 4 2( T] 6 )
+ -___ __ +
P P 2 2nn + “211+“ + l
2
+ '4‘) 'nn+'4'(2n+1) '71—:0'
L
Setting p = O in (3.3.9) and remembering that l-—2n==n,
yields
n2_1.e 540(8)
‘4-2’ 8 °

Somewhat more lengthy computations lead to the third

order result (for the first critical region of the Mathieu

equation):

2 3

2 l 6 6 -— 6 4
=._ + _.- __. __.

n 4._ 2 8 +- + 0(6 )

N

Assume now that n is appreciably different from -%
or - %-. We can thus proceed directly to the second
order perturbational equation. ‘Using the notations of
Section 3.2, we note that in this case m. = mk = m = n:

j m
s = p = 1. We observe that the second order perturba-

tional equation contains terms of the form Csinnt + Dcosnt

vfliich have to be removed from it and transferred to the

53

variational part. We also note, by applying what we learned
from Section 3.2, that small divisors will occur in the
solution of the second order perturbational equation if
(p+s)w — mm = mm + n (n—small); in our specific case this
means 2 - 2n = n.

Thus by using the technique developed in Section 3.1

we obtain the following system of variational equations:

 

r 2 2
" - ' 6 6 int
Y + 21nY = --———- Y - —--—— Y e
O. 2 2 0
3.301 . —
( O) < Y2 - ziny2 = ———J%———-Y2 — ZI2%’IT'Y1e int
2(4n —l) -
(Yd = C + LD)
‘ 2

Since we are concerned here with solutions of the system of
equations (3.3.10) which are correct to second order in 6.

we assume n = 0(6) and drop the terms Y and Y in

 

 

l 2
(3.3.10). From the resulting equations:
. 2 . 2 .
° _ _ 16 16 1nt
Y1 ‘ 4 4 2 Y1 + 8n(2n-1) Yze
(3.3.11) “( n '1)
. 2 . 2 .
' 16 16 -1nt
Y = Y - —-——-—-Y e
2 4n(4n2-1) 2 8n(2n-1) 1

obtained by dropping the terms Y and Y in (3.3.10), we

1 2
can conclude that the omitted terms are at least of third

 

order in 6.
Setting Y1 = Aept+§lnt, ‘Y2 = sept'filnt in (3.3.11)
and requiring A and B not to vanish simultaneously leads

to the indicial equation:

54

 

 

2 2 4
(3.3.12) p2+[-1—n+ 62 1 = 2G 2
4n(4n -l) 64n (Zn-l)

Setting p = 0 in equation (3.3.12) yields

 

2 2
6 6
(3.3.13) n = - : ——-————
2n(4n2-l) 4(2n—l)
Since n = 2-—2n we can conclude from (3.3.13) that
2 2
n2 = 1 - 'E—Z- + 0(E4) or n2 = l + '53-??- + 0(E4) . These

results agree completely with those obtained by Stoker for

the Mathieu equation,see Stoker [24], pages 208-213.

Remarks:

i) It should be mentioned that a more standard form

of the Mathieu equation is
32+ (5+ 6cosZt)x=O

where 5 and 6 are parameters. This equation can be
transformed into the form which we used in this section,
i.e., equation (3.3.1), by setting 2t = T, 6 = 4n2,

e=-4E.

ii) The results which we obtained in this section
agree closely with the ones obtained by using Floquet

theory (Section 1.1), see also McLachlan [14].

3.4. The van der Pol equation.

We consider the equation

(3.4.1) 32+ x= 6(l-x2)x

55

where 6 > O is a small parameter. This equation is

called the van der Pol equation.

Set
N < )
(3.4.2) X = Csint + Dcost + Z: qu q
q=1
where C = C(t,€), D = D(t,€) . We observe that for

E = 0.. C and D reduce to constants.
Substituting (3.4.2) into (3.4.1) yields
.. .. . . 1% q; ,, ((1) ( )
Csint + Dcost+ 2(Ccost -Dsint) + Z 6 (x '+x q) =
q=l

(3.4.3) = €[l-—(Csint+-Dcost)2][Ccost-Dsint+-Csint4-Dcost]+-

+ C(62) .

Restricting ourselves to terms of first order in 6 and

using the perturbation technique develOped in Section 3.1

leads to the following reduced system of variational

 

 

equations
c-§(C-9-i-——CD2\‘
‘2 4 4/
(3.4.4)
D__§(D_23_.Ci>
‘2 4 4

and the perturbational equation

32(1) + x”) = %[(C3 - 3CD?) cos 3t +
(3.4.5)

+ (D3 — 3C2D) sin 3t] .

56

In order to solve the system of first order ordinary

differential equations (3.4.4) we multiply the first equa-
tion of the system by C, the second one by D and add

them together. We obtain

2

2 2
' ' E 2 2 (C +—D )

 

If we now set C = Asine and D = Acose we obtain,

by utilizing the system (3.4.4), that 6 = 60,. where 9

O

is a constant to first order in E . The substitution
C = Asine, D = Acose also transforms equation (3.4.6)
into
(3.4.7) A=§A(4-A2)
and the perturbational equation (3.4.5) becomes
(3.4.8) 32(1) + X”) = %A3sin3(t-8) .
The solution of equation (3.4.7) is
(3.4.9) A2 = €/& + {—35 - IJe-Et

A0
where A02 denotes an arbitrary (nontrivial) initial value
for A? . We note that for t.-—>~¢*,IAI ——> 2,. and that
if A0 = 2, then A = 2 for all t 2_O .

Equation (3.4.8) possesses a particular approximate
3
integral x1 = - %‘2' sin 3(t— 6) , which satisfies the

57

perturbational equation to first order in 6, since the
derivatives of A and 6 are each of at least first order
in e.

For a detailed discussion of higher order approxima-
tions and the physical significance of the van der Pol

equation, see the book by Minorsky [16], page 219-224.

CHAPTER IV

ADDITIONAL APPLICATIONS OF THE GENERALIZED
STRUBLE METHOD

4.1. A First Order Approximation for the Generalized Wehrli

 

System.
In Chapter two we investigated the stability of the

 

system
0. n
(x + azx + E!— Z a coskflx = O
1 1 Lk- k z 2
—O
.0 2 n
(4-1-1) < x + a x + E[ Z a coskflx = O
2 2 k- k 3 1
—O
k = d/dT

which we call the generalized Wehrli system (see also equa-
tion (2.1.9)) by using Floquet theory.

In this section we shall investigate the stability of
the system of equations (4.1.1), or equivalently (2.1.9),
by utilizing the perturbation technique developed in Section
3.1.

We observe first that the system of equations (4.1.1)
can be decoupled by means of the nonsingular transformation

X = PY where:

58

S9

’ _ X1 = Y1
x- (X2). Y (,2)
(4.1.2) ) 1 1
=_1_
KP 2(1_1)

 

The resulting decoupled system is

 

r
u 2 n
(a) y1+a yl+ €[k§ akcosk'r] yl - O
—0
(4.1.3) < .. 2 n
(b) y+ay -E[Z acoskTJy =0
2 2 k=0 k 2
K o = d/dT

From the structure of equations (4.1.3a) and (4.1.3b) it
follows that once we found the transition curves for one of
the equations, the transition curves for the second one are
obtained by replacing 6 by -6.

Using the notation of Section (2.1) we let 0 l l in
equation (2.1.4) and thus a 5 m0.

Since the system of equations (4.1.3) is a system of
the form (3.1.1), we have wj = wk E w and w = l.

0
Using the technique develOped in section 3.1 we set

y{m

N
y1 = CsinaT + DcosaT + Z: Eq , substitute into

q=1

(4.1.3a) and restrict ourselves to terms of first order in

6. ‘We observe immediately that the term EaO(CsinaT +

+ DcosaT) has to be associated with the variational part.

If in addition s-a = a4-n (n—small real number) for some

s(s = l,---,rU , the solution of the perturbational

6O

equation becomes unbounded. We thus transfer the resonance

 

creating terms to the variational part and use the Gener-
alized Struble method (GSM). We thus obtain after some

computations the indicial equation

 

 

 

Setting p = O in the indicial equation yields

__.}.
n-Za (2a +as)€.

0—
Since a - a2k a = azk (1 < s < n see notation
o‘ 1' s 1‘13 — — '
following equation (2.1.8)) and a E go, we Obtain

wokl (2 i us)
n = 2

 

Let n = $016 and recall that the transition curves for
equation (4.1.3b) are obtained by replacing E by --6.
We thus finally obtain

kal (2 2t us)
23

 

(4.1.4) 0 =;:

If in the system of equations (4.1.3) we replace the

co

finite Z: by' Z: , with the agreement that “s —¥> O

for s > n, we obtain

w k

_ O 1
(4.1.5) 01 —i_ s

 

for s = n + l,°--

61

HI

We also note that if a w = g-(p—positive integer) then

0
Q = O .

1
These results are in complete agreement with those
obtained in Chapter two, by using Floquet theory, (see
(2.3.4a,b) and section 2.5).
It is worthwhile to mention the relative ease with
which we arrived at the results, expressed by equations
(4.1.4) and (4.1.5), by using GSM as opposed to the enormous

amount of computation involved by using Floquet theory,

(see Section 2.2).

4.2 A Second Order Result for Wehrli's Original System.

We are interested to obtain the transition curves for
the first critical (resonance) region up to and including
terms of C(62) for a particular case of the general system

(4.1.3). We treat the system

(

ll
0

(a) yl + a2y1 + €(a04-a1c05'r)y1

(4.2.1)( H 2 - = d/dT
(b) Y2 + a yz — 6(a0+a1cos'r)y2 = o

 

k
which we shall call Wehrli's original system.

As already pointed out in Section 4.1 we need only treat

one of the equations of the system (4.2.1). We set
N

y1 = CsinaT + DcosaT + 7‘ quiq), utilize the technique
q=l

developed in Section 3.1 for the first order critical region

(in our case 1 - 2a = n, n—small) and the procedure

62

outlined at the end of Section 3.2 (in order to refine the
details for the first critical region).
After some computations we obtain the following system

of variational equations

€2a2

 

1 la 1 ’ " o 4(2a+l)_l 1 2 2
2 2
6 a Ea .
(4.2.2) < -- . - _ l: 1 ‘J 1 —1n'r
Y2 ’ 21E“Y2 " ’ 6a0 + 4(2a+l) Y2 + 2 Yle
\ c + in = Y
12
where - 2k a - 2k ( e S t' 2 1) W l

the system (4.2.2) by setting

Y1 = AepT+§1nT ' Y2 = BepT—éinT

and require A and B not to vanish simultaneously. We

thus obtain the indicial equation

62 2
(p2-l-n2 + [6a -+-———:l—- — an + i (2a+- )} x
4 O 4(2a+l) p n
62a2
(4.2.3) 2 l 2 [ 1 - _
X(p -Z'T] + an+m]-an-1p(2a+n)}_
2 2
a

-61

 

Setting p = O in equation (4.2.3) and recalling that

n = 1-2a, yields the transition curves for the first
critical region of equation (4.2.la) up to and including

terms of C(62). We thus obtain

63

 

 

Ea 62a2
2 _ l_ - 1 l 3
(4.2.4) a - 4 — an + —§—-- 8 + 0(6 ).
S' - a2k a - a2k we substitute into (4 2 4)
ince aO — 1' 1 — 1H1 . .
and set
2 _ 2 3
(4.2.5) a — 60+ 651+ E 62+O(E)
This yields
2 1 k1(2*“1) k: 2 2 3
(4.2.6) a :4. 8 €+T2—8-[7uli32u1+32]€ +O(€ ) .

The transition curves for equation (4.2.lb) are obtained by
replacing E by -.6 in (4.2.6). In this case the transi-
tion curves are given by

2
k (2th ) k
2 _ 1_ 1 1 1 2 2 3
(4.2.7) a — 4+ 8 €+128[7u1¢32u1+32]€ +O(€ ) .

 

Remarks:

i) In chapter five of our work we shall apply a tech-
nique developed by Porter [18] to the system (4.2.1). By
applying this technique to the system (4.2.1), we will
obtain the transition curves for the first critical
(resonance) region of the system (4.2.1) up to and includ-
ing terms of third order in 6 . We will do this in order
to compare the results, which we obtained in this section
by using the Generalized Struble Method, with those

Obtained by Porter [18].

64

ii) We will also see in.Chapter five, by utilizing a
technique develOped by Rand [19], why we are unable to
obtain a second order result for the first critical region

(transition curves) of the general system (4.1.3).

4.3. The Differential Equation Q + (54-Ecost)-m)c-C).

In a paper by Rand and Simon [21] the differential

equation
(4.3.1) x + (54-6costf4nx.= O,

(m is a positive integer, 5 and 6 are parameters) is
investigated. Particular cases of equation (4.3.1),e.g.,

the cases m = l and m = 3 have been investigated by
Panovko and Gubanova [17] pages 180—193, because of their

physical significance.
We shall treat equation (4.3.1) by using the Gener-

alized Struble Method (GSM), but first a few remarks are

in order.

Remarks:

i) For any given 6 and 6 the point (6.6) is
said to be stable if all solutions of equation (4.3.1) are

bounded for t > O,. and unstable if an unbounded solution
exists.
ii) The stability of equation (4.3.1) is not affected

if we replace 6 by - E .

65

iii) If [5| 3 (5| the differential equation (4.3.1)

possesses unbounded solutions.

(iv) If m is an even positive integer the stability

of equation (4.3.1) is not affected if we replace 5 by -5.

Back to equation (4.3.1), we assume 5 > 0,. O<I6<C< 1,

set

—m 2 6k2(m+1)/'m

(4.3.2) 5 =k , ='é

and restrict ourselves to terms of C(62). We thus obtain

 

-2
(4.3.3) x + kzx = [m-écost - m(m+l)2€ [1+c052t] ] x .
4k
N -q (q)
Let x = Csinkt + Dcoskt + Z: 6 x where C and D
q=l

are variables. Restricting ourselves to terms of first
order in 6, ‘we observe that the solution of the first

order perturbational equation will become unbounded for k

near -% or - %-. Assume l-2k = n (n-small real number).

Since we are interested in refining the details for
the first critical region (i.e., to obtain an improved first
order approximation for k near %), we use GSM developed in
section 3.1, and the ideas outlined at the g2g_of section

3.2 to obtain

Csinkt + Dcoskt + 2k(Ccoskt — Dsinkt) =

(4.3.4) = %?-[Csin(k-l)t + Dcos(k-—1)t] —

Earn m+1)
‘ 4

2 .
2 + 2%111 [Csinkt + Dcoskt] .
k _:

66

Replacing k by (1-n)/2 in the sines and cosines of

equation (4.3.4) , equating the coefficients of sin;- and

cos-3:2— on the R.H.S. and L.H.S. of equation (4.3.4) and

finally setting C + iD = Y1, C-—iD = Y2 yields the follow-

ing system of variational equations

 

 

 

.-- . - _ .3; int
Yl + 21kY1 — - 2 Yze —
_ §3_ m2 + m(m+l)]Y
4 L 2k+1 'k2 1
(4.3.5) <
oo . o _ E -int
Y2 - 21kY2 — - 2 Yle
-2 2
ELJ' m + m(m+1)1
\ " 4 L2k+1 k2 _. 2
Let Y1 = Aept+blnt , Y2 = Bept-ilnt and require A and

B not to vanish simultaneously. This yields the indicial

 

 

 

equation
4 “:13 E2 MLH 1
r - _ __ _
P+2P14 ”+4121? k21+2}
(4 3°94 . 2 -2 2 2-2
-1.- .€_[__rp_ M112 _m 6
+1 4 k” + 4 2k+1+ R21 " 4
L
Setting p = O and replacing n by l-2k in (4.3.6)
yields
- -2
(4 3 7) k2=l+fi..-“‘————262 -M‘“+1)E
'° 4— 2 4(2k+1) 2

4k

Using (4.3.2) we obtain from (4.3.7)

67

(4.3.8) 1

m-
+ m(m+1)€25_(m+2) + 4m2€25-(2m+2)] +

 

L + C(63).
1

. . . EH
Since 6 is smaller than unity we recall that (l+—x)
can be expanded in a convergent power series for x<<l .
We expand (4.3.8) as a power series in 6, restrict our—

selves to terms of second order in 6 and set

_ 2 3
(4.3.9) 5 — 50 + 661 + e 52 + 0(6 ) .

We thus obtain after some computations

_ 5 -.§ ’fii.im .l:] 2 3
6 - 4 + 2 + 4 32 + 8 6 + 0(6 ).
We observe that for m = —Il, we obtain the well known

result for the Mathieu equation (see Section 3.3)

2
_l‘§_§_ 3

Assume now that k is appreciably different from -%

and - %~ and proceed directly to the second order perturba-

tional equation. We observe that the solution to the second

order perturbational equation will become unbounded if

2 2

2 2
k - (k-2) or k - (k4-2) are small, i.e., if k is

near 1 or near -1 . We assume that k is near 1 and

68

set 2-—2k = n (n-small). Following the procedure developed

in Sections 3.1 and 3.2 we obtain for the variational part

r Csinkt + Dcoskt + 2k(Ccoskt-—Dsinkt) =

 

2
(4.3.10) < = e 2[Csinkt+Dcoskt][————— - MflizlL] +
2(4k2 -1) 4k
2
'2 - m m(m+l)
+ 6 [C51n(k-2)t+-Dcos(k-2)t][———————-_
K 4(2k-1) 8k2

Recalling that 2-2k = n, 'we substitute for k in the
sines and cosines of equation (4.3.10) and equates the co-
efficients of sint and cost on the R.H.S. and L.H.S. of

leads to the following system of variational equations

equation (4.3.10). Setting C + iD = Y

 

 

(. u _
Y1 + 2'1le = €2[————— - 35%] Y1 -
2(4k 2-1)
_ -2[__L_ _ M Yzein
4 2k- k2
(4.3.11) ( ( 1) 8
Q2 - 21m = '2[“‘ _ ELM—211 Y2 -
2(4k 2-1) 4k
2 .
L _ §2[ n1 _ m(m+1) ‘Y e-int
4(2k-1) 8k2 1

In order to solve the system (4.3.11) we set

Y1 = Aept+§1nt' ‘Y2 = Bept-iint'

to vanish simultaneously and thus obtain the indicial

require A and B not

equation. Setting p = O in the indicial equation and re-

placing n by 2-2k we obtain:

69

 

 

(k2 = 1 +{ m: _ m(m+21)}§2
(4.3.12) ( 2(4k -1) 4k
K i1fi7ngi‘17 _ m(8m‘:-21)'}-é2
We recall that k2 = 5"“ , g Ek2(m+1)/m and write
6 = 60 + 661 + e252 + C(63) .

Thus by substituting into equation (4.3.12) we obtain

6 = 1 + [924 —M]€ + 0(6 4)

6- +[5m__+__23]ez +O(64).
For m = -1 we obtain

6 = l + %§;-+ C(64)

o=1-§; +o<e4)

which is the well known result for the Mathieu equation
(see section 3.3).

The results obtained in this section agree completely
with those obtained by Rand and Simon [21] in their paper

using a perturbation technique developed by Stoker [24].

4.4. The Differential Equation x4-(5-6coszt)(l-Ecoszt)-1x==0.

 

In a paper by Rand and Tseng [20] the stability of the

solutions of

70

(4.4.1) 3} + f(t)): = o
where
(4.4.2) f(t) = (5 — 6coszt) (1— Ecoszt)-1

is investigated by using a perturbation technique developed
by Stoker [24]. We shall treat (4.4.1) by using the Gener-
alized Struble Method.

We note that for 6==l , we have f(t) E 1 and equa-
tion (4.4.1) has only bounded solutions. We also observe
that for 6 2_l and 6 f 1, equation (4.4.1) possesses
unbounded solutions. In the following considerations we
assume O<6<<l and 5>O.

Since 6<:l, we expand (l--6coszt)-1 as a power
series in E, restrict ourselves to terms of C(62) and

obtain

(4.4-3) (6--Ecoszt)(l-—6coszt)'l =

e 5 + [(5-1)coszt]6 + [(5-1)cos4t] €2+0(€3)-

Substituting (4.4.3) into (4.4.1) we Obtain

2 cosZt

(4.4.4) 3} + k 2

x+ (k2—1)[§(1+coszt) +(%+ +

+ cog4t)€2]

)<==C): k2==5 # l.

N q (q)
We set x = Csinkt + Dcoskt + Z) 6 x where C and D
q=l

are variables and substitute into (4.4.4). Restricting

71

ourselves to terms of 0(6) we note that

(k2-1)e

2 [Csinkt4—Dcoskt] has to be associated with the

variational part.

Solving the first order perturbational equation and
continuing to the second order perturbational equation, we
observe that additional terms have to be transferred to

the variational part. These terms are

_ 11 (k2-1) 62[Csinkt + Dcoskt]

 

32 ‘
If in addition we assume that either k2 - (k+4)2 or
k2 - (k-—4)2 are small, the solution of the second order

perturbational equation becomes unbounded.
Assume 4-2k = n (n—small), in this case the

variational part becomes

Csinkt + Dcoskt + 2k(Ccoskt-—Dsinkt) =

2 '1
(4.45) = .. (k2 — 1){[§+-13i2€—] [Csinkt+DcosktJ +
62
+ g—-[Csin(k-4)t + Dcos(k-4)t]} .

Replacing k by 2 --E in the sines and cosines of equa-
tion (4.4.5), equating the coefficients of sin2t and
cosZt on the L.H.S. and R.H.S. of equation (4.4.5), and

setting C + iD = Y C - iD = Y leads to the following

1' 2

system of variational equations:

72

 

 

 

r " . ' _ 2 .g 116 :[
Yl + 21le — - (k -l){[:2 + 32 Yl +
2 .
) 6 int
(4.4.6) \ + FY28 }
3} — 2ik’i! - — (k2-1){[-§+ 1162] Y +
K 2 2 _ 2 32 - 2
2 .
.§_ -1nt}.
+ 6 Yle
Let Y1 = Aept+b1nt , Y2 = Bept-filnt, substitute into

(4.4.6) and require A and B not to vanish simulta-

neously. This leads to the following indicial equation

 

(4. 4. 7) {p2 -—n2— kn + (k2—1)[§+ ”6 ———222]} + 16p2 =
_ (k2--1)2€4
642

Setting p==O in equation (4.4.7) and replacing n by

4-2k yields

(4.4.8) k2 = 4 + (k?- _1)[- _

 

mm
H
LOH
Mm
|+
mlm
4:.
..J

2
Let k2 = k + 6k + 6 k + C(63) and substitute in equa—

O l 2
tion (4.4.8), we thus obtain

 

2
2 _ _3_§ 1__se
k — 4 -' 2 —. 64 + 0(63 )
(4.4.9) 2
2 _ 3g 216 3

These results are in complete agreement with those obtained

by Rand and Tseng [20].

CHAPTER V

APPLICATION OF PERTURBATION TECHNIQUES
DEVELOPED BY PORTER AND RAND

5.1. A Third Order Result for Wehrli's Original System.

 

In Chapter two we generalized a result obtained by

Wehrli [29] and obtained certain results which were valid
up to and including terms of first order in 6 (see equa-
tions (2.3.4a,b) and section 2.5).

In Chapter four (section 4.2) we decoupled Wehrli's
system of 0.D.E.'s (see Wehrli [29], equations 4.3),
applied the Generalized Struble Method to the system of
equation (4.2.1) and obtained a result which was valid up
to and including terms of second order in 6.

In this section we shall use certain results Obtained
by Porter [18] in order to find the transition curves for
the critical (resonance) regions, up to and including terms
of 0(63),. of the system of equations (4.2.1). 1

In his paper, Porter [18] investigates an equation

of the type
C. m r
rx+[)\{1+ Z Er[ Z ”r cosZs'r]}+
r=1 s=O 8
(5.1.1) ( w r = d/dT
+ Z 6r< 2‘ VrSCOSZS T>]X = O
L r=1 s=O

 

73

74

where l. 6. “rs and Vrs are parameters and 6 < < 1.
By using a perturbation technique develOped by Stoker [24],
Porter [18] obtains the transition curves for the critical

(resonance) regions of equation (5.1.1). His results are

valid up to and including terms of third order in 6.

 

Back to the system of equations (4.2.1), we Observe

that replacing T by 27 will transform the system (4.2.1)

into the system

ll
0

" 2

(a)Y1 + 4a Y1 + 46(aO+-alcos'r)y1

(5.1.2) - = d/dT.
oo 2

(b) y2 + 4a y2 — 46(a0+ alcos'r)y2

ll
0

Without loss of generality (see Section 4.1) it suffices to
treat equation (5.1.2a). It is clear that (5.1.2a) is an

equation of the type (5.1.1) with

_ 2 _
l — 4a “rs — O, r > 1
(5.1.3) “10 = kl “rs = O: r _>_ 1: S > 1
“11 = klul Vrs = O, r 2_l, s 2_O .

Thus using the results obtained by Porter [18] for
equation (5.1.1), we obtain the following transition

curves, for the various critical (resonance) regions, of

the system of equations (5.1.2):

2
H
H

 

 

 

 

For

2
a =

where

blr—I

N= 3 we have

A+

k

_l
- 8(2iu1)6+

3

2
k1[72
128 “1

:l: 32u1+ 32]€2 -

[139u1+ 336ui i 768ul+ 512 ]63 + C(64)

k1
?(2 11111) E 1'

[:t 39p? + 336;;

B.6+C. +
J 6

2
k1

128

2
1

5U2
21116 -ki1[
2
L1
112 3[
12 e -k11

5 2
_“_121 3
62+k1[1

2

u

l 2 3
—2]6 + k1[1

133.63 + C(64) .

[7111* 321.11 +32]62 +

i 7681.11+512]63 + 0(e4).

Suz

“—4'14163 + C(64)
2

u

_4];.63 + C(64)
2

5u

-—4i 63 + C(64)

L12

7} e3 + M?)

j = 1,2

76

9k
_9. _ _ __l_. _ _
A—-4-.B1—-B2—— 4.c1—c2—
9k2 9k3 27u2
_.__l;b_+,_2_ 21'-D - — D - - -—l41 + ———l-+
‘ 4 641413” 1‘ 2‘ 4 64 —
81p3
+——l‘\
-— 512_:'

We observe that in the case N==3 we have two transition
curves for each of the equations (5.1.2a) and (5.1.2b),

respectively.

For N 2_4 we Obtain

 

 

 

 

 

 

 

2 2 2
r 2 N k N k 2
azzgi“ 416+ 41[1+ N2 ”1162'
8(N -1) J
2 3
N k 3N2
_. 41@-+——7——L€]63+<H6%
< 8(N -1)
2 2 2
2 N k N k 2
32=§4‘+ 416+ 41[1+__N2__“fl€2+
8(N -1)
2 3
N k 2
+ 41[1+—3—b21—ui]63+0(64).
k 8(N -1)

We observe that for N 2_4 there is only one transition
curve for the critical (resonance) regions of equations

(5.1.2a) and (5.1.2b), respectively up to and including

 

terms of C(63) .

77

5.2. Anggpplication of Rand's Technique.

 

In this section we shall outline the essentials of a
perturbation technique develOped by Rand [l9] and apply it
to a particular case of the general system of equation
(4.1.3).

Rand [l9] investigates the stability of the solutions of

a class of equations of the form
(5.2.1) 3:: + (A+Bcos¢+CcosZ¢+DcosB‘r)x = O - =d/d'r

where A,B,C and D are parameters. An equation like
(5.2.1) is called a Hill's equation with 4 independent
parameters.

For a given A,B,C and D the point (A,B,C,D) is
said to be stable if all solutions of equation (5.2.1) are
bounded for all t > O and unstable if unbounded solutions
exist. We are interested to find the boundaries between the
regions of stability and instability (transition hyper-
surfaces) in the hyperspace ABCD . We know from Chapter
one (Section 1.1)that corresponding to transition values
from stability to instability, there exists at least one
periodic solution to equation (5.2.1) of period T or 2T.
We note that if B is not identically equal to zero then

T

2n . Thus for B = C = D = 0 transition points can
2
occur only if A = %f1' N

0,1,2,... 0

78

Remarks:

i) For N==O the solution of (5.2.1) is a constant
which might be thought of as a periodic function of period
4n.

ii) For B = C = D = O and A S_O.. equation (5.2.1)
possesses unbounded solutions and thus the entire negative
A axis is unstable.

From these remarks and the observations preceding it,
we expect two transition hypersurfaces to intersect each
of the transition points on the A axis, one behaving like
cos g;- the other like sin g?" for B = C = D = O .

In order to obtain explicit expressions for these
hypersurfaces for small values of the parameters B,C and

D, Rand generalizes a perturbation technique developed by

Stoker [24]. We set

 

f co a) co . u
X(T) = Z: Z: Z: Xijk(T)BlCJDk
i=0 j=o k=O
(5.2.2) <
A: Z Z Z A..kB1C3Dk
k i=0 j=o k=O 13
N2
with AOOO = TI' and substitute into equation (5.2.1). By
k

equating like powers of BICJD , we obtain a linear
differential equation in xijk(T) with constant coeffi-
cients. By requiring Xijk(T) to be periodic, one obtains

a (certain) value for Ai' We also note that for N > 0.

3k'

(T) is taken first as sin nT

nT .
-§-, then as cos ——- Since

X000 2

79

each of these choices gives a different transition hyper-
surface for N = 0,1,2,

We shall now investigate a particular case of the

 

system (4.1.3) and obtain the various transition curves by
a direct application of Rand's technique. The equations

which we investigate are

-- 2 r 1 _
(a) y1 + a y1 + ELkEg ak COSkT Y1 — 0
(5.2.3)
3 ‘1
(b) y2 + a 2-y2 6[k§0 akcosk'tJ y2 = O .

From the structure of equations (5.2.3a) and (5.2.3b) it
follows that once we found the transition curves for one of
the equations, the transition curves for the second one are
obtained by replacing 6 by' -6 (see also Section 4.1).
We note that equation (5.2.3a) is an equation of the

form (5.2.1) with

2 _ _ _
A—a +€aO,B—Eal,C-Ea2,D—Ea3
h - azk a = azk (' - l 2 3 see 150 Section
w ere aO — 1. j 1“j j — , . a
2.1). Hence by substituting into the expressions Obtained

by Rand for the various transition hypersurfaces, letting

2_ 2 3
a — 60 + 661 + 6 62 + 0(6 )

and restricting ourselves to terms of second order in 6

we Obtain for N==1:

80

 

r 2
k
2_ 1 1 1[ 2 12_
a —74_- —8—k1(2:u1)6 + 128 7uli32ul+32J6
(5.2.4)(

k2

L —————l [8u2+3u2+4uu +12uu162+0(63)
3'162 2 3- 2 3- l 2

for the transition curves of equation (5.2.3a) and

k2

2 _ l l l [_ 2 '] 2 _
a —Z-+-8—k1(24_-_u1)6+——1287uli32ul+32 6

(50205) 2
k1 2 2 2 3
-——[8L1+3L1+4L1L1+12L1Ll]€+0(€)
3-162 2 3-— 2 3-— l 2

for the transition curves of equation (5.2.3b).

In a similar fashion we obtain for N==2

2

 

k
f 2__ __1_ 1 2 ]2
(a) a — 1 + 2 k1(2-u2)€+—{32 7u2-32u2+32. e _
k1 2 21 2 3
(5.26%
2
(b)a2-1¥-1—k (2+ )e+:l[7 2+32 +32]62+
‘ 2 1 “2 32 “2 112
k2
L 1[ _ 2 2 2 3
+—60 (5111 113) -7u3]6 +O(6).
Remark:

Equation (5.2.6a) represents the first transition curve
for both of the equations of the system (5.2.3) and similarly.

equation (5.2.6b) represents the second transition curve for

both of the equations of the system (5.2.3).

81

For N=3 we find

k2
2-2. 2 __1 1 2
a —4-8k1(2:u3)€+ 12 87u[2 3+32113+32§6 +
(5.2.7) (

2
81k 2 2

1 2 ‘1
1 16 [5(u1i2u2)—12u2 e +o(e3)

 

for both of the transition curves of equation (5.2.3a) and

 

r 2 9 9 91112 ‘1
a = :4- §k1(2:u3)6+ +—'2—'[7L13 +32u3+32i 62
(50208) g 2
81k
1 - 2 2] 2 3
L +——16 [5(u1+2u2) -12u2 6 +002)

for the transition curves of equation (5.2.3b).

From equations (5.2.4) through (5.2.8) we conclude that
for the transition curves, corresponding to the various
critical regions of the system of equation (4.1.3), we can
obtain a general result (which exhibits a pattern), only up
to and including terms of first order in 6, i.e.,

2 2
2 s s 2

It is fairly safe to assume that one of the many second

order terms appearing'in the expressions for the various

transition curves, of the system of equations (4.1.3),will
have the general form
2 2

S 2 — 0..
'i—2'§'(7llsi32us+32)e , S—l, ,n.

CHAPTER VI

HSU'S METHOD AND ITS MODIFICATION

6.1. Hsu's Method.

In his papers [7] and [8] Hsu discusses the stability

of systems of second order ordinary differential equations

of the form

" ° - ._.d
X + EC(t)X + [Boa-€B(t)]x — 0 dt
where:

X is a column matrix in the variables

X 00.x .
l' ' n

B0 is a n.xn matrix with constant entries.

B(t) and C(t) are two real n.xn. matrices whose

elements bij(t)' Cij(t) are periodic in t with

period T.

6 ) O is a small parameter.

We outline the main features of Hsu's method as applied to

the system

-° 2 S n (S) ‘3
(6.1.1) x. + w.x. + e Z Zak cosswtxk = o -= 312'
3 3 5:1 k=1 3

(for notation we refer to the explanations given below

equation.(3.l.1).

82

83

Let

_ . _ N
(6.1.2) xj =C.e 3 +D.e 3 + Zequ); i =/—l

h C. = C. t, , D. = D. t,6 nd x. = x. t ,
w ere 3 J( E) J J( ) a 3 j ( )

(p=1,'°°,N). We also note that the cj's. and Bj's are

related to the Cj's and Dj's of equation (3.1.2) by the

following relationship

6 .

3 (Dj -1cj)/2 , Dj = (Dj+1Cj)/2.

iswt+_e-1swt)/2 in equation

We also replace cos swt by (e
(6.1.1).

Following Hsu we set

L iw.t 1 -iw.t
(6.1.3) C.e 3 + D.e 3 = O
J J
and obtain
_ 1w.t _ -iw.t N
(6.1.4) x. = iw.(C.e j -D.e 3 )+ Zqugq)
3 3 J 3 q=1
Thus
" ; 1wjt _ —iw.t 2 _ iwjt _ -iw.t
6.1.5 x. = i ». C.e -D.e )-—u). C.e +D.e )+
( ) 3 ULJ< 3 3 3< J J
N ..
q=1

Substituting (6.1.5) into (6.1.1) and restricting ourselves

only to terms of zero order and first order in 6 yields

84

. iw.t - —iw.t
- " J " J --(1) 2 (1) .
6.1.6 1 . C.e -D.e + x. 4— . . +
( ) w3\ J J > E( 3 wa3 )
S n . . _ iw t _ —i t
+ g_ Z: Z‘ags)<elswE-e’lswt)(c e k 4—D e wk ) = O
2 5:1 k=l jk k k

j = 1'...'r1.

An examination of (6.1.6) suggests the following distribu—

tion of terms

L 1wjt _ -iwjt
6.1.7 ". C. -D. > = O
( ) 1DJ< 3e Je
and
"(l) 2 (1)
6.1.8 E x. +— .x. =
( ) ( J m] 3 )
S n . . i t —i t
= - E- Z: Z:a(s)<elswt+-e-lswt)<cke wk 4—Dke wk )

2 5:1 k=1 3k

Remarks:

i) Equations like (6.1.3) and (6.1.7) will be called

variational equations and an equation like (6.1.8) will be

 

called a perturbational equation.

ii) At any step of the process Hsu associates terms
_ iw t ; -iw t
of the form imp(c e p _ D e P ) with the so-called

P P
"offending" terms in the perturbational equation, i.e.,
terms which lead to unbounded solutions of the perturba-

tional equation, and the perturbational equation with the

remaining nonresonant terms.

85

From (6.1.3) and (6.1.7) it follows quite easily that

C. E C .; 5. = 5 ., ‘where the C .'s and the 5 .'s
3 03 J 03 03 O]

are constants for j = 1,°°°, n.
If we assume sm i.wk to be appreciably different

from + wj for any j,k, and s, the perturbational

equation (6.1.8) has the solution

 

_ i(surtu1k)t _ -i(sw+w.k)t
FX(1) = __J; ~28: %a(s)[cke +Dke
3 2 5:1 k=1 3k w; - (sw+wk)2
(6.1.9) 4 E e-i(sw-u1k)t+5 ei(sw-<1).k)t
k k

+

 

w: — (aw-mk)2

 

where the Ck's and Bk's are constants.

Assume now that sw + wk = wj + n (n-small real

 

number) for a certain set of values of j,k and s , say

L,m. and s We also assume wt # gm. ‘Under the above

1'
assumption it follows that small divisors will occur in the

solution of equation (6.1.8). We remove these "offending"

 

terms from equation (6.1.8) and associate them with

.L it» t _°_ —iw t
iwL(C e L - D e L

L L ). We thus obtain:

 

( _ 1m t ; -im t
C e 1 + D e L = o
g z
1w t -im t
- L — L ) _
1w£<Cze Die —
(6.1.10) <
(31) ( ) ( >
Ea r- i s w+w t _ —i s w+w t
_ Lm C e l m + D e l m
2 m m
1 slw + mm = wL + n

Interchanging L and m in equation (6.1.8) and re-
moving the "offending" terms, yields the following system

of variational equations

 

( _ 1(1) t L -10) t
C e m + D e = O
m m
/; lwmt _ -1th
10) (C e — D e )=
m. m m
_ Eamz {E e-1(slw—wz)t + B ei(slm-wz)t>
‘ ' 2 K I. I.
k slw-wL=—wm+n.

From equations (6.1.10) and (6.1.11) we obtain:

87

 

 

 

 

f - . (s ) . -i(2(1) +n)t

" _ 16 1 ' int 1,
(a) Ct _ 4mJc aim (Cme +1)me >

; . (s ) _ 1(2m +n)t _ .
(b) D = — 16 a 1 (c e 2 +13 e'mt)

L 4w! Lm m m

(6.1.12) (

- __ 16 l - -1nt - m )
(C) Cm ’ 4mm amt. (Cze +Dze

. . (s ) i(2w -‘n)t _ .

- _ 16 1 - m int)
((d) Dm -- - 4mm amt. (Cze +DLe .

In order to solve the system of equations (6.1.12) we
take the average value (see Section 1.5) of the right hand
side of equations (6.1.12a) through (6.1.12d) with respect

to WI and uh over a period of Zn. In these computa—

tions CL' DL' Cm and Dm are considered to be constant.
We thus obtain the following system of first order ordinary

differential equations

 

 

 

 

‘ l 16 (S1)- int
(a) CL _ 4(1)‘ aLm Cme
L . (S ) .
_ _ 16 l - —1'r)t
(b) DI. - 4m! 31m Dme
(6.1.13) (
1 - (S ) '
_ 16 1 - ~1nt
(C) Cm ' 4mm amt C18
1 - (S ) -
_ _' 16 l - int
k}d) Dm _ 4w amt Dze '

88

We observe that the system of equations (6.1.13) is com-
pletely similar to the system of equations (3.1.12) with
Y1, Y2, Y3 and Y4 replaced by Et' BL' Em and am'
respectively. Thus the indicial equation obtained by
solving the system (6.1.13) will be the same as equation
(3.1.13) and the analysis of the indicial equation will be
a word for word repetition of the one following equation
(3.1.13).

The main disadvantage of Hsu's method is that we have
to use the technique of averaging in order to solve systems
of equations like (6.1.12). By using this technique we
introduce an error of order 62 (see Bogoliubov and

Mitropolsky [2], pages 392-394) and thus are unable to re—

fine the details for the first critical (resonance) region.

6.2. Modification of Hsu's Technique as Applied to the

Mathieu Equation.

 

In order to gain a better understanding of the short-
comings of Hsu's technique, let us consider the Mathieu
equation

u 2
(6.2.1) x + n x = (6cost)x
where n2 and 6 are constants and O < E < < 1 (see

also Section 3.3).

89

 

C = C(t.6).

zero and first order

2a) and (6. 2. 2b)

We set
N
r(a) x = f+ Z) quiq)
q=l
(6.2.2) <
- N ( )
(b) x = 9+ 2 eqxzq
K qzl
and choose f = Celnt + De-lnt where
Restricting ourselves to terms of
in 6, we obtain from equations (6.2.
(6.2.3) f - g + 6(x{1)—Xél))

Similarly substituting equations (6.2.2a) and

(6.2.2b) into equation (6.2.1),

 

and restricting ourselves

to terms of zero and first order in 6, yields
(6.2.4) g + n2f + 6(xé1)+-n2x{l)-fcost) = O .
Let us now analyze equations (6.2.3) and (6.2.4). Set
((a) iil) _ X£1) = 0
(6.2.5) (
k(b) iél) + nzxil) — fcost = O .

Thus equations (6.2.3) and (6.2.4) reduce to

(6.2.6)

9O

 

If we let g = in(Ce1nt-De-lnt) and remember that
f = Ce1nt + De-lnt, then we obtain from (6.2.6) the
system of equations

((a) Celnt + De—lnt — 0
(6.2.7) <

L(b) in(Ce1nt-De_1nt) — 0
whose solutions are C 5 CO and D 5 D0,. where CO

and D0 are constants.

We also observe that the system of equations (6.2.5)
can be solved easily and its solutions are bounded, as long

as n is appreciably different from i_l-

2 0

Assume now that n is near -%, set l-2n = n
(n-small real number). We choose f = Celnt + De-lnt, set
21(1) — xél) = O and let g = in(Ce1nt-De-1nt) in equa-

tions (6.2.3) and (6.2.4), respectively. We thus obtain

(restricting ourselves to zero and first order terms in 6)

(a) Celnt + De—lnt = 0

(6.2.8) (b) in(Ceint-De—int)

+

+ 6(x(l)+-n2x{l)-fcost) = 0.
Since l-2n = n and n is small, it follows that the

term fcost will give rise to small divisors in the

solution of:

91

i(n—1)t*_ e-i(n—l)t

( ..
€(xil) + n2x{1)) = g-[Ce D +

(6.2.9) (

 

+ Ce1(n+l)t+-De—l(n+l)t] .

L

i(n-1)t e-i(n-l)t]

We remove the offending term -§[Ce -tD

from equation (6.2.9) and associate it with

in(Ceint-De-int). We thus obtain the variational system
(a) Ceint + De—int = O
(6-2-10) (b) 1n(éeint..be‘int) —
=_§[Cei(n-tL+De—i(n-l)t].

The system of equations (6.2.10) cannot be solved exactly

and we have to use the method of averaging in order to solve

 

it. Therefore the reasoning employed in this case leads us
back to figu;§_method (see Section 6.1) as applied to the
particular case of the Mathieu equation.

The analysis of equations (6.2.8) through (6.2.10)

compels us to conclude that the shortcoming of Hsu's tech-

 

nique might lie in the requirement to set Celnt + fie'lnt

equal to zero, which is restrictive in the sense that we
require terms up to and including 0(6) to vanish.

We shall thus try to modify §§3;§_method, as applied
to the Mathieu equation in the case l-2n = n, (n-small)

by requiring:

92

(a) A variational system of equations which can be

solved exactly.

(b) Following requirement (a) a perturbational equation
whose right hand side is of 0(n).

We return to equations (6.2.3) and (6.2.4) and let

f = Celnt + De_1nt: g = in(Ce1nt-De—1nt)

with C = C(t.€) and D = D(t,6). We also make the follow—

ing assumptions:
i) l - 2n = n (n—small real number).

ii) C = Aeqt+IXt7 D = Beqt-1Xt where A and B are

arbitrary constants and l is some real number. We also

 

write
( , qt+1s t qt—is t
(a) xil) — xél) = plAe l + p28e 1
, qt+is t qt—is t
(b) xél) + nzxil) — %{Ae 1 ~tBe 1 +
(6.2.11) 4
qt+is2t qt—iszt
+ Ae 4-Be 1 ==
qt+islt qt—islt
\ =
qlAe + qZBe
whe1:e s1 = l + (n-l), $2 = l + (n+1) and q, pj, qj,
(3 == 1,2) are coefficients which will be determined

in time subsequent analysis. We observe that small divisors

Will occur in the solution of (6.2.11) if:

93

(qiisl )2 + n2 = om).

2

Assume that l = (1-2n)/2 = n/2 and q = 0(n) . From
these assumptions it follows that (q+isl)2 + n2 = 0(n)

and the perturbational equation (6.2.11) becomes

’ '(l) (1) - qt—éit qt+§it
(a) xl - x2 — plAe + p2Be
(6'2'12) °(1) 2 (1) 1 qt—tit 1 qt+§it
(b) x2 +n x1 = (q1+§)Ae + (q2+-2-)Be +
1 q“ 3 2t “-3—? \
+ 3 (A6 + Be ) .

We will now determine the p.'s and q.'s in such a
way that the solutions of the system of equations (6.2.12)

will not contain small divisors and that the variational

system

( 0 _ - .
(a) f - 9 + €(p1Aeqt élt+p23eqt+ht) = 0

(6.2.13) <

 

(b) 6 + n2f4-€(qlAeqt-élt4-queqt+ilt) = o
k

can be solved exactly.

Differentiating equation (6.2.12a) with respect to t

and adding it to equation (6.2.12b) yields:

94

:41) . .241)

_ l_. l. qt-éit
- [plq - 2 1p]. + (ql + 2) JAE +

(6.2.14) 1
. t+ 't
+ [p2q+~% 1p2+-(q2+~%)]Beq £1 +

31t 3.t
1 “*7 qt‘—2"
+ -2-(Ae + Be )

 

Since q = 0(n) we require

1

(a) - % ip1 + (q1+§) = om)
(6.2.15)
(b) lip + (q +l) =0(n)
2 2 2 2 °

We emphasize again that the pj's and qj's (j==l,2)
have to be chosen in such a way that the system of equations

(6.2.13) can be solved exactly.

_ _1_ _ .1_. .. _L
q2 = --%, recall that f = Celnt + De-lnt and
g = in(Ce1nt-De-lnt) then we obtain by using equations

(6.2.3), (6.2.4) and (6.2.11) the following system of varia—

tional equations

(a) éelnt + De-lnt =_fiEmeHn-Dt__De-1(n-l)t)
(6.2.16)
(b) Ceint _ De—int =-%§(Cel(n-l)t+-De-1(n-l)t)

Equations (6.2.16a) and (6.2.l6b) lead to a system of equa—

tions completely similar to the system of equations (3.3.4)

95

with Y1 replaced by -C and Y2 by D. Solving the

system of equations (6.2.16) leads to the indicial equation

 

 

Thus the solutions of (6.2.16) will be bounded if we re-
. 1 2 62
quire -'Z'T] + 2
l6n

equation in this case is

 

to be negative. The perturbational

"(1) 2 (1) _ _1_ . pt-Qit
x1 + n xl — 8n[-A(21p4—n)e +

(6.2.17) + B<Zip-n)ept+ht]+

Bit Bit
1 WT Pt"?
+ -2—<Ae + Be )

and similarly for xél)

We wish to emphasize that the modification of Hsu's
technique could not have been carried out without the use

of equation (6.2.2) where use of two perturbation param-

x{Q) and xéq)

eters has been made (see also Stevens

[23]).

CONCLUSIONS

The main aim of our work was to develOp a perturbation
technique for certain classes of second order ordinary
differential equations with periodic coefficients. The
need for developing approximation techniques of this kind
becomes obvious in light of the large amount of computation
involved if we use classical Floquet theory.

We applied Floquet theory to a prdblem in mechanics
(see Chapter II) which we generalized following a paper by
Wehrli [29]. The amount of work and computational diffi-

culties arising if we want to find the characteristic expo—

 

23233, or even make statements about their size (whether
they are in absolute value less than or equal to one) are
clearly exhibited by the example, which we treated in
Chapter II. see also Magnus and Winkler [12] and Magnus [11].
In Chapter III, Sections 3.1 and 3.2, we developed a

perturbation technique for systems of 0.D.E.'s of the type

 

(3.1.1) and thus generalized a technique developed by

 

Struble and Fletcher [26] and [27] for the Mathieu and

van der Pol equation, respectively. We apply the method
which we developed in Sections 3.1 and 3.2 of Chapter III to
various examples (see Sections 3.3,3.4 and Chapter IV) and

obtain results which compare favorably with results obtained

96

97

by rigorous mathematical methods (Floquet theory), 25
with those obtained by other approximation techniques dif-
ferent from ours, see Stoker [24], pages 208-213.

The advantage of our perturbation technique over
Struble's and Fletcher's [26] and [27] or Hsu's [7] and [8]
(see also section 6.1), is that we are able to solve certain
second order 0.D.E.‘s exactly without neglecting certain
terms as the first two authors do, or use the method of
averaging as Hsu does.

The stability of the solutions of the 0.D.E.'s, which
we call variational equations (borrowing the terminology
from Struble) reduces in each case to the investigation of
the zeros of polynomials of degree ‘2 2 with complex co-
efficients.

We wish to point out that in 1966, Stevens [23] de-
veloped a perturbation technique for first order 0.D.E.'s

of the form
(ST) X + [A0 + 6A(t)]X = o . = d/dt

where X is a column vector in the variables x1,"', xn,

A0 is a diagonal matrix with constant entries, A(t) is
a n.xn matrix with periodic coefficients of period Zn/w.
Applying his perturbation technique to equations like

(ST), Stevens [23] obtains variational equations which can

be solved exactly. Using the transformation xj==yj1+yj2

98

and xj = iwj(yjl-yj2) an equation like (3.1.1) can be
reduced to a system of Stevens' type (ST).

A question of interest to be pursued by us in the.
future is to develop a perturbation technique for 0.D.E.'s
of the type (3.1.1) with wj = wk for j #'k which cannot

be transformed to a system of 0.D.E.'s of the form

(D) X+A(t)x=o, x=§ , -=d/dt

X
n

where A(t) is a diagonal matrix with periodic entries. An

example of an equation of this kind can be found in Wehrli

[30].

APPENDICES

APPENDIX A

THE INFINITE DETERMINANT A(O)

The infinite determinant mentioned in Section 2.2 is

of the form

A(o)==

 

where

i)

ii)

iii)

iv)

Note:

diagonal elements of

Ga \ Ga 6a
5

0 s
0 2A x x x 1 7f—- 0 x x x 0 2A ...
v v v
2Av Av \ 2AV

denote zeros

a = 0 for s

s > n 2_1
x — denotes that nonzero (zero) and zero (nonzero)

elements are alternating

Av = [a2 - (0+V)2]

The sum of the absolute values of all the non-

A(O) is

 

99

...—¢-.

 

100

Since this series is uniformly convergent with the excep-
tion of the points 0 = — v;: a, it follows that A(o)

is a meromorphic function of o.

APPENDIX B

EXPANSION OF A(O) UP TO TERMS OF C(64)

The infinite determinant mentioned in Section 2.2 has
ones on its main diagonal. The expansion formula for a

determinant of this kind is

 

 

 

 

Ia..| = 1 + Z: Z: 0 a. i ... a. .
13 j22 -m<i <--~<i.<m 11 2 111j
l J
a. i 0 .
121
a. . ..... 0
Hence:
2 2
2+” a0 1 n a
A<°>=1-€ Z [7+2'ZX—1—1“
V=-OD AV 3:1 VAV+j
E4 +m m (4a§"ai)2 n n (a at .-2aLaO)
*6 {Z 22 +312 J _ 2 1+
v=—oo i=1 2AD 3:1 L=l+j ABD

101

102

 

 

 

2
n n n (aa .-aa .)
+Z[X Z kL-J Lk-J 1+
j=1 ‘k=2 £=l+k ABCD
n 0° on (a.a!’k—akaJ6 .)2
+X[Z Z J—ABCD -3 +
j=1 'k=2 t=l+k
(aa aa )2
' -k'- k-' 1 6
+ J£A3c6311+0(6)
where
Av=a2- (0+v)2
A = Av—l
B =ij-l
C =A\»+k--1
D = A

Vtt-l

APPENDIX C

THE COEFFICIENTS Kl AND K2

The coefficients K1 and K2 (see section 2.2,

equation (2.2.10)) are computed by using the following

1 ,r a£35.[(o+v—a)213(<5)]g=—v+a

79
ll

7C
ll

n2[(o+-v-a)2A(O)]g=-v+a

In the expansion of A(O) (see Appendix B) we observe

that terms of the form

2

[a - (O+V+p)2]o P=112I...

are raised at most to the second power. Thus the term

(2a-—p) will appear in the denominator of K1 raised at

rnost to the third power and in the denominator K2 raised
at.most to the second power. These facts enable us to

arrive more easily at equation (2.3.3).

103

BI BL IOGRAPHY

10.

ll.

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Fleckenstein, J 0., Ueber eine verallgemeinerte
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Hamer, K. and Smith, M.R., Stability of General Hill's
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