M

This is to certify that the
thesis entitled

VOLTERRA-LOTKA SYSTEMS
WITH SMALL PERTURBATIONS

presented by

Mr. Larry Gordon Blaine

has been accepted towards fulfillment
of the requirements for

Ph o D a degree in Mathematics

{LN/ml, LLJ

Major professor

Shui-Nee Chow

 

0—7 839

 

 

 

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VOLTERRA-LOTKA SYSTEMS
WITH SMALL PERTURBATIONS

BY

Larry Gordon Blaine

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1985

ABSTRACT

VOLTERRA-LOTKA SYSTEMS
WITH SMALL PERTURBATIONS

BY
Larry Gordon Blaine

The subject of this dissertation is the behavior of
solutions of the classical Volterra-Lotka system of
differential equations under certain small perturbations.

The first four sections of Chapter I deal with the
question of subharmonics: given a T/k-periodic
perturbation depending on a parameter u. do there exist
T—periodic solutions near the unique T-periodic solution of
the unperturbed system? In Sections 1 and 2 this problem
is discussed in an abstract context and certain bifurcation
manifolds in u-space are described. In Section 3 this
theory is applied to the Vblterra Lotka system. Section 4
is devoted to examples and numerical calculation.

In Section 5 of Chapter 1 we show that if the
perturbation consists of damping only, then, under certain
‘mild hypotheses, all solutions approach the unique critical
point. In particular, no nontrivial periodic solutions

exist.

Larry Gordon Blaine

Chapter II concerns systems with a perturbation
consisting of linear damping and linear forcing in each
term. using the resonance overlap criterion of Chirikov,
we offer a qualitative description of the “transition to
chaos” of solutions as the forcing parameters are

increased.

TO THE MEMORY OF MY FATHER

ii

ACKNOWLEDGMENTS

I would like to thank Professor Shui-Nee Chow for much

help and encouragement, and for suggesting the topic of

this dissertation.

iii

INTRODUCTION .

TABLE OF CONTENTS

0 O O O O O O O O O O O O O O O O 0

CHAPTER I: ON THE EXISTENCE OF SUBHARMDNICS . . .

Section 1.

Section 2.

Section 3.

Section 4.

Section 5.

Bifurcation Equations for
Periodically Forced Two Dimensional
Systems . . . . . . . . . . . . . .

Analysis of the Bifurcation
Equations . . . . . . . . . . . . .

Application to the Volterra-Lotka
System . . . . . . . . . . . . . .

Examples . . . . . . . . . . . . .

Predator-Prey Systems with
Damping . . . . . . . . . . . . . .

CHAPTER II: RESONANCE . . . . . . . . . . . . . .

Section 1.

Section 2.

Section 3.

Section 4.

BIBLIOGRAPHY .

Preliminaries . . . . . . . . . . .

Formulation of the Problem in
Action-Angle Coordinates . . . . .

Resonance: a Formal Calculation. .

Application of Resonance Overlap. .

O O O O O O O O O O O O O O O O O 0

iv

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25
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41

54

54

61
65

71

77

Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure

Figure

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

LIST OF

FIGURES

Page

16
17
19
20
36
40
43
47
49
52
53
59
60
68
71
73
75

INTRODUCT I ON

In the broadest terms, a predator-prey system is a
two-dimensional first-order system of ordinary differential
equations with the first quadrant (or the closure of the
first quadrant) as the phase space, satisfying the

conditions

(i) x (the 'prey") tends to decrease with

increas ing y.

(ii) y (the 'predator") tends to increase with

increasing x.

Perhaps the best-known example, and one of the first to be
studied, is

x = (A - By)x
(1)

5r - (Cx - D)y .
where A, B, C, D are positive constants. Equations (1)
are known as the Volterra-Lotka system, after two
scientists who studied them in the 1920's (See Volterra
[l], Lotka [1].) This system was formulated as a model for
the fluctuations in population of two interacting species,
with the following simple interpretation: x is the prey

population, y is the predator population, and the number

of encounters between predator and prey is jointly

proportional to x and y. We have:

(i) In the absence of predators, the population of

prey increases exponentially.

(ii) In the absence of prey, the population of

predators decreases exponentially.

(iii) The population of prey tends to decrease in joint

proportion to the number of encounters.

(iv) The population of predators tends to increase in

joint proportion to the number of encounters.

In fact Volterra proposed (1) in an attempt to explain
population cycles of certain species of fish in the
Adriatic Sea.

We can summarize the phase portrait of [l] as follows:
there is a unique critical point in the first quadrant, at
(D/C, A/B). All other solutions in the first quadrant are
periodic. The limiting period at (D/C, A/B) is Zn/JKD.
The period increases as the initial condition moves away
from (D/C, A/B) along any ray. (This last was proved
only very recently).

From the point of view of applications, [1] has serious
drawbacks. Initial conditions, no matter how extreme, are
reattained infinitely’ often, .Also, it is structurally
unstable: the slightest perturbation can drastically alter
the phase portrait. For these reasons, we investigate

systems of form [1], with small perturbations added. These

3

perturbations are often combinations of damping (large
populations tend to decrease, in some sense) and periodic
timm-dependent forcing (changes in population are affected
by some factor independent of sizes of population, such as
seasons or reproductive cycles).

A question of great interest is the existence or
non-existence of periodic solutions of the perturbed
system, In this thesis we will consider mainly the
question of ‘the existence of subharmonics, namely
T-periodic solutions of_ perturbed systems *with
T/k-periodic forcing where k is a positive integer.

We now outline the organization of the thesis, Chapter
One deals with the existence of subharmonics locally; i.e.
near solutions of the unperturbed system with the same
period. The first two sections discuss a general scheme
for finding subharmonics of two-dimensional equations of

form

(2) 0 = fW) + g(t,u,u)

where g is T/k-periodic in t, n. 612'“ is a small
parameter, and g(t,w,0) a 0. The treatment follows
closely that in Chow and Hale [1]. Our hypotheses are more
general however and our conclusions more detailed. Section

3 applies the theory to the Volterra-Lotka system: i.e.

(A ’ BY)!
(3) f(XIY) ‘ .

(Cx - D))'
In Section 4 we look at some specific perturbations
g(t,x,y,u) and do some numerical computations.

Section 5 deals with systems with damping only. The
conclusion here is that under quite general assumptions,
there are no periodic solutions at all; all solutions tend
to the critical point. Some comment is in order here. In
mechanical. systems (e.g. a. two-dimensional. harmonic
oscillator) “damping“ generally pushes against motion,
toward the equilibrium. It is easy to see the effect such
a perturbation has on the Volterra-Lotka system:
construction of a. ldapunov function shows that all
solutions tend to the critical point. But for systems
whose damping tends to push everything toward the origin,
the conclusion is not at all clear a-priori.

In Chapter Two we take a more global point of view.
Given a Volterra-Lotka system with periodic forcing
depending on a parameter u 6 R3, we expect (on the basis
of previous experience and numerical evidence) that as In!
increases past a certain threshold, we experience a
”transition to global stochasticity', i.e. chaotic
solutions appear whidh wander throughout large regions of
the phase space. Various schemes have been proposed to

characterize this transition (none with complete success

however) and we apply one of them, the so-called resonance

overlap criterion of Chirikov to our equations.

CHAPTER I

ON THE EXISTENCE OF SUBHARMONICS

Section 1. Bifurcation Equations for Periodically Forced
Two Dimensional Systems

Consider the systems

(1.1) a} = f(x)
and
(1.2) 5: = for) + g(t.X.u) .

where fuzz -. R3 is smooth, gm x R3 x V a R2 is smooth,
V is an open neighborhood of 0 in R‘“, g(t,x,0) = 0
for all t and x, and g is periodic with least period
T/k in t, for all x and u. (T > 0 and k is a
positive integer.) Suppose that system (1.1) has a
solution p of least period T. Given small u. e V, we
would like to characterize the T-per iodic solutions of
(1.2), if any, which lie near p. These solutions are

called subharmonics of order k.

Note 1.1. In applications it may be necessary to
consider systems of form (1.1) and (1.2) in which

f : U -» '23, gm x U x V 4 R2, where U is some open

7
subset of R2. The theory that follows requires no

essential modification in this case.

We begin by considering the variational equation to

(1.1) around p:
(1.3) 9 = Art)y .

Here A(t) is the derivative Df(p(t)). It is evident

that p is a solution of (1.3); we make the hypothesis

(Hl) All T-periodic solutions of (1.3) are scalar

multiples of p.
Now consider the adjoint equation to (1.3):
' T
(1.4) y = -A.(t)y .

(The superscript T denotes transposition). Let Q be a
nontrivial T-periodic solution of (1.4) for which

T
(1.5) [0 IQ(s)l2 ds = 1 .

Here I I denotes the usual Euclidean norm. Since (1.4)
has only a one-parameter family of T-periodic solutions,

(1.5) determines Q uniquely.

Definition 1.2. Let h be the smooth, T/k — periodic
function

T
h(a) - J0 Q(s) - g5 (s — a.p(s).0)ds .

We shall see that the properties of h are crucial in

determining the existence or nonexistence of subharmonics.

Lemma 1.3. Let a e R. Suppose the functions x and

z from R to R2 are related by

(1.6) x(t) = p(t + a) + z(t + a) .
Then x satisfies (1.2) if and only if
(1.7) é(t) = A<t>z + P<t.z.n.a) .

where

(1.8) F(t,z,u,a) = f(z + p(t)) - f(p(t)) - A(t)z

+ 8(t - a'2 + P(t)rfl) 0

Proof: trivial.

Note that if x and z are related by (1.6), then x
is T-periodic if and only if z is T-periodic. Thus the
problem of finding subharmonics of (1.2) whose graphs lie
near the graph of p is equivalent to that of finding
small T-periodic solutions of (1.7)

We now explain the role of the parameter a. Denote
the graph of p by I‘. Construct a transversal line L
to I' at p(O), orthogonal at p(O) to p(O). (See
Figure (1.1)). If x is a T-periodic solution of (1.2)
whose graph lies close to I‘, an elementary continuous
dependence argument shows that there is a unique a e [0,T)
for which x(-a) lies on L. (Just start at x(O) and

“follow x backwards in time" until x crosses L.)

Next we note that if x is a T-periodic solution of

(1.2), then so are the functions x(t + mT/k), for

m - l,2,~-,k—l, and these are distinct. (They have the
same graph in the plane however.) Thus to find all
T-periodic solutions of (1.2) near r, it suffices to find
those for which a e [0,T/k). This in turn is equivalent
to finding small solutions 2 of the system (1.7) for
'which 2(0) is orthogonal to inc).

 

 

 

Figure 1.

10

Definition 1.4. PT is the Banach space of continuous,

T-periodic functions R -9 R2, with norm

 

l(x.y)lT = sup¢x2(t) + Y7(ti .
Definition 1.5. P is the projection of PT onto the

span of Q; i.e.

T
Pb = [I0 Q(s) . b(s)ds Q .

(It is easy to see that this actually is a projection.)

Now consider the equation
(1.8) é = A(t)z + b(t) ,

where b 6 PT. It follows from the Fredholm alternative
theorem that (1.8) has a solution in PT if and only if
Pb 8 0, in which case it has a one—parameter family of

them.

Lemma 1.6. Given b e ker P, there is exactly one
solution 2 of (1.8) for which 2 6 PT and

2(0) - p(O) = 0.

Proof: If Pb is any solution of (1.8) in PT, then
all such are of form cp + ¢br where c e R. In order

that z(O) - p(O) = 0, we must have

9(0) - ¢b(0)
lp(0)l2

Ca

 

Definition 1.7. If b and z are associated as in

Lema 1.6, we write 2 = Kb. Thus K is a map from ker P

to PT.

11

The following may be found in Hale [1].
Lemma 1.8. K is bounded and linear.

we can summarize at this point by saying that (1.7) has
a solution 2 6 PT with z(O) - p(O) - 0 if and only if

2 satisfies the following:

a) z - K o (I - P)F(°.z,u.a) = 0
(1.9)
b) PF(-,z,u,a) = 0 .

Definition 1.9. Let F be a map PT x U x R 4 PT
defined by

F(zrura) = Z - K ° (1 " P)F(’:Z'ura) 0

Note 1.10. PT x Rm x R is a Banach space if we impose

the norm

l(z,u,a)l = IzlT + lul + lal .

For any a e R we have F(0,0,;) = 0 and
ar/az(o,o,2) - Id. (The partial derivative is a Frechet
derivative of course: recall that Frechet derivatives pass
through linear operators, and note that
F(-,z,0,3) e 0(IzIT3)). Thus we can apply the Implicit
Function Theorem and deduce the existence of a function
z*(u,a) 6 PT, defined for small u, and a near 3, for
which z*(0,E) = 0 and such that F(z,u,a) = 0 if and
only if z a z*(u,a). Using the fact that F is periodic

in a and an elementary compactness argument, we get a

12

function 2*(u,a) defined for all a e R, and small u,
for which F(z,u,a) - 0 if and only if z - 2*(u,a), and

such that z*(0,a) - 0 for all a.

Now our problem has been reduced to that of finding

solutions (a,u) of (1.9-b) with z - z*(a,u); i.e.

T
(1.10) I Q(8) - P(s.z*<a.u)(s).u.a)ds - o .
0

We can put (1.10) in a more tractable form. First expand

g(',p 4- 2,11.) around (-,p,0) and use the fact that
8(t'X.0) = 0 to get

(1.11) 8(°,p + z.u) - [gg ( .p.0)] - u + 0(|u|2) .

Since p, z* e PT we get a uniform estimate

(1.12) 3(8 - a.2*(u.a)(8) + p(s).u) =

.. [g3 (s - a,p(s),0)] - u + oumz) .

Also f(p + z*) - f(p) - A(-)z* e O(|z*l2), and
2*(a,u) e 0(lul) uniformly in a. Combining the above
estimates and recalling definition (1.9), we get

F(5.z*(a.u).u.a) = [g% (s - a.p(5).0)] ° u + 0(Iulz)

I

uniformly in s and a, and (1.10) becomes
(1.13) n . h(a) + é(a.u) = o .

where B e 0(lul2), uniformly in a. It is convenient to

make the definition

13

2301.63): if C 7‘ 0

(1.14) é<a.a.c) =
0, if C = 0

and we consider the equation

(1.15) a - h(a) + Emma) ——- C(a.a.c) =- o .

It is easy to see that there is a one to one correspondence
between the solutions (mu), u. f 0 of (1.13) and pairs
of solutions (man), (a,-B,-() with IBI - 1, C i 0 of

(1.15). The next section is devoted to an analysis of

(1.15).

14

Section 2. Analysis of the Bifurcation Equation

We begin our analysis of equation (1.15) by considering
the simplest possible case, namely m = 1. We make the
assumption that h does not vanish except possibly at a
finite number of points «1 -:~- an, and that h'(aj) f 0
for j - 1,2,"-,n.

If h never vanishes, then it is easy to see that
(1.15) has no solutions, for C sufficiently small.

Now suppose that h(;) - 0. Then for 3 = :1 we have
C(Z,’E,0) - o, and aC/aa('&,§,0) - E - h'di) y o. By the
implicit function theorem, there is a function a*,
defined near (3.0) for which c(a.5.c) = 0 if and only
if a - a*(B,C) (locally). In particular we restrict 3
to be :1, and letting u. = (B we see that for
sufficiently small u, there is a unique a near 2 such
that (a,n) satisfies (1.13).

Applying the above argument to each a1, and noting
that 11 .must be an even number, we conclude that for all
sufficiently small 11., there exist exactly n a's in
[0,T/k) for which (mu) satisfies (1.13). It follows
that for each u sufficiently small, there are exactly nk
subharmonics of (1.2).

Next suppose that m - 2 and that {h(a),h'(a)} is an
independent set for all a. (Geometrically this means that

h traces out a curve around the origin.)

15

Theorem 2.1. Under the above hypotheses, there exists
a small neighborhood N of the origin in R2 such that
for all u e N ‘ {0}, equation (1.2) has precisely
Zkllndh(0)l subharmonics near p. (Indh(0) is the number
of times h(a) goes around the origin for a e [0,T/k].
To be precise, regard h as a function into C and define

Indh(0) to be the index of 0 with respect to path h.)

‘Proof: We outline a proof for the case Indh(0) = 1;

 

the necessary modifications for Indh(0) f 1 are almost
obvious. Let E e S'. Then there exist exactly two 3‘s
in [0,T/k) for which h-h(3) a 0. Single out one of them
by rotating 2 90° clockwise and projecting to the graph of
h. (See Figure 1) We have then 05.3.0) - 0,
aC/aa(;,§,0) f 0, and the implicit function theorem gives
us neighborhoods U1 of 3 (on 8'), U, of E, and J
of 0, and a smooth function a*: U1 x J a U; such that
«(3,0) - E and C(a,B,C) - 0 (a e 0,: a e 01, c c J) if

and only if a - a*(£,c).

Now, since h is T/k-periodic, we identify
[0,T/k) with a circle. Apply the above analysis to every
pair (3,?) and use compactness arguments to arrive at the
conclusion that there is a unique function a*:S' x (—0,0)
a [0,T/k) such that C(a.£.C) e 0 with a e [0,T/k), B e

S', C s (-6,6) if and only if a a a*(£,().

\\\\ 1

Figure 2.

 

Now let N be the open disk in R3 with radius 0.
If u e.N ' (0} then there are two elements (3.6) of S'

x (~0,0) for which CB - u, namely i: (u/Iul, u). The

result follows.

Note 2.2. In a later section we will discuss the

system
i - A(1 - y)x + ”1 x sin.wt

y - D(x - 1)y + “2 y sin wt

It turns out that the graph of h is an ellipse around the

origin with index 1.

17
Next we consider the case m.- 2 with hypotheses

(i) h does not vanish

(ii) (h(a),h'(a)) is an independent set for all but
a finite set of a's in [0,T/k). Call these
exceptional points «1, a3, --- an.

(iii) {h(aj),h'(aj)} is independent for
j - 1,2, --° n.

We impose a further hypothesis which is not essential but
which is technically convenient:

(iv) {h(a1),h(a3)) is independent for i f j;
1' j 6 (192'... 11}.

Note that these hypotheses are generic. Before stating a
theorem, we discuss the geometric meaning of (i) - (iv).
Hypothesis (1) is obvious. Hypothesis (ii) says that a ray
from the origin to h(a) will not be tangent to the graph
of h, except in a finite number of cases. Hypothesis
(iii) says that this tangency is non-transversal in all

cases. Hypothesis (iv) excludes behavior like that in
Figure 3.

hm.-
he-

i U

Figure 3.

 

 

18

Theorem 2.3. Under hypotheses (i) - (iv), there is a
small neighborhood N of the origin in R2 (we may take
N to be a disk), and n smooth curves C1 through the

origin with the following properties

(i) The Ci's don't intersect except at the origin.

(ii) C1 is orthogonal at the origin to h(a1).

(iii) As u crosses any of the Ci's away from the
origin the number of subharmonics changes by
precisely 2k. (The number of subharmonics always

even).

Furthermore, 2kllndh(0)l is a lower bound on the number

of subharmonics.

Note 2.4. We easily conclude that n is an even

number. Of course this can be proved directly.

lggggf. First let 2 a aj for some j a 1,2,...,N and
let "it s' such that 3 ~ h('&) - Z - h'(;) - 0. Then
5 - h-(E) ,4 0. Note that aC/aa(E,§,0) - 3 ~ h'(a) - 0,
and 32C/aa(3,3,0) - Z . h'(a) ,« 0. We apply the implicit
function theorem to aC/aa and conclude that there exists
a function a*(B,C) defined near (5,0) such that
a*(§,0) - 3, and aC/aa(a,£,() - 0 with (emit) near
(3,2,0) if and only if a = a*(fi,(). Thus for fixed 3
and c, C(~,£,() has an extremum at a*(B,C). Denote the

value at this point by M(B.C) - C(a*(B,(),£,C). We want

to characterize the zero set 2 of M, locally.

19

we compute (3,0) e z and

$ (3.0) - m3) «- 0.

Thus 2 is a two-dimensional manifold (z is a subset of
S x (-0,0), where 3 is a neighborhood of '3 in R2 and
0 ) 0). The normal to 2 at (3.0) is (h(;),*) for

some constant *, and it 'follows from elementary

transversality arguments that

21 - ((3,;) e z | IBI - l)

is a one-dimensional manifold, and that

ze - {(3.6) e 21 l c - c)

is a zero—dimensional manifold (i.e., a point) for any
c s (-6,6).

Now ‘we can conclude that there exists a function
3*: (-0,0) e s1 such that 3*(0) - 2. and (s,() e 21 if
and only if B - 3*(C)-

One possible situation is shown in Figure 4 below.

 

    

m

Figure 4.

 

 

 

 

 

20

‘We imagine that the extremmm.at a* is a minimum, and that
M is negative in the shaded region, zero on the curve
(graph of fi*) through 3, and positive above this curve.
('above" and 'below" depend of course on how we
parameterize S'.) If M is positive, then locally there
are no solutions a of C - 0; ‘ if M is negative there
are two.

In terms of our old variables (u,a), we have a curve
u(c) - 63"“): as u. crosses this curve away from the
origin the number of solutions of (1.13) changes by two.
Note that aim) -s*(0) ii, so u“) is orthogonal at the
origin to h(3). (See Figure 5).

#1

 

fl.

 

Figure 5.

21

Now consider 3 e [0,T/k) such that E f aj for every
j e {1,2,...,n}. Choose 3 e S' as before such that

s - hdr') - 0. Then '5 ~ h'('&) ,1 0. Since C(Z,Z,0) =
E . h('&) - 0 but aC/aa(3,'§,0) =- E . h'(a) ,4 0. Apply the
Implicit Function Theorem to get a function a*(£,()
defined near (3,0), a*(§,0) = 3, such that locally
C(a,£,£) - 0 if and only if a = a*(B,C). In terms of our
original variables (a,u), we have the following: for all
u in a small double wedge, excluding the origin, there is
precisely one a near 3 for which (am) satisfies
' equation (1.13).

Now if we piece together these results in a

straightforward way, we complete the proof.

We now look at the situation when m = 3. First supose
that {h(a),h'(a),h'(a)} is independent for all a. (This
hypothesis is not generic!) Given 3, there exists 3 e 83
such that E-hG) = 0. (There is a circle of such 3'3
actually.) Either Z-h'm) ,- 0 or §~h'(E) = 0. This
latter occurs for exactly two E‘s antipodal on S2 for
any given 3, and in this case our hypothesis implies that
33h'(3) # 0.

If E-hGi) = 0 and Zoh'Ci) a 0, the implicit
Function Theorem implies the existence of a function
a*(£,€), defined near (3,0), such that C(a,B.€) = 0 if
and only if a - a*(£,£). Of course a*(§,0) = E. To look

at it another way: in a small neighborhood of (3,0), the

22

number of solutions a of C(°,£,() near 3 does not
change.

If E-h(E) - Zoh'(3) - 0, then our assumption implies
that §~h'(3) it 0. As in the case m - 2, we apply the

Implicit Function Theorem to g-g at (3,3,0) to get (1*
such that C(°,B,C) has an extremum at a*(£,(), and
define M(£,C) -= C(a*(B,(),B,(). Arguing just as before
(the manifolds involved are one dimension higher) we find
that for any fixed, small c, the set {(£,c)IM(B,c) - 0,
IBI - 1} is a one-dimensional manifold.

Translating back to (mu) co-ordinates, we find that
there is a two-dimensional manifold defined near the origin
and consisting of two components (corresponding to c
positive and c negative). As this manifold is crossed,
the number of solutions near 3 of u-h(°) + B(-,u) - 0
changes by two. The ”limiting normal“ at the origin is in
the direction of h(E).

Now, as a varies through one period of h, the
points 3 for which B-h(a) - B-h'(a) - 0, Isl - 1,
trace out a pair of (antipodal) closed curves on 83.
Apply the previous analysis to each point on these curves,
and use compactness to conclude that in a small
neighborhood of the origin, excluding the origin, there is
a two-dimensional smooth manifold (actually an “immersed

manifold“ since it may cross itself) which is tangent at

the origin to the cone (in the obvious sense) defined by

23

the antipodal curves. As this manifold is crossed the
number of solutions a changes by 2k.

If we relax the hypothesis that {h(a),h'(a),h'(a)) be
independent and replace it with the generic hypotheses

(i) (h(a),h'(a)} independent for all a

(ii) {h(a),h'(a),h'(a)) is independent except for a

finite number of «'8, say a1,-°-,an,

then it is straightforward to show that the bifurcation
surface may no longer be smooth: it may have singularities
corresponding to the ai's. It is possible to describe
these singularities under various further hypotheses on the
derivatives of h, but such a study would take us too far

afield here.

If m h 4 the situation becomes quite complicated and
we give only a.1ocal description of the bifurcation sets.
‘We assume that {h(a),h'(a),h'(a)} is independent for all

a. (This hypothesis is generic.) Given a, let

w1('a') - sm-Jl n {h(5)}"
W’G) = 3"H n {11(52).h'(3)}JL

w3('&) = s"H n (h(E),h'(E),h'(E)-L.

The dimensions of W1, W3, W3 are m - 2, m - 3, and
Ini- 4, respectively.
For E 6 W1(;) but ,8 t WzCi), we use the Implicit

lEunction Theorem as before to show that there is a function

24

a* such that C(a,B.C) = 0 for (3,6) near (5,0) if
and only if a - *(s,c). If E 6 wide?) but 3 g w3('&),
we proceed as before to construct an m - 1 dimensional
manifold (in u—space) near the origin of Rm; as we cross
this manifold, the number of solutions (I of (1.13) near
a changes by 2k. The points 2 5 W303) correspond to

singularities of the bifurcation sets.

Note 2.5: In many applications, the graph of ti lies
in a cone. In this case, it is easy to describe a region

of u—space in which there are no solutions of (1.13).

25

Section 3. Application to the Volterra-Lotka System

We want to apply the theory of the previous sections to

the equations

i - (A - By)x
(3.1)

y - (Cx - D)y .

Here A, B, C, D are positive constants. We are
interested in the behavior of this system in the first
quadrant. There is a unique critical point (D/C, A/B) in
the first quadrant and all solutions there are periodic.
(See for example Hirsch and Smale [1]).

We begin by making the trivial change of coordinates

] x
]y.

UH)

x—~[

>Hn

y.__,[
and then (3.1) becomes

5: - A(l — y)x
(3.2)

y - D(x - l)y ,

which is the equation we discuss throughout.

Let (p,q) be some fixed T-periodic solution of
(3.2). (This is a slight change of notation). We write

the perturbed system in form

i - A(l - y)x + gl(t:x,y;u)
(3.3)

it - D(x —- l)y + 82(t3x.y:u) .

26

“liil

satisfies the hypotheses of Section 1.

where

An easy computation shows that

A(1 - q(t)) -Ap(t)
(3-4) Mt) "

DGU?) D(p(t) — 1) .

Our task is to show that (H) is satisfied, i.e. that
(p q) is the only' T-periodic solution (up to scalar
multiplication) of the variational equation

(3.5) s = A(t)z .
First we introduce some notation.
Qgginition 3.1. Let p(t,a) be the value at time t

of the solution of (3.2) with initial value (a,1). (We

assume a a 1.) Componentwise we write

‘1
.-[ 1.
‘2
Let 1(a) be the period of ¢(-,a). Let (ao,l) be the

initial value of (p,q). Let E - D(ao - 1).

ILemma 3gg. A fundamental matrix for (3.5) is

27

¢ (t.a0) R(t)/E

¢(t) =

l
a
2
‘3

(t.a0) é(t)/E .
(The subscripts denote partial differentiation.)
Furthermore, 0(0) is the identity and the first column of

O is not T—periodic.

Note 3.3. It follows that (p,q) spans the space of
T-periodic solutions of (3.5).

Proof. It is easy to verify that the second column of

0 has the right properties.

Let us temporarily* denote the vector field on the
right—hand side of (3.1) by f(x,y). We have then for all

a h l
¢(t.a) ‘ f(¢(t.a)) -

Differentiating this with respect to a gives

¢a(t.a) - [Df(¢(t.a))1¢a(t.a)

If we let a - an we see that ¢a(:,ao) is indeed a

solution of (3.5), since

P
¢('rao) ‘ [ l .
q

Next we note that since

a
¢(Ora) a [ l i
1

28

we have

[1]
¢ (Ola) a 3
a 0

and in particular we can let a - an. This shows that O
is a fundamental matrix and that O(0) - Identity.

To see that ¢a(-,ao) is not T-periodic we note that
for all a > 1:

a
¢(1(a),a) = .
l
Differentiation with respect to a yields

1
¢<r(a).a) ~ ‘r'(a) + ¢a(r(a).a) -- l 0 I

‘Let a - a0 and get

1
¢ (Tia ) a
a 0 [ -ET'(a0) I .

It is known (see Waldvogel [2]) that 1"(a) ) 0 for

all a > 1, and the result follows.

We compute next the corresponding fundamental matrix

Q - O‘T for the adjoint equation to (3.5). Easily

 

5: -E ¢§
(3.5) t - .1 .1 2 .
‘3 . q — '3 p _b E ‘1 '

 

29

and it follows that the space of T-periodic solutions of

this adjoint equation is spanned by

(3-8)

 

Fortunately this can be simplified. Since

a: q - a: p a E det o ,

Liouville's formula gives us

t
—] tr A(s)ds
e 0

 

1 a}.
*I . 2 - E
:a q - fia p t
1 -I [A - Aq(s) + Dp(s) — D]ds
E
t . .
l —I {p/p + q/qi
- _ e o
s
=32 [W1 J
E p q '
so (3.8) becomes
an é/pq a0 D(l - g)
"E’ . ‘75? 1
’P/Pq A(l-a) .

Definition 3.4. The positive number n is defined by

the relationship

T
[0 Mail2 68 = 1.

30

 

 

where
° 1
q/pq D(l - —)
Q = .1; s .1; p
W , n l
’P/Pq A(1 ’ a)
Now we have
(3 9) Ma) = 1 IT (D(l - 1 ) Ml — -1 ))~
’ n 0 5735' ' quT
1

2%.. (s - a,p(s).q(8):0)

2
gg— (8 - a.p(s).q(s).0) ‘

 

 

ds.

31
Section 4. Examples
Example 4.1. Consider the system

x - A(1 - y)x + ulxz + uzx sin wt
(4.1)
9 - D(x - l)y

which represents a. predator-prey interaction.‘with small
damping and small periodic forcing confined to one species.
This system has proved to be of interest in recent studies
of population dynamics (see Blom, et. a1. [1] and
references therein). Here w is a fixed constant,

ulx2 + uzx sin wt

8(t:X:Y:fl1:u2) a r
0

and so

p2(s) p(s) sin w(s — a)

(5 ’ a.p(s),q(s),0,0) ' .

:16:

Thus

h(a) = % I: [A[1 - 5%37] , D[l - fi%§)]] .

92(8) p(s) sin w(s - a)
. ds
0 0

T

I

 

2 T
(p (a) — p(8))ds . I A p(s) sin w(8 — a)ds
O 0

32

Lem 4.2. The first component of h is strictly

positive.

Proof. From the Cauchy-Schwarz inequality we have
T 2 T
(4.2) [I p(s)ds] < T I p2(s)ds
0 0

(Strict inequality, since p is not constant.)

Since (p,q) is a solution of (3.3) we have

so +1
and so
T T .
f0 p(s)ds - % I0 q(:) ds + T .
But
T é(s)
[0 3??? ds = ln(q(k)) — in(q(0)) = 0 .
30
T
(4.3) I p(s)ds - T .
0
Thus (4.2) becomes
T T
(4.4) f0 p(s)ds < f0 p2(s)ds .

and the conclusion follows.

33

Note 4.3. Equation (4.3) is of interest in its own
right; it shows that the mean value of p is precisely l.
The same holds for c; of course. This is independent of

our choice of (p,q).
Definition 424.

T
v =5; (132(8) — p(s))ds .
0

If we simplify the second component of In. *we arrive

at

(4.5) h(&) = (7, Cl sin wa + C2 cos we) ,

where

T
A
C1 = — n I0 p(s) cos we ds

C

T
.A
2 - 5.I0 p(s) sin us ds

we have immediately

h'(a) = w(0, C1 cos wa - C2 sin ma)

and

h'(a) = —w2(0, C1 sin ua + C2 cos ma) .

It is easily seen that B . h(a) - 3 ° h'(a) = 0 is

possible only when h'(a) - 0, i.e. when

 

i‘lij“

34

nn
(4)

a = g;- Arctan [:1] +
2
for some integer n. It is also evident that if
B - h(a) - 0, then B : h'(a) f 0.

We conclude that there is a small disk around the
origin in the (unuz) plane and two bifurcation curves
passing through the origin and perpendicular at the origin
to h(a1) and h(a2) respectively, where

C
1 l

and

1

a- Z
2 0:

C1
a Arctan E— + w .
2
These curves divide the disk into two regions. If (u1,u2)
lies in one of these regions, there are precisely 2k
subharmonic solutions of (4.1), and no subharmonics if
(111,143) is in the other region. (If we are discussing
subharmonics of order k, we have u :- Zak/T).

Ifwechoose T-8, A-D-l, and w=n/2, we

find the following (all to four—place accuracy):

35

8 2
f (p (s) - p(s))ds = 11.1003
0

n = 20.6143
8 178
p(s) sin —-2- ds = —1.1028
0
8 178
I P(8) cos ’2 ds = 2.8635 .
0

It follows that

7 = .5385
C1 = -.1389

C2 a -.0535 .

And so we have

h(a) - (.5385, -.l389 sin 1; -.0535 cos 1;) .

The maximum (minimum) of the second component of h is
c.1489. Thus the tangents to the bifurcation curves meet
at the origin at an angle of approximately 31'. Referring
to Figure 6, we see that if (111,142) lies in the shaded
region, there exist 4 subharmonics, of order 2, and if

(u1,u2) lies in the other region, there are none.

Note 4.5. We see that for u; - u, f 0, there are no
subharmonics. This could be shown directly by considering

the perturbation

36

 

 

M

 

 

Figure 6.

uxz + ux sin 7-,;-

s(t:x.y:u) - ,
0

where u e R. We compute
1m 1m
Ma) .5385 .1389 sin .-2 .0535 cos “—2 .

It is easy to see that h never vanishes. Now we apply
the argument at the beginning of Section 2 and the result

follows.

37

Note 4.6. It is evident that the technique of this
example can be extended. For example, if our perturbation

is of form

z(t.X.y1.ul.u2)

where f is a T/k periodic function of mean value zero,

then we find that

7
h(a) = .

T
1 -
5 I0 p(s)f(s a)ds .

 

 

L

In particular, f might be a trigonometric polynomial (a
truncated Fourier series of some other T/k periodic

function for example), in which case we will have

h(a) =

 

7
T(a) ] '
where T is a trigonometric polynomial.
Example 4:1: Let us consider the system
x = A(1 - y)x + ulx sin wt

(4.6)
y - D(x - l)y + uzy sin wt

Which represents a predator-prey interaction with small

Eneriodic forcing in each variable. we compute

38

Ma) - % I: [All — SIB] , D[1 - 3713‘)” .

p(s) sin w(s — a) 0

0 q(s) sin w(s - a)
T T
= i [I Ap(s) sin w(s — a)ds, I Dq(s) sin w(s - a)ds]
7’ 0 0
= (Cl sin wa + C2 cos ma, C3 sin wa + C4 cos wa) ,
where C1 and C3 are as in the previous example, and

T
_ _ D
C3 - i-IO q(s) cos we ds

T
D .
5 IO q(s) sin ws ds .

‘We can write h more concisely:

(4.7) h(a) = (A1 sin(wa + b1), A2 sin(wa + b2))
where
_ 2 2
Al - «C1 + C2
2""?
.A2 -1/C3 + C4
C
2
tan b - ——
1 C1
C
4
tan b2 6-3- 0

As long as b1 7‘ b; + nn for any integer n, h describes

aux ellipse around the origin. we conclude that there is a

39

small disk U around the origin in the (111,112) plans
such that if (111,112) e U and (111,112) ,1 (0,0), then
system (4.6) has 2K subharmonics.

The graph of h(a) for T - 8, A s D = 1, and w =

n/Z is shown in Figure 7.

4O

 

 

 

Figure 7.

41

Section 5. Predator—Prey Systems with Damping

In this section we discuss Volterra-Lotka systems with
damping only, and indicate how the results obtained can be
extended to more general predator-prey systems. The main

result is

Theorem 5 . 1 . Let f and g be smooth funct ions
[0,m) a [0,«) such that

(1) f(0) = 3(0) = 0
(ii) f'(x) > 0 and g'(x) ) 0 whenever x > 0

(1111);), f(x) - I33 g(x) = ..,

Then if e and 7) are positive and sufficiently small,

the system
i = A(l — y)x - exf(x)
(5.1)
5' = D(x - l)y - nysw)
has a unique critical point (xr,ya) in the first

quadrant, and for every solution p of (3.2) with initial

condition in the first quadrant, limtqm p(t) = (xs,yt).

Note 5.2. In Robinson [1], the case f(x) = g(x) = x
is considered, and our proof uses extensions of the method
therein. In regard to this problem, see p. 204 of Hirsch

and Smale [1] where it is stated that "It is not easy to

42

determine the basin of [the critical point], nor do we know

whether there are any limit cycles“.

Proof: (1) First we show the existence of the
critical point in the first quadrant. Let F:R4 a R2 be
defined by

(A(1 — y) - ef(x))x
(5-2) F(XoY'€.n) ' .
(D(x - 1) - ”8(Y))Y

0
Then P(1,l,0,0) - [ l and
0

”I _
azx'yj (1.1.0.0) - D 0

The Implicit Function Theorem. now guarantees that for
(e,n) small, there is a unique (x: yr) near (1,1) for
which F(x:,ya,€.n) ' 0. In particular we will assume that

e, n are positive.

(ii) The derivative matrix of the vector field of

(5.1) at (x: ya) is

—ex,f‘(x*) —Axt
(5.3) .
Dy, -ny.s'(y)

It is easy to see that the eigenvalues of this matrix have

negative real parts and thus that (x: ya) is

43
asymptotically’stable.

(iii) We now make some geometrical construction (Refer
to Figure 8). The zero set of [A(1 - y) - ef(x)] in the
first quadrant can be regarded as the graph of a decreasing
function you with 9(0) - 1 and 9m, - 0. where
p(xo) - A/e. We assume that e is so small that xo > 1.
The zero set of [D(x - 1) - ng(y)] in the first quadrant
can be regarded as the graph of an increasing function
5"(r) where §(1) - 0 and 11m,” §(x) - on. Now choose
x3 > xo such that y, - y(x3) > 1. Construct a rectangle
R with corners at (0,0), (0,y,), (x, y,) and (x,,0).
Construct a line L - ((r.y.) I Xe < x}. It is easily seen

that ‘L is transversal.

7
y.

 

 

 

 

.— >2: T x. 'x. X

 

Figure 8.

44

(iv) It is easy to show that R is positively
invariant and that every solution of (5.1) with initial
condition in the first quadrant mmst eventually enter It
It follows from the Poincare-Bendixson theorem that the
w—limit set of such a solution must be either (Xt,yx) or
a limit cycle inside R. (Elementary arguments show that
the w-limit set cannot be the critical point at (xo,0) or
at (0,0).)

(v) Suppose that there is a limit cycle, intersecting
L at the point (i.y.). Then by continuous dependence, we
can define a Poincare map n : L a L, where L is a small
neighborhood of (i.y.) in L, say L =
{(x,yu)l Ix - in < O}. In what follows we regard n as a
map from (i — b, i + O) to (xs,o), and don't bother to

write the second co-ordinate ya.

(vi) We claim that in fact II is well-defined on
(i — Ohm). The proof is elementary: just start at

(x,yx), with i < x, and follow the flow around.

(vii) We now recall that the derivative of the
Poincare map is given by the formula

(5.4) fl'(x) -

[D(x-1) - fl8(Y*)]Y*
ID(U(X)-1) - 08(Y*)IY*

 

1(X)

l I.

where F is given by (5.2), p is the flow for (5.1) and

div F(¢(8.X.y*))ds ] .

f(x) is the ”first return time“; i.e. the time it takes a

45

solution with initial condition (x,yn) to return to In

(See Robinson [1], or Andronov, et. a1. [1] for a more

general formula.)

Let
M111?) = A(1 - Y) - ef(X)
and let
N(x.y) = D(x - 1) - ns(y) .
We compute

div F(x.y) - MIX.y) + N(x.y) - (exf'(x) + nys'(y)) .

Along solutions this is

- [exf'(x) + nys'(y)] .

VF<°

div F(x,y) - ; +

Note that the expression in brackets is strictly positive.

Integrating, we find

f(x)

I div P(¢(a:x.y*))ds =
0
we we _
= in x(s) - in x(a) - u .
O 0

where i > 0. But y(r(x)) = y(0) - ya (we are assuming

an initial condition on L), so we find that

T(x) -”
exp div F(¢(s;x,y*))ds a [Eéil - e “ < §§§l .
0

It follows that

46

“(1‘) [N(X;‘Y*)]

(5‘5) n'm ‘ [h(n(x).y;)

for all x e (i - 0. on).
Let h(s) - N(s,ya)/s, for all s e (i - 0.»). (Note
that h > 0. Of course we assume xx < i - 0). Let H be

an anti-derivative of h.

(viii) From (5.5) it follows that
(5.6) h(II(8))II'(6) < h(s)

for all s e (i - O,~).

(ix) Suppose that i < x, and integrate (5.6) from i

to x. we get

H(n(x)) - H(n(i)) < H(x) — H(i) .

But D(x) - i. and H is strictly increasing, so we

conclude that
(5-7) D(x) < x .

It follows from this that there can be gt mpg; one limit
cycle. For if there were two, we could apply the above
argument to the inner one to get a contradiction. (See
Figure 9).

Figure 9 cannot be correct because we have shown that

D(x') ( x'.

47

 

 

.=.. r,
>< X

 

Figure 9.

(x) Now suppose that i — O < x < i. and integrate

(5.6) from x to i. The result is

H(H(i)) - H(n(x)) < H(i) — H(x): i.e.
H(x) < H(n(X))3 i.e.

(5.3) x < I'I(x) .

Now (5.7) and (5.8) together imply that the limit cycle we
have assumed in (v) must be stable. Thus we have a unique,
stable limit cycle surrounding a unique, asymptotically
stable critical point. This is a contradiction, and we
conclude that there is no limit cycle. This proves the

Theorem.

Note 5.4. Our main interest in this thesis is the

Volterra-Lotka system and its perturbations, but it is

48

interesting to note that the arguments above can be applied

to very general systems. Consider

i a x ° M(XIY)
(5-9)
9 = Y ° N(X.y)

We make the following hypotheses:

(H1) M, N are smooth real-valued functions defined
on some open set containing the closure of the first

quadrant: in the first quadrant we have

3N BN

3"

5i ( 0 .

6M
<0, 8(0'

(H1) reflects the predator-prey interaction (x tends to
decrease with increasing y, y tends to increase with
increasing x) and a damping mechanism (x tends to
decrease for increasing x, and y tends to decrease with

increasing y).

(H2) M(0,yo) - M(xo,0) - 0 for some positive x0 and

Y0-

(H3) N(x1,0) .. N(erY2) - 0 for some x1 ( x0 < x,

and Y0 < Yr

(H2) and (H3) are technical hypotheses imposed to guarantee
the existence of a critical point (x*,ya) in the first

quadrant. This critical point is unique.

49

Now, standard arguments using the implicit function
theorem show that the zero set of M in the closure of the
first quadrant is [(x,q(x)) I 0 o x ‘ x0] for some
decreasing smooth function 6;, 51(0) - yo, q(xo) - 0, and
the zero set of N in the closure of the first quadrant is
[(x,p(x)) I x; d x < a) for some increasing smooth
function 5 with p(xl) - 0. p(xz) - y,.

Now construct R and L as before (See Figure 10),

and argue just as before to get the following result:

Theorem 5.6. Suppose hypotheses (H1), (H2), and (H3)
hold. Then for every solution a of (5.9) we have

11%». Nb) " (Xt,y*).

 

)6 <

 

 

 

>'<. X. x x.

 

Figure 10.

50

Note 5.6. If it were known a-priori (for some
equations) that the Poincare map were well-defined on all
of 1L, the foregoing analysis would be considerably
simplified. This is not always the case however. For
example, if the eigenvalues of the linear part of the
vector field at (x:,yr) are real and negative, then
solutions crossing L near (Xt,y*) may never return to
I“

It is instructive at this point to consider some

examples with numerical computations.

Example 5.7. Consider the system

i = (l - y)x - .041:2

(5.10)

9 = (x - l)y - ~04Y2

This system obviously satisfies the hypotheses of
Theorem. 5.1. The critical point (x: ya) is at
(1.04/.9984, .9568/.9984). ‘we have computed a solution
using the IMSL routine DVERK with tolerance set at 10".
The initial condition is (l,e3). We have plotted the
solution over the range t 6 [0,100] at time intervals of

length .05. The results are shown in Figure 11.

Example 5.8. Consider the system

“a
II

(1 — y)x - .07 ln(x + l)
(5.11)
(x - l)y - .07 in(y + l) .

‘<
ll

51

We choose initial condition (9.974, 1.0) and compute as
before over t 5 [0,150]. The results are shown in Figure

12.

52

 

 

C.
to

Figure 11.

3.00

 

 

 

 

 

Figure 12.

CHAPTER I I

RESONANCE

Section 1. Preliminaries

In this chapter we confine our attention to systems of

form

x = A(1 — y)x + “1x2 + uzxf(t)
(1.1)
Sr -- D(x - l)y + u3y2 + u4ys(t)

i.e. systems with damping and forcing in each variable. We
assume that f and g are 211/0 periodic, analytic, and

have mean value 0.

Note 1.1: The assumption that f and g have mean
value 0 causes no loss of generality. For suppose that

f has mean value M1 and g has mean value [4,. Define

f-f-Ml and g-g-Mz, so I? and g havemean

value 0. Then system (1.1) becomes

r - [(A + uZMl) - Ay]x + ulx 2+ uZx-f(t)
(1.2)

2

Y - [Dx - (D - u1M2)]Y + n3y + u4Y§(t) .

54

55

If u, and u‘ are small enough then A + u,M1 and D —
11.14, are positive and we can change variables as in
Chapter 1, Section 3.
For system (1.1) we have
u x2 + u xf(t)
1 2
(1-3) 8(t:X:Y:u) ‘
2
u3y + #4Y8(t)

and so

T
l l l
“W‘s-Io (1'5731'1'61‘1‘51’

p2(a) p(s)f(s - a) o o

2 ds
0 0 q (a) q(a)8(s - a)

1 T T
= 5171. I0 p(s)f(s - a)da. 72. I0 q(a)s(8 - a)d8 .

where

T T
71 - IO (92(8) - p(s))ds. 72 - I0 (q2(8) - q(s))ds .

Recalling Lema 1.4.2, we see that 71 and '73 are
positive. We will have subharmonics when the damping
(u1,u3) is small in comparison to the forcing (u,,u4).

‘We have mentioned (in Section 3 of Chapter 1) that the
period is an increasing function of initial condition for
system (1.3.2). Precisely, if 1(a) is the period of the
solution of (1.3.2) with initial point at (a,1). then

56

f'(a) > 0 for all a ) 1. We will need something more,

namely an estimate on how fast the period increases.

Lemma 1.2: There exist numbers a, > 1 and K ) 0

such that if a a a, then f'(a) h K.

Proof: First make the change of coordinates
(1.6) u=1nx, v=lny.

Then system (1.3.2) becomes

0 - A(1 - ev)
(1.7) u .
v - D(e - 1)

Now (1.7) is a Hamiltonian system with energy function

(1.8) H(u,v) - A[ev — v - 1] + b[eu - u - l]

and so its solutions trace out level sets of H, iue. if
(u,v) is a solution to (1.7), then
H(u(t),v(t)) - h - constant. It is easy to see that the
solution of (1.3.2) with initial condition (a,l), a h 1,
corresponds to the solution of (1.7) with energy
h-D[a -ina- 1].

In Waldvogel [2] it is shown that if P(h) is the

period of the solution of (1.7) with energy h, then

57

 

 

u/Z
P'(h) - —~$— I R1(/E7D cos ¢)R(/E7A sin ¢)~(cos ¢//D)d¢
(IA-EH 0
(1‘9) n/2
+ 1 I R(/E7D cos ¢)R'(/H7A sin ¢)° §£2_£] dd
«m 0 «x

where the function R has the properties
(i) R(0) ) 0
(ii) R is C' on [0,w)

(iii) R' is positive and nondecreasing on (0,”)

For convenience, we assume A a D = l; the proof for the
general case is almost exactly the same. Fix a, 3, and

ho such that 0 < a < B < n/Z and ho > 0. Then we have
(1.10) P'(h) > ;%_I8 R'(¢fia’cos s)R(¢fi'sin a)cos p d¢
a

+ _l I3 R(¢fi'cos s)R'(¢fig'sin a)sin a d¢
“Ea

whenever h a ho: i.e.

(1.11) P'(h) > —l [M1 x(yfi sin a) + M2 R(/fi cos 3)] .
ME

where M1 = (B - a)cos B R'(/h~; cos B) and M2 =

(a — a)sin a R1(¢E; sin a). It follows easily that

58

KlRm/fi)
(1.12) P'(h) > ____.._..._..

for some positive constants K1, m.

From properties (i), (ii), (iii) it follows that there
exist c > 0 and x0 > 0 such that R(x) h Cx whenever
x 5 x0. Thus there is hl > 0 such that R(m/h) i Cm/h)

whenever h h hl. So we conclude that
(1.13) P'(h) > ka1 = K2

when h h hl.
Finally, 1(a) - P(a - in a - 1), so there is an a 2

such that if a h an,
(1 l4) 1"(a) > K - (l - l) a 1:
° 2 a I 2 °

Now let X = Kz/Z, and the proof is complete.

We can interpret this lema as follows: given any
perturbation function of form (1.3), there are solutions of
(1.3.2) with period (21r/fl)k, for all integers k h k0,
where k0 is determined by the limiting period of (1.3.2)
at the critical point. Furthermore, these solutions are
spaced “linearly or closer".

Figure 13 displays 1(a) versus a for 1 ea ‘ 8 in
the case A - D - 1. In Figure 14, A - 10 and D - .10.

Computations were done using the scheme in waldvogel [l].

59

 

n)
O

‘7’

5.)
rd

Pu
rd

C)

nip

‘1)

PJ

ma

 

C)
vs
0
L?
C) .4
pa
L)

 

igure 13.

'F‘

L

60

Du

C)

 

 

14.

Flgure

61

Section 2. Formulation of the Problem in Action-Angle
Co-ordinates

If we take system (1.1) and perform the change of

coordinates (1.6), we get

u = A(l - ev) + uleu + u2f(t)
(2.1)

V

6 a D(eu - l) + u3e + “43(t)

we often refer to (2.1) in the case #1 = u3 = 0:
. v
u = A(1 — e ) + u2f(t)
(2.2)

6 = D(e“ — 1) + #48(t)

System (2.2) is Hamiltonian, with Hamiltonian function

(2.3) H(u,v,t) = H(u,v) + v(u.v.t) .
where
(2.4) v(u.v.t) = uzvf(t) — u4ug(t) .

‘We can write system (2.1) in form

: ' u
u = Hv(u,v,t) + ule

(2-5)
v

v = -Hu(u,v,t) + use

Now (1.7) is the transformation. of the unperturbed
Vblterra—Lotka system (1.3.1). Its phase space is all of
R3, all orbits are periodic, and it has a unique center at

(0,0). Thus (1.7) can be used to set up a system of

62

action-angle coordinates (J,¢). (For a discussion of

action—angle coordinates, see Arnold [1]).

Denote the transformation to action-angle coordnates

by 8:
(2'6) S(U,V) 3 (31¢) r
and denote the ”transformed Hamiltonian” by H:

(2.7) H(J.¢.t) - H(s’1(a.¢).t)

a H(u,v) + v(u.v.t) .

An important characteristic of S (or of any
canonical tr ansf ormat ion) is that it preserves

Hamiltonians. System (2.5) transforms to

] l H (J 4.1) uleu

-HJ (J ¢.r) u3ev

C...

(2-8)

‘0

Here D801,” denotes the derivative of S at the point

(u,v) - S'1(J,¢). Since

(2-9) fi(JIth) =

H(s'1(a.¢)) + u2v(a.¢)t(t) — u4U(J.¢)8(t) .

and by the construction of act ion-angle coordinates we have

(2.10) H . 31(0. ¢) - IJ [% ] dx .

63

(T(J) is the period of the solution of (1.7) corresponding

to action J), system (2.8) can be written

C...

v¢(a.¢.t) P(J.¢)
(2.11)

‘0

0
8 4'
TIgT —VJ(J,¢,t) G(J.¢)

Here F and G are defined by

P(J.¢) ule “

(2.12) s Ds(u,v)

6(J.¢) u3ev

where (u,v) = S‘1(J,¢).

It is easily seen that F and G are Zn-periodic in
d, and that V is Zn-periodic in p and 2n/n - periodic

in t. Now define a new cyclic variable a by
(2.13) 9 = nt .
and we have

5 - v‘(J.¢.9/n) + P(J.¢)

2n "

(2.14) i 6 - T737 - v5(J.¢.e/n) + G(J.¢)

9 = fl

 

L

The functions on the right-hand side of (2.14) are

2n—periodic in s and in 9. We expand in Fourier series:

64

(2.15) a) v(J.¢.a/n) = z

“(J)ei(n¢ - m9)
m'nu—fl

a
m:

b) P(J.¢) - E bn(J)e1“‘

n-“

d) G(J.¢) - z came1M .

n”...
where

2n 2n ~
(2.16) a) ammw) -3172]; I0 V(J.¢.a/n)e’1(“"m”d¢d0

2n

b) bn(a) = 2%‘I P(J.¢)e“‘"‘ d4
9
2n

c) cn(a) - III G(J.¢)e’m¢ d¢ .
0

Note that ao'n(J) - 0 for all n and all J.

Now the system (2.14) becomes

I a

5, )3 inamm(0)e1(""’“°) + I: bn(J)ein¢

m'nn—Q nae—Q

m'ns-m nan-es

 

65

Section 3. Resonance: A Formal Calculation

System (1.1) is said to be in resonance when J is
such that the “natural period” T(J) is an integral
multiple, say H, of the forcing period 217/0. Given 0.
there will be infinitely many such J's. Since for small
u. we have p ~ 2n/T(J) we see that (2.17) is near
resonance when Ha - 9 ~ 0. For the terms of (2.17) this

corresponds to (m,n) - k(h-,1) for any integer k.

Note 3.1: It is possible to consider “higher order
resonances“ which occur when J is such that mT(J) =1
H(zn/n) for relatively prime integers 10,5 with m i 2,
but we will not do so in this thesis.

Suppose that we are given 0, J, a such that T(J) -
5(217/0). We want to examine the behavior of solutions of
(2.17) near the graph of the solution of (1.7)
corresponding to action J, i.e. the behavior of solutions
of (2.17) near resonance. We begin by averaging system

(2.17) over the non-resonant terms. The result is

° - a - __ ik(h¢-9)
J kg." ikn ak(n'1) (J)e

(3’1) I i " T%T ' RE ‘1':(E.1)(J)eik(""°)
0 = n

 

66

Note 3.2: In this context, “averaging“ means replacing

terms of form R(J,¢,9) with their averages

21! 217
l
7] I R(J.¢.a) dd d9 .
4” 0 0

For a detailed discussion of averaging, including error
estimates, see Sanders and Verhulst [l], particularly
Chapter 5. We emphasize that this is done to non—resonant
terms only. It is not hard to see that the averages are
all zero here.

For purposes of illustration, let us drop all terms
except the “primary resonances" k - 11 in (3.1). (See
note 3.3). Recalling that a_m,_n(J) -- m and
simplifying, we get in this case

IJa-Zh [Beast-tasinr]
(3.2) 3‘T%)' 2 [a' cost—8' sin y]
é=n

 

where r - r-w - 9, and 3.6.103) = a + is. Since r =-

np - 6, (3.2) can be written

r

5=-2E[scosy+asinv]

(3.3) 1 i - [H[T%g) - n] - 23 [a cos at + s' sin y]

 

La-n.

A phase shift for r transforms (3.3) to

67

 

. J a 23 A1 sin(v + '0)
(3.4) -i-3[T%-R+HA2cosr
éan .
Here

 

Now regard Sun/T(J)) - n as constant and
differentiate the second equation of (3.4). Dropping terms

of high order (A13), we have
(3.5) ;‘+A2Asinr=0,

where K - 5(H(2n/T(J)) - (1). Finally change the time
scale t a t/JK;, and we get

(3.6) ;+Asinr-0.

The first equation of (3.4) becomes

A1

VA;

(3.7) .‘1 .. 2'6 sin (y + to) .

 

 

Equation (3.6) is the well-known equation for the
motion of a pendulum. We illustrate the separatrices of

this motion in phase space in Figure 15 below.

68

/ a
\/

 

Figure 15.

It is easy to see that L s 4J3, a bound for the
variation of r for the bounded motions of 7. From (3.6)

we get

1 r

A

(3.8) sin y a -

 

 

Substituting this in (3.7) and integrating, we have

The maximum variation of J for bounded r is thus

69

(3.9) AJ - __ - A VA

 

 

 

- fr? 0 (¢|(uz.u4)f)

Note 3.3. To see how closely solutions of (3.2)
approximate those of (3.1), it is necessary to estimate the
sizes of the Fourier coefficients ak(fi'1)(J) and
a'(k('fi'1)(J) for lkl h 2. In applications they usually
fall off quite rapidly with Ikl. But if we wish to retain
all terms of (3.1), we can proceed as follows: let
a(kfi'k)(J) - a], + ink, and let it - Ea - 9. Then the
equation for '7 is

(3.10) <i - [B[T%§7 — n] - 23 §(r)

where P is a Zn-periodic function of mean value zero

given by
§(r) - £21 (a; cos xv - Bi sin kt)
(3.11)

= )3 008(k)? - r)
k_1 Alt k

 

where Ak -1/(a-k)3 + (B'k)3, and the rk's are constant.

Now dropping terms of order l(u,,u‘)l3 as before, we have

70

(3.12) ("r .. 2’6[E[-flz—g—)] - n]i5(r)
where
(3.13) 5(v) = gel kAk ain(kv - vk) .

Now in general when we have an equation of form
(3.14) i + g(x) - 0

where g is a periodic function with mean value zero, the
phase portrait contains elliptic and hyperbolic alternating
along the x-axis, the hyperbolic points being connected by
separatrices (See Hale [1], Chapter 5), i.e. the phase
portrait of (3.14) resembles that of the pendulum (3.6).
The separatrix width L is 2J2- (ll—4.377), where

x
M - max I: g(x)ds and m = min I0 g(s)ds ,

and so we know the maximum variation of x for bounded
motions of x.

Proceeding as before, it is still possible to obtain an
upper bound for the maximum variation of J, similar to

(3.9).

71

Section 4. Application of Resonance Overlap

In the J - 7 plane the motion, near resonance,
estimated in the last section has the form of a ”translated

pendulum” (see Figure 16).

J

 

 

 

 

Figure 16.

The center corresponds to exact r esonance . We recal 1
that for any n. there are infinitely many resonances.
It is known that the true metino includes the

following features:

a) For small _ rturbat ion strength, there is a thin
“stochastic layer” around the separatrix, i.e. a small
region around the separatrix through which solutions wander

in a chaotic fashion.

72

b) The center is surrounded by invariant (topological)
circles. These circles are intersections with the J - r
I plane of invariant tori, the so-called KAM tori, in
J - r - 9 space. (For a discussion of these facts, and as
a general reference for this chapter, see Lichtenberg and
Lieberman [1], Chapters 1 - 4).

Numerous numerical and physical studies indicate that
for perturbed Hamiltonian systems, as the perturbation
increases, there occurs a (fairly abrupt) transition: for
perturbations below a certain strength, chaotic solutions
are confined to narrow regions bounded by KAM surfaces; for
stronger perturbations, chaotic solutions occupy a much
larger proportion of the phase space and may wander
throughout large regions.

There has been a great deal of interest in estimating
this threshold perturbation strength and varidus techniques
have been proposed. One of the earliest and simplest, due
to Chirikov, is the so-called “resonance overlap criterion"
(see Chirikov [1], [2] for a. discussion and numerical
examples). Since the separatrix width W’ (see Figure 16)
tends to increase with increasing perturbation strength,
and the separatrix is surrounded by a stochastic layer, it
seems reasonable that the transition to global chaos should
occur when the separatrices corresponding to adjacent

resonances, calculated separately, begin to overlap. This

73

happens when the sum of the half-widths of the separatrices
exceeds the distances between the resonances. (see Figure

17)

La

r

 

 

 

Figure 17

Note 4.1: In most numerical examples, the tr reshold
value computed by the resonance overlap criterion appears

to be slightly too large.

In most problems involving resonance overlap, the
procedure is as follows: start with a system in which the
forcing depends linearly on a parameter u, estimate the
distances between resonances in J - r space, estimate the
dependence of the separatrix width on in. then calculate
the transition value of Iul. We take a slightly different

point of view: ,we regard the forcing parameter as fired

74

and let the frequency increase. Specifically, we suppose
that f and g are Zn—periodic functions and let f(t) a
f(nt) and g(t) - g(nt) in (2.1). We look at some
bounded portion of the phase plane of (2.1) say
[(u,v)IT(J) ‘ 100). Increasing the forcing frequency (1

has two effects:
a) By (3.9), the separatrix widths tend to increase.

b) There are more resonances, and therefore adjacent
resonances are closer. For example, doubling the forcing
frequency causes separatrix widths to increase by a factor
of f2 and adjacent resonances are on the average half as

far apart.

Now we expect on the basis of (3.9) that the separatr ix
widths increase with the strength of the perturbation and
with the square root of the order of the resonance.
Applying this to (2.1), we discuss two cases.

(1) Suppose that f and g are fixed, and look at
some bounded r eg ion of the phase space , say
[(u,v)IT(J) ‘ M] for some positive constant M. It is
imediate from (3.9) that for |(u2,u4)| sufficiently
small, there will be not resonance overlap for any of the
adjacent resonances, and if l(u3,u.)l is sufficiently
large, all adjacent resonances will overlap.

(ii) Suppose we fix [1 and increase the perturbation
frequency in some bounded region. Precisely, let f,g be

’fixed l-periodic functions, and let f(t) a f(nt), g(t) s

75

EUR.) in (2.1). As n increases, two factors change:
there are more resonances (and so the distances between
adjacent resonances are smaller) and the resonance numbers
‘N' are larger. For example, doubling the forcing frequency
causes separatr ix widths to increase by a factor of V3,
and adjacent resonances are approximately' half as far

apart. One possible situation is shown in Figure 18.

 

 

 

 

 

 

 

 

Figure 18.

76

We can sumarize this case by saying that resonance
overlap tends to increase very strongly with forcing

frequency.

B I BLIOGRAPHY

BIBLIOGRAPHY

Andronov, A.A., E.A. Leontovich, I.I. Gordon, and
A.G. Maier [1], Theory 9; Bifurcations pg Qypamic
Systems pp p Flapp, John Wiley and Sons, 1973.

Arnold, v.1. [1], Mathematical Methods 9; Classical
Mechanics, Springer-Verlag, 1978.

 

Blom, J.G., R. de Bruin, J. Grasman, and J.G. Veriver [l],
Forced Predator-Prey Oscillations, Preprint TW 201/80,
Mathematisch Centrum, Amsterdam, 1980.

Chirikov, B.V. [l], A Universal Instability of Many—
Dimensional Oscillator Systems, Physics Reports,
52 (1979), 263-379.

Chirikov, B.V. and G.M. Zaslavski [1], Stochastic
Instability of Nonlinear Oscillations, Soviet Physics,
14 (1972), 549-568.

Chow, S.N., and J.K. Hale [1], Mgthodp pf Bifurcation
Theory, Springer-Verlag, 1982.

 

Hale, J.K. [1], Ordinary Differential EquatIOQ§, Krieger
Publishing Co., 1980.

 

Hirsch, Mew., and S. Smale [1], Differential Equations,
Dypgmical Systems, and Linear Algebpg, Academic Press,
1974.

 

Lichtenberg, A.J., and M.A. Lieberman [1], Regular and
Stochastic Motion, Springer—Verlag, 1983.

Lotka, A.J. [1], Elements pf Mathematical Biology,
Dover, 1956.

Robinson, C. [1], Phase Plane Analysis Using Poncare
Maps, Preprint, 1984.

Sanders, J.A., and F. Verhulst [l], The Theory pf
Averagipg (in preparation).

 

77

78

Volterra, V. [1], Locons sur la Theorie Mathematique
93 lg Lutte pour lg Vie, Gauthier-Villars, 1931.

waldvogel, J. [l], The Period in the Volterra-Lotka
Predator-Prey Model, SIAM Journal of Numerical
Analysis, 20 (1983), 1264-1272.

[2], The Period in the Lotka-Volterra System is
Monotonic, Preprint, 1984.

  

 

will

minim/ii

III".

93

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