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MICHIGAN

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University

NIVERSITY LIBRARIES

 

 

 

 

 

This is to certify that the
dissertation entitled

Geometry of the Melnikov Vector

presented by

Masahiro Yamashita

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Mathematics

 

 

{L/R)- Lilo

Major professor

 

Date August 8, 1988

 

MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771

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TO AVOID FINES return on or baton 6‘. duo.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Affirmative Action/Equal Opportunity lnstltwon

 

GEOMETRY OF THE MELNIKOV VECTOR
by

Masahiro Yamashita

A DISSERTATION

Submitted to
Michi an State University
in partial ful lllment of the requirements
of the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics
1988

Squ

Cp- C-‘

ABSTRACT

GEOMETRY OF THE MELNIKOV VECTOR

By

Masahiro Yamashita

The Melnikov method is developed for higher dimensional systems, and
the transversal and tangential intersection of the stable and unstable
manifolds are discussed. Hamiltonian systems are discussed as a special
case of the general theory. The theory is then extended to the case of a
heteroclinic orbit to invariant tori which includes systems with

quasi—periodic perturbations as a Special case.

This thesis is dedicated

to the memory of my father,

Seiichi Yamashita

1912—1984

ACKNOWLEDGEMENTS

I am sincerely greatful to Professor Shui—Nee Chow, my advisor, for his
advice and encouragement. Without his constant help and attention this

work would not have been possible.

Chapter

§1
§2
§3

§4
§5
§6
§7

§8

§9

§1o
§11
§12
§13
§14

TABLE OF CONTENTS

Introduction
Formulation of the Problem

Basic Results from the Theory of
Exponential Dichotomy

The Stable and Unstable Manifolds
The Fredholm's Alternative
The Melnikov Vector

Transversal Intersection of the Stable
and Unstable Manifolds

The Index of 7, the Fredholm Index
and the Dimension of the
Melnikov Vector
Computation of Higher Order Terms
The Linear Melnikov Vector
Hamiltonian Systems
A Heteroclinic Orbit to Invariant Tori

Three Examples

Exponentially Small Spritting of
Stable and Unstable Manifolds

References

14
23
28

38

46
49
53
61
70
87

101
103

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13

TABLE OF FIGURES

vi

§1. INTRODUCTION

The notion of a homoclinic point was introduced by Poincare [18].
To recall this concept, consider a diffeomorphism in R2 with a
hyperbolic fixed point p. A point q is called a homoclinic point of p
if q is in the intersection of the stable and unstable manifolds of p.
The point q is called a transversal homoclinic point of p if the
intersection of the stable and unstable manifolds is transversal, i.e. the
tangent spaces at q to the stable and unstable manifolds span the
whole space. We note that if one homoclinic point exists, there must be
infinitely many homoclinic points.

Poincare already observed that the existence of homoclinic points
implies complexity of the dynamics of the diffeomorphism. Later G.D.
Birkhoff [3] proved that every transversal homoclinic point of a
two—dimensional diffeomorphism is accumulated by periodic orbits. The
results by Smale [20], now called the Smale—Birkhoff theorem, extend the
Birkhoff's results both in two dimentional and to higher dimensional
cases and assert that the existence of a transversal homoclinic point
implies the existence of an invariant Cantor set in which the periodic
orbits are dense. See also Moser [14]. Moreover Newhouse [15] has
proved that there is a much more complicated dynamical behavior
associated with a homoclinic tangency. Thus the dynamics of
diffeomorphisms with transversal or tangencial homoclinic points are fairly
well understood.

However to apply the above abstract theories for diffeomorphisms to
a system of differential equations, we need to know the existence of a "

homoclinic point of a diffeomorphism induced by this system. More

precisely since we shall deal with an autonomous system with a
time—periodic purturbation, the above diffeomorphism appears as a
time-one map, called a Poincare map, induced by the flow of the
system.

Our problem is the following: an autonomous system of ordinary
differential equations with a time—periodic perturbation is given and
assume that the unperturbed autonomous system has two hyperbolic
critical point (not necessarily distinct) and a homoclinic or hetercolinic
orbit connecting them. Find computable conditions under which the
Poincare map induced by the perturbed system has a transversal
homoclinic point. See §2 for more precise definitions of these notions
and for a precise formulation of the problem.

Poincare [17], Melnikov [12] and Arnold [2] deveIOped such
conditions for two—dimensional analytic Hamiltonian systems and it is
now called the Poincare—Melnikov—Arnold method or simply the Melnikov
method. The Melnikov theory has been studied by several authors, e.g.
Chow, Hale and Mallet-Paret [4], Holmes [9] and Palmer [16], and
generalizations to higher dimensional cases have also been studied, e.g.,
Holmes and Marsden [10] and Gruendler [6]. The key of these theories
is the use of the Melnikov function which measures the splitting distance
between the perturbed stable and unstable manifolds.

One of the purposes of the present notes is to clarify the geometry
of the Melnikov function (now should be called the Melnikov vector) in
higher dimensional cases and to extend the previous theories for the
two—dimensional case to higher dimensional cases.

Our theory is based on the theory of exponential dichotomy. We

shall recall basic results on exponential dichotomy in §3. Palmer [16]

showed that the linear variational system along the homoclinic orbit of
the unperturbed autonomous system has exponential dichotomies on
half—lines. Using this fact we shall derive explicit expressions of the
local stable and unstable manifolds of the perturbed system. This is the
content of §4. Then the Fredholm's alternative, given in Chow, Hale
and Mallet—Paret [4] for the two—dimensional case, in Palmer [16] in
heigher dimensional cases and explained in §5, is used to derive the
Melnikov vector in §6. In §7 we examine conditions for a transversal
homoclinic point and introduce the notion of the index of a homoclinic
or heteroclinic orbit which is useful to classify the cases that can occur
in higher dimensional cases. In §8 we discuss a relation between the
dimension of the Melnikov vector and the index of the homoclinic or
heteroclinic orbit. Numerical aspect of the Melnikov vector is discussed
in §9. In §10 we consider several Special cases in which the Melnikov
vectors take simpler forms, and also we discuss the tangency. We apply
these general theories to Hamiltonian systems in §11. In §l2 we extend
our theory to the case of a heteroclinic orbit to invariant tori and as a
by—product we drive a formula which guarantees the transversal
intersection of the stable and unstable manifolds of a two—dimensional
system with a quasi—periodic perturbation. See also Meyer and Sell [13]
and Wiggins [21]. Several interesting examples are discussed in §13 and
finally in §14 we show a serious limitation of the Melnikov method by
using an example for which the Melnikov method does not work. This
difficulty comes from the nature of Melnikov method as a perturbation

theory.

§2. FORMULATION OF THE PROBLEM

Consider a system of differential equations

(2.1) x = f(x)

and its perturbed system

(2.2) St = f(x) + £g(t,x)

where xcan, th, cell and |e|<<1. The vector fields f and g are
assumed to be sufficiently smooth and bounded on bounded sets. The
vector field g is periodic in t with the least period T(>0).

Assume that system (2.1) has two hyperbolic critical points x+
and x_ (not necessarily distinct). Also assume that there is an orbit

7(t), th, of system (2.1) which connects the critical points x and

+
x_. That is,
(2.3) 7(t) -+ xat as t —i i 00.
If x + = x_, the orbit 7 is called a homoclinic orbit. Otherwise 7

is called a heteroclinic orbit.
Let x(t:x0), xoclin, be the solution of system (2.1) with the initial

data x(0; to) = x0. The stable manifold w3(x+) of the hyperbolic

critical point x of system (2.1) is defined by

+

(2.4) Ws(x+) = {x0 6 Rn: x(t;x0) -» x as t -i he},

+
and the unstable manifold Wu(x_) of the hyperbolic critical point x_
of the system (2.1) is defined by

u _ n, , _’ _’
(2.5) W (x_) — {xoc R . x(t,x0) x_ as t —w}.
Then we have

(2.6) 7 c w3(x+) n w“(x_)

from the above assumption.

Since the critical points x are hyperbolic and system (2.2) is

:l:

periodic in t, there exist unique T—periodic solutions gm) of

system (2.2) such that

(2.7) lim gm) = -i(t,0) = x

:t
£-+ 0

uniformly in t. For details see Hale [7].
It will be shown in the next section that there exist sets Wiocfiy‘)
and w‘l‘oc(i'_,e) in s"x{0} c Rnle, where Rnle is the extended

phase space of system (2.2), such that

(2.8) wheat) = {(x0.0)c R"x{0}=|x(t;0,xo) -
— xi(t;c)| -v 0 as t-» +00 and

x0 is in a sufficiently small neighborhood

of 7}

and

and x0 is in a sufficiently small

neighborhood of 7},

where x(t;r,xo) is the solution of system (3.2) with
x(r;1',xo) = x0, xoc Rn.

If we define the time dependent stable and unstable manifolds,
WS(—+;6) and W“(§_,t), of system (2.2) by

(2.10) WS(§+,t)={(xo,r)e Rnle: |x(t;r,x0) — i t;c)| .. 0

+(
as t-i +00}

and

(2.11) W“(§_,c) = {(xo,r)c Rnle: |x(t;r,x0) — x_(t;e)| -+ 0

as t-t-oo},

then WIoc(i+") and w‘foc(3z_,t) are the local cross sections at

t= 0 of W8(§+,c) and Wu(3€_,c) respectively. That is,

(2.12) wfoc(§+,e) c WS(§+,e) n (121140))

and

(2.13) w‘foc(sz_,e) c W“(§_,t) n (anx{0}).

Since system (2.2) is periodic in t, its extended phase space can be
regarded as IRnxS', where S1 is the unit circle, and then W?OC(E+5)
and Wllloc(x_,c) are the local stable and unstable manifolds of
hyperbolic critical points it
He: Rn -+ IRD, which is defined by the flow of system (2.2) as follows:

a ii(0;c) of the Poincare map

(2.14) H€(xo) = x(T;0,x0), X06 IRn.

Now we state our problem.

Problem I. When does system (2.2) have an orbit x(t), t c R, so

that x(t) -i xi(t;c) as t -+ $00 ?

Following the above argument, it is clear that Problem I is equivalent to

Problem 1'. When do W? (x+,c) and W‘1 (23) defined

0c loc

above intersect each other?
Then next natural question would be

Problem II. When do W?0c(§+,c) and WlllOC(§—’6) intersect

transversally?

Here the transversal intersection means that tangent Spaces to
s - u - . . .
Wloc(x+,c) and to W1 0c(x__,c) at a pomt of intersection Span the

whole space IR“.

\

 

 

 

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Figure 1

§3. BASIC RESULTS FROM THE THEORY OF EXPONENTIAL
DICHOTOMY

In this section we recall the definition and basic results of
exponential dichotomy which will play a key role throughout the paper.
For details on the exponential dichotomy, see COppel [5], Palmer [16] and
Hale and Lin [8].

Consider the system

(3.1) a = A(t)z, zan

where A(t) is assumed to be a continuous nxn real matrix function
on R. Denote by <I>(t,s) the transition matrix of system (3.1).
Definition 3.1. System (3.1) is said to have an exponential

dichotomy on [to,oo), t fixed, if there exists a projection

o
szn-llin,K21 and a>0 suchthat

(3.2) |<I>(t,t )P<I>(t ,s)| 5 Ke’a(t‘s), t g s g t,
O O O

and

(3.3) |<I>(t,to)(I—P)<I>(t0,s)| g Ke—a(S—t),to g t s s

Similarly system (3.1) is said to have an exponential dichotomy on
(-oo,t0] if there exists a projection Q: It“ -i R”, L 2 1 and b > 0
such that

10

(3.4) |<I>(t,t0)Q<I>(to,s)| s Le_b(t-s), s s t s to,
(3.5) |<I>(t,to)(I—Q)<I>(to,s)l s 15““), t s s s to-

Roughly speaking, the exponential dichotomy is hyperboliCity on
half-lines. More precisely, from (3.2) and (3.5), we see that the range of
P, denoted by $(P), is the (exponentially) stable subspace at t 0:

(3.6) $(P) = {fit It“: <I>(t,t0)€ -» 0 as t -» +00},

and 3(I—Q) is the unstable subSpace at to:

(3.7) $(I—Q) = {5: It“: <I>(t,to)£ e o as t .. as}.

 

 

 

Figure 2

We also note that 'the stable projection' <I>(t,tO)P<I>(to,t) at time

ll

t(2 to) is uniquely determined by P since <I>(t,to)P<I>(tO,t) is a

solution of the matrix system.
(3.8) X = A(t)X — XA(t)

with the initial value P at t = to.
The similar statement holds for the unstable projection
<I>(t,to)(I—Q)<I>(t0,t) at time t($ to).

The key fact on the exponential dichotomy which shall be used is
the following: If system (3.1) possesses an exponential dichotomy on
[to,oo) with projection P and if P' is a projection such that
52(P) = 58(P'), then system (3.1) also possesses an exponential
dichotomy on [to,oo) with projection P'. Similarly if system (3.1)
possesses an exponential dichotomy on (mo,t0] with projection Q and
if Q' is a projection such that SKI—Q) -—- $(I-Q'), then system (3.1)
also possesses an exponential dichotomy on (-oo,t0] with projection Q'.
Thus the stable subspace 58(P) and the unstable subspace $(I—Q)

are uniquely determined but their complementary subspaces can be any.

Next the adjoint System of system (3.1) is defined by

(3.9) ¢ + A*(t)¢ = 0

at
where A (t) is the transpose of A(t).
If system (3.1) has an exponential dichotomy on [to,oo) with projection

P, then the adjoint system (3.10) automatically has the exponential

12

* *
dichotomy on [t 0,011) with projection I—P where P is the adjoint

Operator of P. That is, (3.2) and (3.3) imply

it _ :1: * __ ..
(3.10) |<I> (t,t0) l(I—P )o (t0,t) 1| g Ke ”—5), to s s s t
and
at: __ :1: at: _ __ _
(3.11) |<I> (t,t0) 1P <I> (t0,t) 1| g Ke 3“” t), to g t s s.

Similarly system (3.1) has an exponential dichotomy on (mo,t0] with
projection Q, then system (3.9) has the exponential dichotomy on

a:
(-oo,t0] with the projection I—Q . That is, (3.4) and (3.5) imply

(3.12) |(I)*(t,to)-l(I—Q*)<I>*(t0,t)—1| g Le_b(t—S), s g t s to
and
(3.13) I<I>*(t,to)”1cl*<1>*(t0,t)‘ll s LENS“), t s s s to'

Note that if <I>(t,s) is the transition matrix of system (3.1), then
<I>*(t,s)'"1 is the transition matrix of system (3.9).

The key geometric fact, which will be used essentially in later
sections, is the following relation between the space of bounded solutions
of the adjoint system (3.9) and the spaces of bounded solutions on [0,00)

and on (-co,0] of system (3.1).

13

(3.14) {n t It”: <I>*(t,tO)—1 1] .. o as t a a co}
= sea—1”“) n and“)
= {$(P) + SRO-CD}l
= [{5 c Rn: <I>(t,to)§ -1 0 as t -i +oo} U
{5 c R“: <I>(t,to)£ .. o as t e soul.

This is clear from (3.10) and (3.13).

14

§4. THE STABLE AND UNSTABLE MANIFOLDS

Since we would like to describe the local cross sections onc(—+’f)

and w‘foc(§_,c) of the time dependent stable and unstable manifolds

W3(§+,t) and Wu(x'_,c) of System (2.2) as part of the 'perturbed
manifolds' of Ws(x+) and Wu(x_) of system (3.1) respectively along
the orbit 7(t), we let, for a fixed will,

(4.1) x(t) = 7(t+a) + cz(t+a).

Then system (2.2) becomes

(4.2) i = A(t)z + g(t-a, 7(t)) + h(t,z,a,c)

where

(4.3) A(t) = Df(7(t)).

and

(4.4) h(t,z,a,€) = % {f(7(t)+62(t)) — «7(a)
— ch(7(t))z(t) + cg(t—a,7(t) + cz(t))
— 6g(t-a,7(t))}-

We note that

15

(4.5) |h(t,z,a,c)| = 0(6) uniformly in t,z and 0.
Since x+(0;c) is hyperbolic, by (2.3), (2.7) and (2.8)

(4.6) 7(0) + 65 E w§m(§+,t)

if and only if the solution z(t;a,£) of system (4.2) with
z(a;a,§) = 6 is bounded on the time interval [a,oo). Thus we have, by

changing adR, that

(4.7) W?OC(-)E+,f) = U {7(0) + 653: the solution
aclR

z(t;a,§s) of system (4.2) iS bounded on [a,oo)}.

Similarly we have

(4.3) w‘foc(§_,t) = [is {7(0) + g“: the solution z(t;a,£u)
a

of system (4.2) is bounded on (m,a]}.

We remark that adR works as a 'sweeping' parameter along the orbit
7. See Figure 3.
Now as the orbit 7 is assumed to be a homoclinic or heteroclinic

orbit to hyperbolic fixed points, the linearized sytem

(4.9) i = A(t)z

16

of system (2.1) along the orbit 7 has exponential dichotomies on [a,oo)
and on (ma). (See Palmer [16].) Let P(a): R” -+ R“ and (2(0): R“
-+ R11 be projections for exponential dichotomies on [a,oo) and on
(—oo,a] reSpectively.

Fix aclR. Then from the variation of constant formula, the

solution z(t;a,{) of system (4.2) satisfies

t
(4-10) Z(t;a,€) = ¢(tia)P(0)€ + [I ¢(t,T)S(T){g(T-as’r(r))

+ h(T,Z(T;a,€),a,6)}dT + ¢(t,a)(I-P(a))€

t

+11 @(tiT)(I—S(T)){g(T-a,7(7))

+ h(T,Z(T;a,€),a,6)}dT

where <I>(t,r) is the transition matrix of system (4.9), and 8(7) is

defined by
(4.11) S(r) = <I>(T,a)P(a)<I>(a,r).
It is easy to Show that z(t;a,€s) is a bounded solution of system

(4.2) on [a,oo) if and only if z(t;a,£s) satisfies the following integral

equation:

17

(4.12) z(t;a,{s) = <I>(t,a)P(a)§s
+ <I>(r,a)P(a) [t 2(atr){g(r-a,7(r))
+ h(T,z(r;a,§3),a,c)}dr
+ <I>(t,a)(I—P(a)) a): <I>(a,r{g(r—a,7(r))

+ h(1',z(r;a,§s),a,c)}dr.
Here we used
(4.13) <I>(t,r)S(r) = <I>(t,a)P(a)<I>(a,r).

Let ”s = P(a)§5. Then it can be shown by the contraction

mapping principle that integral equation (4.12) has a unique solution

z(r)S)(t) 5 z(t;a,€9(ns)) for Ins|<<| where {S = {3(178) is a function
Of 173. By letting t = a, the function {3 = {8(173) is given by

S a
(4-14) 63 = 77 + (1-P(0)){l ‘I’(aiT)g(T-ai’r(7))d7

a

+1 ‘NCUWT,Z(ns)(T),a,6)dT}-
We remark that
(4.15) [0 <I>(a,1')h(r,z(ns)(r),a,c)dr = 0(6).

uniformly in 118. Similarly z(t;a,§u) is a bounded solution of system

(4.2) on (-oo,a] if and only if

18

(4.16) C“ = n“ + mama <I>(a,r)g(r-a,7(r)dr
a —m
+1 <I>(a,'r)h(r,z(n“)(r).a,c)dr}

—m

where nu c $(I—Q(a)), Inu|<<1 and

z(1)u)(t) a z(t;a,§u(nu)) is the unique solution of

(4.17) z(t;a,§u) = <I>(t,a)nu

t
+ ¢(t,a)(I-Q(a)) [y ¢(O,T){s(T-ai7(‘r))

+ h(T,z(T;a,€u),a,6)}dT

t
+ <I>(T,a)Q(d) l @(a,r){g(T-a,7('r))

-co

+ h(r,2(na,€u),a,6)}d7’-

We also remark that

a
(4.13) J <I>(a,‘r)h(r,z(1)u)(T),a,c)d1' = 0(a).

—00

Thus we have Shown that W?Oc(§+,c) and w‘foc(§_,r) have the
u

following expressions as functions of a, n3 or n .

Proposition 4.1.

(i) The local cross section w§m(§+,5) at t = 0 of the time

dependent stable manifold WS(§+,E) of system (2.2) is given by the

following:

19

(4.19) wifocfijrn) = at)“ {7(0) + cMS(a,nS,c)}

where

(4.20) Mswfc) = n8 + (I-P(a))[la<P(a,T)g(T-a.7(r))dr

a S
+ I <I>(a,'r)h(r,z(17 )(r),a,c)dT]

and use $P(a), [nS|<<1 and 2(178) is the solution of equation (4.12)
with 178 = P(a)€s.

(ii) The local cross section w‘foc(§_,r) at t = 0 of the time

dependent unstable manifold W“(§_,r) of system (2.2) is given by the

following:

(4.21) w‘lloc(i‘r_,c) =01)“z {7(a) + 6M"(a.n“6)}

where

(4.22) Mu(a,nu,t) = nu + (Mama q’(aar)8(7'ai7(7))d7

—00

a
+1 2(arr)h(7,2(nu)(7),a,€)drl

"(D

and nuc$(I—Q(a)), [nu|<<1 and z(1)u) is the solution of equation
(4.17). o

20

    
  

£/
mum): (Maw)

 

L))=- tR( 11-91(4))

 

 

 

( 1: (tub-e)

Figure 3

21

Beka 4.2. Notice that 7(a)c$(P(a)) n .92(I—Q(a)). Since we
consider the cross sections of the time dependent stable and unstable
manifolds in a vicinity of the orbit 7, it is sufficient, by the tubular
neighborhood theorem, to consider coordinates in the normal bundle

D

U TfKMIRn of the submanifold 7, where 167(0)": stands for the

016R
normal vector subspace, in the tangent space T ’7( 00R”, to the

one—dimensional vectorsubspace spanned by 7(a), i.e.,

Tao)“ 2 T 7( a)Rn/span{7(a)}. Hence from now on, we assume, for
"SCT7(G)IRD, fluCT’Ka)Rn and 06R, that

4. s 1 n

( 23) 77 c T7(a)fli n .9BP(a)

and

(4.24) nut Tfy(a)an n .92(I—Q(a)).

We also assume, for aclR, that

(4.25) sea—mo» c WMRH’ $Q(a) c was“.

- . . s — _ _
Under these assumptions, each pOint in W10C(x+,c) or W1110c(x_,£) is

uniquely expressed in terms of the coordinates a, 713 or 7)“.

22

Remark 4.3. The higher order terms

(I—P(a>) Ia Marlinnaturism and

a 11
(42(0) 1 ‘I’(aiT)h(T,Z(fl )(T),a,€)dT

“TD

in (4.20) and (4.22) are of order 6 uniformly in 0. Though these
terms include the solutions z(ns)(r) and z(r)u)(r) of equations (4.12)
and (4.18), these solutions can be approximated in an arbitralily high
order of accuracy by an iterative scheme. In fact, second iteration is
enough to obtain all information we need to determine the transversality

of wfoc(§+,r) and Wioc(§—’c)' (See §9).

23

§5. THE FREDHOLM'S ALTERNATIVE
Suppose that the following system
(5.1) s = A(t)z, zc Rn

has exponential dichotomies on [0,oo) with projection P and on (-oo,0]

with projection Q. Consider the inhomogeneous system
(5.2) 2 = A(t) z + g(t)
where g(t) is bounded and continuous on IR.

mm: find a condition under which system (5.2) has a bounded
solution on R.
Let <I>(t,s) be the fundamental matrix of system (5.1). Then it is
known that the solution z(t;0,§s) of (5.2) is bounded on [0,oo) if and
only if

(53) 68 = 178 + (I-P) [0 “01030)“.

and the solution z(t;0,{u) of (5.2) is bounded on (—oo,0] if and only if

(54) t“ = n“ + Q JO 2(0,t)g(t)dt,

-oo

24

where ascflP) and fluefll—Q).
Thus the set of initial data 55(5“ respectively) which gives a bounded
solution on [0,oo) ((—oo,0]) constitutes the hyperplane which is a shift by

the constant vector

0 o
(I-PH <I>(0,t)g(t)dt (QJ <I>(0,t)g(t)dt)

-00

from the unperturbed Space x(P) ($(I—Q)).
Let

(5-5) 3 = Q lo <I’(0,t)g(t)dt - (I-P) lo @(Ost)g(t)dt-

Then we have the following lemma which is the starting point of the

paper.

Lemma 5.1. System (5.2) has a bounded solution on R if and
only if

(5.6) d c 5?,(P) + 52(I—Q).

Proof. This is obvious because condition (5.6) is equivalent to say

that two hyperplanes defined by (5.3) and (5.4) intersect. o

 

25

The geometrical statement in Lemma 5.1 can be expressed analytically by

using bounded solutions of the adjoint system of (5.2):

(5.7) 4 = 4*(t) 4

where A*(t) is the transpose of A(t).

Recall, from §3, that {gm—p") n 53(5)} = (2(1)) + sea—my and
{3(1—P*) n 52(Q*)} is the subspace of initial points at t = 0 of
bounded solutions of the adjoint system (5.7). Therefore condition (5.6)

q
is equivalent to say that d is annihilated by ¢(O) where ¢(t) is
any bounded solution of the adjoint system (5.7).

61(1- P)

/ Woo

/ a (zomxiamit

 

 

 

 

 

 

 

Let m = dim{.9t(P) + sea—ow and let {¢1(t),...,¢m(t)} be a

complete set of bounded solutions of system (5.7) which satisfies that

26
(5.3) {$03) + 941—er = span {41(0).....¢m(0)}-
Then we have the following analytical restatement of Lemma 5.1.
-i
Lemme 5.2. d c $(P) + 52(I—Q) if and only if

(D

(5.9) I ¢:(t)g(t)dt = , i=1,...,m.

Proof. dc$(P) + $(I—Q) if and only if 330) d = o,

 

i = 1,...,m.

Since

q*¢,(0) = (I—P*)¢,(0) = «4(0)

and

4,3) = (<I>’I(t,0))*¢i(0) = 930.0440),

0 =¢:(0)d
.. 0 .. o
= ¢i(0)Ql <I>(0,t)g(t)dt — ¢i(0)(I-P)I <I>(0,t)g(t)dt

oo

0 * 0 *

l ¢,(0)<I>(o,t)g(t)dt -I ¢,(0)<I>(o,t)s(t)dt
0 it 0 *

= I ¢i(t)g(t)dt - I ¢i(t)g(t)dt

= 1‘” ¢§(t)g(t>dt. u

27

We remark that Lemma 5.2 had been proved in Chow, Hale and
Mallet-Paret [4] for two dimensional case and in Palmer [16] for general
case. However our proof is different and more geometrical. We shall
apply in the next section the method of proof of Lemma 5.2 to the

tangent space at each point of a homodinic or heteroclinic oribit.

28

§6. THE MELNIKOV VECTOR

Our purpose in this section is to deveIOp necessary concepts which
are useful to derive computable conditions under which the purturbed
stable and unstable manifolds intersect transversally. To do so, we
would like to measure the 'distance' between WIoc(§+’€) and
w‘lloc(i_,t). Define, for simplicity, the following quantities in' the

expressions in (4.20) and (4.22):

(6.1) mSo) = (I—P(a)) Ia <P(a,t)g(t-a,7(t))dt,
.. a
(6.2) ms(a,ns,€) = (I—P(a)) l <I>(a,t)h(t.z(123)(t),a,6)dt,
(6.3) mum) = (2(0) la <I>(a,t)g(t-a,7(t))dt,
(6.4) humus) = (4(4) 1“ <I>(a,t)h(t,z(n“)(t),a,e)dt.

Then we have
(6.5) Ms(a,ns,6) = as + m8(a) + amuse)
and

(6.6) Mu(a,nu,c) = n“ + m“(o) + mu(a,nu,c).

29

.. -l
Finally we define the distance vectors (1 and d by

(6'7) 8(a,flsnu,€) = Mu(aiflui€) — M8(ainsi£)
and
(6.8) 3(a) = mu(a) - ms(a).

Recall that we are working on the normal bundle U TfK Rn.
06R 0)

Fix 06R and let 178,17ucT7(a)Rn. Consider the following decomposition
1 11,
Of T 7( (QR .

(6.9) TfKMRD ={5t(l-Q(o)) n .92P(a) n T;(a)ii:}
‘9 {SKI—(2(a)) 0 3043(0)) 0 Ilka“? 11)}
‘9 {$(Q(a)) 0 3PM) 0 T7(a)R n}
e {.52Q(a) n 58(I—P(a)) n T 7( a)an}'

According to this decomposition of TfKMRD, the vectors Mu(a,nu,t)

and MS( a, 778,6) are decomposed as follows:

11 u u u mu ~u u ~u
(6°10) M (0:77 )6) = ("117/29m1 + m1, m2 + m2)

and

S s ~s
(6'11) M (0277 90207191118 + “11,772,111 2 + m2)
See the diagram below.

30

 

 

 

 

 

 

 

 

o<
(RU-cm) *
7' 7:5 (RC Po»
72:: mi + 171:
Metal) 1"? + R 72': Wk 9(4))
Tn: + 17): mi+ iii:
Figure 5

To get familiar with the decomposition defined above we give an example
in Figure 6. Here consider a homoclinic orbit 7 in R3 and assume

dim 52 P(a) = 2 and dim 58(I—Q(a)) = 1.

31

GM 1- Phil)

 

echo

0—!

 

 

 

at; L'Ii'mi‘fi?
m) ' ' ‘ ‘ ~ ~ -

 

“p---—
. 1"
I . I s
s

    

(R( Pm

Figure 6

32

Now it is clear from (4.19), (4.21) and (6.7) that W?m(x+,c) and
W‘lloc(§_,f) intersect each other in the hyperplane 7(a) + TRORH if

and only if d(a,ns,nu,c) = 0 for some 0,173 and 7)“. Since

- s -8 ..
(6.12) d(a,ns,nu,é) = (17111 - vi, 1712‘ - (m1+m1), (m‘f+m‘f)

- n3, (mgmg) 41113533»,

d(a,178,r)u,f) = 0 if and only if there exist a,u(= 17513 = 17111), 1); and

111.21 such that the following three equations are satisfied.

(6.13) 77121 — {mi(a) + mi(a,u,ng,c)} = 0,

(6.14) 173 — {m111(a) + mlll(a,u,n121,c)} = 0,

(6.15) {mg(o)+mg(o,u,ng,t)} — {m;(o)+mg(o,u,ng,t)} = 0.
From (6.13) and (6.14),

(6.16) F(n‘2’,a,V) 713— {mi(a)+mi(a,u,m‘f(a)+m‘f(a,u,ng,c),c)}

= 0.
We notice that
ems am“
(6.17) -6—u F(n121,a,u) = I — ——;- —11—i
0172 6112 3172

is nonsingular for 6 small enough because

33

ans am“
(6.18) I 1 1|

3% «3?;—
Hence, it follows from the implicit mapping theorem that
(6.19) 9‘21 = ng(a,1/,£)

for [V[<<1. Similarly we have

(6.20) a; = flaunt/,6)

for [ll[<<1.

Therefore, by (6.15),
(6-21) 3(a,nsnu,6) = 0
if and only if

(622) {1113(5) + 1113(a,v,n‘2‘(a,v.e),c)}
- {1103(0) + ringer/saunas» = 0.

To rewrite (6.22) in a more convenient form, we utilize bounded

solutions of the adjoint system

(6.23) d + A*(t)o = o

34

of system (4.9). Let

(6.24) m = dim{.9P(P(a)) + x(I—Q(a))}*

and let {¢1(t),...,¢m(t)} be a complete set of bounded solutions of

system (6.23) which satisfies

(6.25) mm» + some»? = span{¢,(a)....,¢m<a)}.

Then (6.22) is equivalent to the following equations:

(6.26) citalllmgwl + sataungwwslcll
— {mg(a) + mg(a,u,n§(a,u,c),c)}] = 0, i=1,...,m,

where ¢:(a) is the transpose of ¢i(a). Since

(6.27) ¢i(t) = <I>*(a,t)¢i(a), i=1,...,m,

and

(628) chance) = ¢:(a)(I-P(a)) = dim). i=1....,m,

(6.26) becomes

35

(6.29) o = ¢:(a){Q(a)l: «atlas-analldt
+ Q(a) {: <I>(a,t)h(t,zu(l/)(t),a,f)dt
— (I—P(a)) JoaQ(a,t)g(t—a,7(t)dt
— (I—P(a)) 0100 <I>(a,t)h(t,zs(u)(t),a,c)dt}
= 1:9:(t)s(t-a,7(t))dt

* a
+ ¢i(a){ l <I>(a,t)h(t,zu(V)(t),a,c)dt

+ [md>(a,t)h(t,zs(u)(t),a,c)dt},
a

i=1,...,m. Here Zu(V)(t) a z(t;a,§u(u,n121(a,u,t))) is the solution of

(4.17) and 17‘; is given in (6.19). Similarly

z8(i/)(t) s z(t;o,§s(u,r,g(o,u,c))) is the solution of (4.12) and 73 is

given in (6.20). Thus we have derived a bifurcation equation (6.29) and

now it is reasonable to define the following quantities.

Definition 6.1. The Melnikov vector M(a,u,t) for system (2.2) is

defined by

(6.30) M(a,u,c) = (M1(a,u,c),...,Mm(a,1/,c))

where

36

(6.31) M,(a,u,c) = l°°¢§<t>g(t-a,7(t))dt
+ ¢:(a){ )0 <I>(a,t)h(t,zu(l/)(t),a,6)dt

+ l°° staslhttzstvlmmat},
a

i=1,...,m.

Also the linear Melnikov vector M(a) for system (2.2) is defined by

A

(6.32) M(a) = (M1(a),...,Mm(a))

where

A 00

(6.33) Mi(a) =1 ¢:(t)g(t—a,7(t))dt, i=1,...,m.

Remark 6.2. M(a,z/,6) = M(a) + 0(6) uniformly in V.

Remflk 3.3. The above argument to derive the Melnikov vector is
essentially the same as the Lyapunov—Schmidt reduction. However we
employed the above more elementary and geometrical argument which
will be useful when we derive the condition for the transversal
intersection.

The next proposition follows from the definition of the Melnikov vector.

Proposition 3.4. WS(§+,t) and w“(i_,c) intersect each other if

and only if M(ao,uo,f) = 0 for some 010 and ”0'

37

hoof. If M(ao,[10,6) = 0 for some ac and ”0’ then it is
obvious from the definition of the Melnikov vector that W?OC(X—+,£)
and w‘foc(i_,c) intersect each other. Conversely once w5(i+,c) and
Wu(§_,6) intersect, then there is a bi—infinite sequence {pi}?=_m of

points of intersection which approaches x+

x -» mm respectively. Hence for sufficiently large |i|, picW?oc(x+,6) n

and i_ as x-++oo and

u — . . . _
Wloc(x_,6) which implies that M(aO,V0,6) — 0 for some do and

V .
O D

38

§7. TRANSVERSAL INTERSECTION OF THE STABLE AND
UNSTABLE MANIFOLDS

In this section we will prove our main result which gives conditions
for transversal intersection of the stable and unstable manifolds.

Recall from Remark 4.2 that W‘I‘OC(§_,c) and w3

loc(;+’£) are

diffeomorphic to the graphs Fu(a,n‘ll,n‘2l) and Fs(a,ni’,n3) respectively
in a tabular neighborhood of 7 which are given by

 

 

(7-1) Fu(a.n‘1’,n‘2’) = ’0 T j 6 R
7‘11 6 Range(I-Q(a))
"‘21
m‘fta)+m‘,‘(a,n‘,‘.ngl 4 Range (4(a)
313(0)+m‘2’(a,n‘1’,n‘21)d

and
\
(7.2) Pitching) = ’a ] .11
vi
m‘](a)+m§(a.ni,n3) 6 Range P(a)
17; 6 Range (I—P(a))
\n§(a)+m§(a,n§n§)

 

 

Hence to Show the existence of transversal intersection, it is sufficient to

Show that column vectors in the following matrices D u 11 F11 and
(0,771,712)

39

 

 

 

 

D( s S)FS Span the whole Space R“.
0,711,772
/ \
(7.3) D F“ = 1 o 0
(0,711,012!)
0 I 0
0 0 I (a)
6 u ”u 0 ’u 6 "u
~ m +m ) — m — m (b)
336 1 1 317111 1 0,7121 1
0 u "u 6 ”u 6 ”u
~ m +m ) — — (c)
83A 2 2 677111 In2 6773 m2 ,
(1) (2) (3)
(7.4) D s S F3 —l1 0 o
(077,177’2)
0 I 0
6 s ‘S 6 'S 8 ”s
m +m ) — m — m (d)
35( 1 1 017513 1 an; 1
0 0 I (e)
0 s 's 6 ”s 6 's
m + ) — m — m (f)
Ed 2 m2 (977? 2 all; 2

Now we have the main result in this paper.

Theorem 7.1. Assume that system (2.1) has two hyperbolic critical

points x and x_ (not necessarily distinct) and has an orbit 7

+

connecting them: 7(t) -+ x as t -i on and 7(t) -+ x_ as t _, —oo.

+

40

W k = dim {59(1-Q(a)) n $(P(a))}s m = dillfl{5lt(I-Q(a)) +
.9B(P(oz))}l and let V = (ul,...,uk_l) 6{5B(I—Q(a)) n $(P(a))} n
Tfflafln, where P(a) and Q(a) are respectively the projections of
exponential dichotomies on (—oo,a] and on [a,oo) of system (4.9) and
satisfy the conditions in (4.25). Consider the perturbed system (2.2) and
define the Melnikov vector M(a,u,6) and the linear Melnikov vector
114(6) by (6.30) - (6.33). Then

(i) the cross sections Ws(§+,6) and Wu(§_,6) at t = 0 of the
time dependent stable and unstable manifolds of system (2.2) intersect
each other if and only if M(ao,uo,6) = 0 for some (10,120 and small
6.

In this case,

11
10c

if there exist m nonzero column vectors in the mxk matrix

(ii) the intersection of W (x'_,6) and W?OC(§+,€) are transversal

[3%, MM.) 33 Mops/0.4)].

13303. (i) This is PrOposition 6.4.
(ii) Consider column vectors in matrices (7.3) and (7.4).
Firstly it is clear that all column vectors in blocks (3) and (6) are
always linearly independent. Secondly by (6.26), M(ao,uo,6) = 0

implies that

(7.5) lm‘2’(ao) - use,” = I513(007V0,n;(a0,6),6)

" u u

41

(7.6) 331474,): [41' (aOXmgool-mgoon' + [41‘ (apagtmgmp—mgtaoh

 

 

 

 

_¢;(ao)(m‘21(ao)—mg(ao))d l¢1:i(ao)'dg(m12l(ao)—m;(ao))r

1

= '¢I<a.>si<m‘i<a.>-m3<ao>> + 0(2-

 

 

_¢,;lao)y§<mg(a,)-m;(a,))]

Therefore if 32 M(ao) at 0, then agmgmokmgmofl at 0 and hence
column vectors (1) and (4) are linearly independent at the point of

intersection. Finally

(7.7) 33—] M(ao,uo,6)

‘

/
at: 3 ~ 3
= ¢1(ao)-57{ml21(ao,uo,”Smog/0,6)6)—mg(ao,uo,n2(ao,uo,£)6))}
J

*‘ . 8 ' u u ‘ s s
¢m(ac)37{m2(ao’”o’772(ao’uo")0- m2(ao’Vo’n2(ao’Vo’f)c)} ’
J

\ /

 

 

j=l,...,k-1.

Therefore if '53; M(ao,uo,6) at 0, then 3%(mg-mg) at 0 and hence
j—th column vectors in block (2) and (5) are linearly independent at the

point of intersection.

Now consider the following decomposition of T7(a)Rn.

(78) small = {58(I—Q(a)) n 52(1’(01))}€9{590-900)n sea—Pom
0{58(Q(a)) n 58(P(a))}s{$(Q(a)) n sea-Pom

42

with the following dimensions:

dim{RU-Q00) 0 “(PM)” = k,
dim{WI-QM) 0 $(I-P(a))} = “—11.49
di114530201» 0 5B(1’(0))} = Dar—ks
di114530200) 0 5B(I-P(flr))} = m,

 

 

 

 

 

 

 

where n_ = dimWS(x_) and 11+ = dimWS(x+).
See the diagram below.
«(I-ow) k
n-n--k (RCPWU
71,- k (RUE-PHD
R C aw) .
m
Figure 7

From the above argument, (n—n_—k) + (n+—k)(= n—m—k) column
vectors in block (3) and (6) are linearly independent. The condition
of the theorem and the above argument imply that there are

(k—m) + 2m = k + m column vectors in block (1), (2), (4) and (5)
which are linearly independent and so we have (n—m—k) + (k-l—m) = n

linearly independent column vectors in matrices (7.3) and (7.4) o

43

The condition for transversal intersections in part (ii) of the
theorem can not be, in general, the necessary and sufficient condition for
transversal intersections since the Melnikov vector is the projection of the
distance vector (6.12) to the subspace R(Q(a)) n $(I—P(a)). See the
decomposition in (7.8). In other words, the Melnikov vector contributes
only to the last components of column vectors, i.e. components (c) and
(f), in matrices (7.3) and (7.4). See example 3 in Section 13. However
if

(79) $(P(a)) = 59042(0)),

the components (a), (b), (d) and (e) of column vectors of matrices in
(7.3) and (7.4) are missing and hence the Melnikov vector can be used
to give a necessary and sufficient condition for transversal intersection.

Thus we have the following

Corollary 7.2. Under the same assumption as in Theorem 7.1 and
. . . u — s — .
assumption (7.9), the intersection of W10c(x_,6) and Wloc(x+,€) is
transversal if and only if M(a0,i/O,6) = 0 for some (70 and V0 and
there exist in nonzero column vectors in the mxk matrix

(1 ‘ 0
[33 M(ao) 617 M(ao,uo,6)]. D
We remark that condition (7.9) implies m = n — k.

Reka 7.3. It can be easily shown, by the implicit mapping

theorem, that if M(al) = 0 and 3'5, M(a1) if 0 for some 011, then

M(a,u,6) = 0 for some a and u.

44

Now we define the following quantity. Let 7 C Wu(x_) n W8(x+) be a

homoclinic or heteroclinic orbit in it“.
Definition 7.4. The Splitting index 6(7) of 7 is defined by

(7.10) 0(7) = dim w3(x_) — dim w3(x+).

Notice from (6.24) that

(7.11) m = n — [{n—dim WS(x_)} + dim Ws(x+) — k].
Thus m,k and 6(7) satisfy the following relation.

(7.12) m = k + 6(7).

Remark 7.3. If 7 is a homoclinic orbit, then the Splitting index
6(7) is always zero. Thus by theorem 7.1(ii), all of k column vectors
in [35 M '3; M] must be nonzero to guarantee the transversal

intersection. This Situation is only the case in the homoclinic case.

Remark 7.6. Assume that 7 is a heteroclinic orbit in Rn. Then
we have three different situations.

(i) 3(7) > 0, i.e., dim w3(x+) < dim w3(x_). Then
in = k + 6(7) > k. Thus theorem 7.1(ii) implies that that there is no

transversal intersection because the matrix [35 M 3-1; M] is of the

45

Size mxk. A reason for this is that dim Ws(x+) < dim Ws(x_) is
equivalent to say that dim Wu(x_) + dim Ws(x+) < 11.

(ii) 0(7) = 0, i.e., dim w3(x+) = dim w3(x_). In this case
we have the same situation as in the homoclinic case.

(iii) 6(7) < 0, i.e., dims(x+) > dim Ws(x_). Then
in = k + 6(7) < k. Thus transversal intersection is possible.
In this way, we can classify the possibility and impossibility of

transversal intersection by using the Splitting index 6( 7).

46

§8 THE INDEX OF 7, THE FREDHOLM INDEX AND THE
DIMENSION OF THE MELNIKOV VECTOR

Consider system (2.1) with the same assumptions in §2 and system
(8.1) x = f(x) + 6g(t).
The linearized system of (8.1) along 7 is given by
(8.2) z = A(t)z + 6g(t),
and the system 2 = A(t)z has exponential dichotomies on half—lines
[0,oo) and (mo,0] with projections P and Q respectively.

We recall that the dimension m of the (linear) Melnikov vector

of system (8.2) is given by

(3.3) m = dim{.z P + 57 (14.3)}i
= dim{5t (I-P*) n 57 {3*},

which says that the dimension of the Melnikov vector is the same as the
number of independent bounded solutions of the adjoint system
3+AMM=0

We also defined the splitting index 6(7) by

(3.4) 0(7) = dim WS(x_) — dim w3(x+).

47

Though the splitting index 6(7) is defined by local data, that is, the
dimensions of stable manifolds of hyperbolic critical points, 6( 7) is
global in nature since it can be used to distinguish homoclinic and
heteroclinic orbits, and also used to classify heteroclinic orbits.

Furthermore the relation
(8.5) m = k + 6(7)

shows how the dimension of the Melnikov vector depends on 7.
In this section we Shall clarify the relationship between m,6(7) and
index L which is the Fredholm index defined as follows.

Define an operator L: Balms“) e BC°(R,Rn) by

(8.6) (LZ)(t) = i(t) - A(t)z,

where BCl(R,an) is the Space of bounded C1 functions from R to
R11 and BCO(R,Rn) is the Space of bounded continuous functions from
R to Rn. As we Shall Show, L is a Fredholm Operator. See also
Palmer [16]. The Fredholm index is defined by

(8.7) index L = dim(ker L) — codim (Range L)

Propoeition 3.1. 6(7) = — index L.

48

emf. Define a bounded linear operator A: BC°(R,Rn) .. Rm

by

(8.8) As = (lm¢](t)g(t)dt,m, l°°4,’§,(t)a(t)dt)

-oo

where ¢i’ i = 1,...,m are independent bounded solutions of the adjoint
system it + A*(t)¢ = 0.

Then by Lemma 5.10, z is a bounded solution of (8.2) if and only if

g 6 ker A. Thus Range L = ker A, which means that Range L is
closed and

(3.9) codim (Range L) = m,

and hence L is a Fredholm Operator.
Thus
index L = dim(ker L) — codim (Range L)
= k — m

= 4(7). 0

Remark 8.2. The Splitting index 6(7) in (8.4) was defined in
Sacker [19] and PrOpOSition 8.1 was also proved there. However his
definition is for linear systems. Our definition Of the Splitting index is
tO relate a local information about eigervalues to a global information

about a homoclinic or a heteroclinic orbit.

49

§9 COMPUTATION OF HIGHER ORDER TERMS

In the case Of dim{5£(P)(a)) n $(I—Q(a))} > 1 for
n—dimensional systems (n 2 3), we need to know nonlinear terms in
expression (6.31) Of the Melnikov vector to examine the transversality
condition. TO this end, we consider again bounded solutions on [a,oo)
and on (-oo,a] of system (5.2). We use the same assumption of
exponential dichotomies as in §4. These bounded solutions are given as
unique solutions of integral equations (4.12) and (4.17) respectively.

Let ”8652(P(a)), I773|<<1, and let z(r)S)(t) be the unique
solution of (4.12) which is guaranteed by the contraction mapping
principle. That is, z(ns)(t) is the solution of the following integral

equation:

(9-1) Z(t) = %(US)(g(t-ar7(t)) + h(trZ(t),arf)),

where the Operator Pans) is defined by

(9-2) ~73(178)(8(t-0s7(t))+h(t,Z(t)ra,€))

t
= 00,7073 + <I>(t,a)P(a) 1 <I>(a,r){g(T—ar,7(r)) + h(r,z(r),a,6)}dr

a
t
+ 9(tsa)(1-P(a)) l 9(arr){s(T-a,7(r)) + h(Trz(T)sar€)}dT°

To approximate z(7)s)(t), we use the following interation scheme

(9.3) z§n+1)(nsl(t) = strings—mm) + httzgnknsxthac».

50

Set z§0)(178)(t)5 0. Then
z§1)(7s)(t)= 5;,(08)(s0-ae(0)
and
42670 = r9;,(ns)(g(t-an(t)) + h(t,3,,(ns)(g(t—ar7(t)),arc))-
Notice that .9§(ns)(h(t,%(ns)(g(t—a,7(t),a,6)) = 0(a) and hence
2:244:70 = zglltnsm) + 694730)

for some function 2:1)(ns)(t). The true solution z(7)s)(t) Of (8.1)

satisfies

(9.4) 20250) = 297750) + 0(3)
= 417736) + 65,1)(770 + 0(42), t 2 a.

Apparently z§1)(ns)(t) is the bounded solution on [a,oo) of linear
system 2 = A(t)z + g(t-a,7(t)).

Similarly define 571(7)“) by

(9.5) 3;,(n“)(g(t—a,7(t) + h(t,z(t).a.c))

t
= 90,0001] + 9(tra)(I-Q(a)) [y 9(arr){g(T-ar7(7)) + h(Tvz(T)aa’a‘)}dT

t
<1>(cr.r){g(r-ar7(r)) + h(r,Z(T)rar6)}dr,

l

+ ‘1’(tra)Q(01)

51

where nu656(I—Q(a)), [nu] << 1 and let z(nu)(t) be the unique
solution of (4.17). That is, z(r)“)(t) is the unique solution of

(9.6) 20) = mules-ans» + madame».
We use the following iteration scheme.
(9.7) 23““)(7‘90) = embed—4,710) + htt,z,§”)(r“)(t),a,c)).
By setting 21(10)(77u)(t) s 0, we have

25,1)(7‘90) = «9;,(n“)(3(t-an(t)),

232)(r“)(t) = 25,1)(7‘50) + 63%“)(0,
and the true solution z(nu)(t) of (8.6) satisfies
(9.8) 4770 = 4397,1170 + effkrultt) + 00:2), t 2 a.
We notice that, by taking u 2 773 = 17116530301» n $(I—Q(a)),
z§1)(u)(t) and 23%;) give the linear Melnikov vector, and 2:1)(1/Xt)
and 21(11)(1/)(t) give the term Of order s or higher in the Melnikov

vector. Thus we have derived the following expression Of the Melnikov

vector:

52

(9.9) Mi(a,u) = Mi(a) + 6¢:(a){ja <I>(a,t)h(t,z1(ll)(u)(t),a,6)dt

+ [00Ma,t)h(t,zg1)(u)(t),a,6)dt} + 0(5").
0

In this way we can compute the Melnikov vector in arbitralily high order
Of accuracy. It is also clear that it is sufficient to consider the first two
terms in expression (9.9) for the transversality condition in Theorem 7.1.
More accurate expressions than (9.9) are needed for the tangency

condition.

53

§10 THE LINEAR MELNIKOV VECTOR

In this section we consider Special cases in which the linear
Melnikov vector gives a sufficient information for transversal and
tangential intersection of the stable and unstable manifolds. We consider
system (2.1) and (2.2) under the same assumption as in §2.

Case (i). Suppose

(10.1) k = 1 and 6(7) = 0.

This means that 52(P(a)) n $(I-Q(a)) = span{7(a)} and m = 1.

 

\ :
\

‘
v

i

Figure 3

Note that the Melnikov function in this case is

A

(10.2) M(a,6) = M(a) + 0(6).

Proposition 19.1. Assume (10.1) and suppose that there exists 6706
R such that

54

A

(10.3) M(ao) = o and ga M(ao) ,4 0.

Then wiloc(x—") and W?OC(X+,€) Of system (2.2) have a point Of

transversal intersection.

Proof. By the implicit function theorem, condition (10.3)

 

combining (10.2) implies M(a,6) = 0 and g5 M(a,6) 1t 0 for some a
near 070. Hence this prOposition follows from Theorem 7.1. 1:1
Note: Condition (10.3) can not be a necessary and sufficient

condition for transversal intersection.

Apparently the two—dimensional case satisfies condition (10.1). In this

Special case we have the following corollary.

Corollary 10.2. Suppose that system (3.1) and (3.2) are
two—dimensional. Then W?OC(X+,£) and W1110c(x_,6) Of system (2.2)
have a point Of transversal intersection if and only if there exists 006R

SO that
(10.4) M(a ) = 0 and (313 M(ao) 4 0.

Prmf. 'If' part is a Special case Of Proposition 10.1. Conversely if
transversal intersection exists, then by Corollary 7.2, there exists 041
d .
such that M(al,6) = 0 and HE M(al,6) a]: 0. USing (10.2), the

conclusion follows from the implicit function theorem. 0

55

 
 
  

MOO

R( 1‘ P<¢ll= R (Q00)

 

(RC P<uo>= ROI-9(a))
Figure 9
Case (ii). Suppose that

(10.5) w“(x_) n Ws(x+) = {7(t,l/): th, yrs c it“)

where S is a Open subset of Rk—1 and 7(t,u) is a homoclinic or

heteroclinic orbit connecting x_ and x+ for each VCS.

In other words the 'homoclinic or heteroclinic manifold'

Wu(x_) n Ws(x is parametrized by (t,l/)£ RxS. This case can occur

1)
when the system and its perturbation have some symmetric prOpcrtics.

See example 2 in §13.

Figure 10

In this case the linear Melnikov vector has the form

56

(10.6) Mi(a,u) = 1m ¢:(t)g(t-—a,7(t,v))dt, i=1,...,m.

Note that

A

(10.7) M(a,1/,6) = M(a,l/) + 0(6).

Hence we have

Pronosition 10.3. Assume (10.5) and suppose that there exist a

and V0 such that

(10.8) M(ao,l/O) = 0

and

(109) rank [d M(a V) 6 M(a 1/ )] = m
’ 63 o’ o 617 o’ o '

Then Wllloc(x_,6) and W?m(x+,6) of system (2.2) have a point Of

transversal intersection.

Prmf. By the implicit mapping theorem, we have M(al,l/1,£) = 0
6 6
and rank [317 M(al,V1,€) 63 M(al,ul,6)] = m for (011,121) near

(00,110). Then the statement follows from Theorem 7.1. a

&mark 13.4. In the case m = 1, the rank condition (10.9) gives

a necessary and sufficient condition for transversal intersection.

57

Next we turn to the tangency condition. Here a tangential
intersection of the stable and unstable manifolds means that the tangent
spaces of the stable and unstable manifolds at a point Of intersection do
not span the whole Space. Our discussion of tangency is based on
Corollary 7.2. Since Corollary 7.2 gives a necessary and sufficient
condition for transversality, we consider the situations in which the
condition in Corollary 7.2 is violated.

We consider the following system with parameters.

(10.10) x = f(x) + 6g(t,x,p)

where xcRn, ”(RN

, 6 << 1, f and g are sufficiently smooth in all
arguments, and g is periodic in t. Assume, as before, that the
unperturbed system (6:0) has a homoclinic or heteroclinic orbit 7(t).
Recall, first Of all, that
m = k + 6(7).
Thus it is clear that if 6(7) > 0, then intersection is always tangential
(see Remark 7.6).

Assume 6(7) 5 0 and assume that

(Hill) $(P(a)) = 304.2(3) (=10.

We consider only several special cases here. Extension to more general
cases is straightforward.

(i) Assume m =1.

58

This case includes e.g. k = 1, 6(7) = 0 in R2 and
k = 2, 6(7) = -1 in R3. We also assume that N 2 k.

Proposition 10.3. Suppose that

A

(10.12) M(0’01fl0) = g"; M(aovl‘0) = 01

A

(10.13) M(a0,p0) at 0

343a

and

A

(10.14) 3!; M(a0,p0) has rank k.

Then there exists a point Of tangential intersection for sufficiently small

6.

Proof. Define

F(a,u,p,6) = (M(a,u,p,6), 36 M(a,u,p,6), “(8)17 M(a,u,p,6)).
Note that F: 32““ .. Rk+l and F(ao,0,p0,0) = 0.
Since conditions (10.12), (10.13) and (10.14) imply the matrix

_ a . a ‘
D(a’#)F(ao,0,p0,0) " “3'5 M(001l‘0) 3E M(aov#0)
82 A 02 a
a? M(010sfl0) m M(00rll0)

 

59

has rank (k+1), by the implicit mapping theorem there exist functions
a(u,6) and p(u,6) such that

FM“), V7 11046)) = 0
for sufficiently small u and 6. Hence the condition in Corollary (7.2)
is violated and the statement follows. 0

See Wiggins and Holmes [22] for a similar result.

(ii) Assume that in = 2 and k = 2 (and hence 6(7) = 0).

Assume also that N 2 3.

Ezopositiop 13.3. Suppose that

A

(10.15) M(ao,u0) = 35 moose) = o

and the matrix

 

 

/ \
a ‘ a ‘

(10.16) 35M(ao,uo) meoruO)
‘92 191(0 ) 32 no )
$2 Mo Won Mo
K /

has rank 4.

Then there exists a point of tangential intersection for sufficiently small

6.

"U

roof. Define

 

6
F(a,z/,p,6) = (M(a,u,p,6), 35 M(a,u,p,6)).
Then the proof is identical to the one in Proposition 10.5. c

60

Next we consider a more special case.

(ii)'

condition ( 10.5).

Promsition 10.7.

A

(10.17)

and the matrix

 

K
(10.13) %M %M
62 * a2 '
\521“ 77—3—qu

has rank 4 at (670, V0, [10).

Suppose that

A

 

01'

a
M(007V07#0) = 65 M(009V09u0) = 0

 

Thus the Melnikov vector satisfies (10.7).

Assume that m = 2, k = 2 and N 2 2, and assume

a a '
317M 35M
62M 62
317 W

 

191
n

Then there exists a point Of tangential intersection for sufficiently small

6.

Proof.

 

Similar to PrOposition 10.6.

D

61

§11 HAMILTONIAN SYSTEMS
In this section we assume that the unperturbed system
(11.1) it = XH(x)

is completely integrable, and we consider its non—Hamiltonian and

Hamiltonian perturbations
(11.2) x = XH(x) + 6g(t,x),
and
(11.3) it = XH(x) + £XG(t,x).
We shall derive the Melnikov vectors for system (11.2) and (11.3).
We first recall some basic facts from Hamiltonian systems. Let
HcC°°(R2n). Then the Hamiltonian vector field XH with Hamiltonian

H on R211 is defined by

(11.4) XH(x) = JVH(x)

6
where J = [3 (1)] and VH(x) = E

 

 

62

Let F1, F26C°°(R2n). The Poisson bracket {FI’F2} of functions F1
and F2 is defined by

(11.5) {F1,F2}(x) = dF1(x)XF2(x), x312”

where dF1 is a differential l-form on R211.
One of the key facts on the Poisson bracket is the following:

{F1,F2} = 0 if and only if Fi is invariant under the flow Of XFj

where (i,j) = (1,2) or (2.1).

We suppose that system (11.1) has two hyperbolic critical points x_

and x + (not necessarily distinct) joined by an orbit 7(t) of system
(11.1): lim 7(t) = x+. Then the linearized system of (11.1) along the
t-iioo

orbit 7 is given by
(11.6) x = A(t)z, A(t) = DXH(7(t)).

Since A(t) = JD2H(7(t))r A(t) is infinitesimally symplectic for each
t 6 R. Namely

(11.7) A*(t)J + JA(t) s 0, trIR

at:
where A (t) is the transpose of A(t). Now let us define the adjoint
system of (11.6) by

(no 4=—¢M0

63
where ¢(t)6T:;(t)R2n e (Rznf.
We note that z(t) is a solution Of (11.6) if and Only if
¢(t) = (J-lz(t))* is a solution of (11.8). This is clear from (11.7).

We have the following

Promsition 11.1. Let Pec°°(172"). If {F,H} = 0, then XF(7(t))
is a solution Of (11.6) and hence dF(7(t)) is a solution Of (11.8).

Prmf. Let <°,-> be the standard inner product on Rzn. Then
{F,H} = < VF, JVH >.

Since
0 = V < VF, JVH >
= D2F JVH - D2HJVF
_ 2 2
we have
Hence

gt? XF(7(t)) _—. DXF(7(t))XH(7(t))
= DXH(7(t))XF(7(t))
= A(t)XF(7(t)).

64

Pmuw

(J‘lxptml))* = dF(7(t)). n

One of the special situations Of the Hamiltonian nature appears in the

Splitting index 6(7) Of 7. Since DXH(x+) = lim A(t), DXH(x+)

t-ioo

is infinitesimally symplectic. Hence if A60(DXH(x+)), then X, —A,
—X60(DXH(x+)), where 0(DXH(x+)) is the spectrum Of DXH(x+)
and X is the complex conjugate of A. This symmetry property
implies that both of the stable and the unstable subspaces of DXH(x+)
have dimension 11. Similarly the stable and the unstable subSpaces Of

DXH(x__) = lim A(t) have dimension n and hence we have
t4 ~00

PrOpoeition 11.2. Suppose that Hamiltonian system (11.1) has a

homoclinic or heteroclinic orbit 7. Then 6(7) = 0. 0

Now we suppose that Hamiltonian system (11.1) is completely
integrable. That is, there exist 11 C°° functions F1 = H, F2,...,Fn

on '1211

which are in involution, namely {Fi’Fj} = 0 for 1 S i,j S n,
and dFi, i = 1,...,n, are linearly independent everywhere in l12Il —
{xi}. We recall that dH(x*) = 0 in our case.

Let Fi(7(t)) = fi 6 R, i=1,...,n, and define the set
Mf = {x c I22“ — {xi}: Fi(x) = fi, i=1,...,n}. Then the Liuville
integrability theorem (cf. Arnold [1]) asserts that Mf is a

R2n

n-dimensional smooth manifold in which is invariant under the

flow Of each

65

Hamiltonian vector field XF.’ i=1,...,n. Therefore the components of
i

the stable and unstable manifolds both Of which contain the orbit 7

coincide each other and it is contained in Mf Furthermore XF.(7(t)),
1

i=1,...,n, constitute a basis Of T 7(0M?
Now since system (11.6) has exponential dichotomies on [a,oo) and
on (—oo,a], 676R, we denote projections at t=a by P(a) and (2(0)

respectively. Then we have the following

Proposition 11.3. Suppose that Hamiltonian system (11.1) is

completely integrable. Then
(i) mm» = 304.2(3) = T7(0)Mf’

and

(ii) {dFi(7(t)); i=1,...,n} forms a complete set Of bounded
solutions of the adjoint system (11.8).

Prmf. These are clear from the above argument and proposition

11.1. n

Thus for completely integrable Hamiltonian systems in R2", we have
k = m = n where k = dim{5£(P(a)) n $(I-Q(a))} and
m = dinu{59(P(a)) + 3(I-Q(a))}*-

66

Now we will give Special forms for the Melnikov vector Of system
(11.2) and (11.3). Let x(t) = 7(t+a) + 62(t+a). then system (11.2)
and (11.3) become

(11.9) 2 A(t)z + g(t-a,7(t)) + h(t,z,a,6)

and

(11.10) 2 = A(t)z + XG(t—a,7(t)) + Xé(t,z,a,6)

respectively. Here

A(t) = mines»,

h(t,z,a,6) = % {f(7(t)+6z) — f(7(t)) - 6Df(7(t))z
+ 630-0670) + 62) - 630-017(0)},

O(t,z,a,6) = % {H(7(t) + 62) — H(7(t)) — 6DH(7(t))z
+ 6G(t—a,7(t) + 6z) — 6G(t—a,7(t))}.

Theorem 11.4. For system (11.2), the linear Melnikov vector

A

M(a) and the Melnikov vector M(a,i/,6) are given by

A

(11.11) Mi(a) = in dFi(7(t))g(t—a,7(t))dt, i=1,...,n

-00

and

67

(11.12) Mi(a,u,6) = Mi(a) + ((1 dFi(7(t))h(t,zu(V)(t),a,6)dt

+ 1‘” dFi(7(t))h(t,zs(u)(t),u,6)dt, i=1,...,n.
a

where zu(u)(t) and ZS(V)(t) are bounded solutions on (—oo,a] and on

[a,oo) respectively Of system (11.9).

m. These are Simple consequences Of Proposition 11.3 (ii) and

Definition 6.1 Of the Melnikov vector. u

Corollary 11.3. For system (11.3), the linear Melnikov vector

M(a) and the Melnikov vector M(a,l/,6) are given by
A 00
(11.13) Mi(a) =[ {Fi,G(t—a,-)}(7(t))dt, i=1,...,n
'11)
and

(11.14) Mi(a,I/,6) = Mi(a) + )0 dFi(7(t))Xé(t,zu(1/)(t),a,6)dt

+ (I: dFi(7(t))Xé(t,zs(p)(t),a,6)dt, i=1,...,n

where Zu(l/)(t) and zs(p)(t) are bounded solutions on (-oo,a] and on

[a,oo) reSpectively Of system (11.10).

68

Roget. These are simple consequences Of PrOposition 11.4 and the

definition of the Poisson bracket. n

The expression of the Melnikov vector in a more Special case was

given in Holmes and Marsden [10]. See also Wiggins [21].

Remark 11.6. The dimension of the Melnikov vector for a
completely integrable Hamiltonian system is the same as the degree of

freedom Of that system.

A

Remark 11.7. Since the linear Melnikov vector Mi(a) can be

written as
(11.15) Rita) = l°° {F,,G(r,-)}(7(t+a))dr.

we have

(11.16) g5 We) = (m M {Fi,G(7,-)}(7(t+a))dr

83

=1 {HrinG(Tr°)}(T(t+a))dT

3

= 1‘” {H,{F,;G(t-a,-)}(7(t))dt-

-00

Here we used the following fact: for f 6 C°°(R2n),

69

3.

(11.17) g; (th) = {H,th}

‘-
where Ft is the flow of XH and th is the pullback Of f by Ft'

More generally we have

k . 00
(11.18) I??? M440 = .3 innit... dampen—4,.)}(v(t))dt.
-times

70

§12 A HETEROCLINIC ORBIT TO INVARJANT TORI

In this section we extend our theory deve10ped in previous sections
to the case of a heteroclinic orbit to invariant tori. The system we

consider is

(12.1) 51 = f(x), den

and its perturbed system

(12.2) it = f(x) + 6g(x), |c|<<1

where f and g are sufficiently smooth and are bounded on bounded
sets. We assume that system (12.1) has two normally hyperbolic smooth
invariant tori M1 and M2 on which the flow of f is quasi—periodic,

and also that system (12.1) has a heteroclinic orbit 7 to M1 and
M2. That is, there exist orbits {xi(t): th} C Mi, i=1,2, such that

”(1)-111(1)] -»0 as 14—00

and
2
|1’(t) -x (t)| -1 0 as t-1 +00.
If M1 = M2, we have a homoclinic orbit to invariant torus M1 as a
special case. We remark that M1 and M2 could be any normally

hyperbolic smooth invariant sets with no stationary points. However we

71

assume the above conditions for its simplicity and applications. We also
remark that the normal hyperbolicity is nothing but the hyperbolicity to
the normal direction.

To deve10p an anologous theory to the one in previous sections, we
need the expressions of the stable and the unstable manifolds. For this
reason we shall decompose system (12.2), in neighborhoods of invariant
tori Mi, into the tangential and the normal components and apply the
theory of exponential dichotomy to the normal components.

Let dim Mi = di (i=1,2) and we assume for i=1,2 that Mi is

given by the embedding
(12.3) ui: Ti .. Mi c IRn

where T1 = Sl><...xS1 (di—times) is a standard di—dimensional torus
with coordinates 0' = (”1,...,021} From now on we supress i and
1

assume that M stands for M1. The case of M2

is done exactly in
the same fashion.
Let TIRDIM be the ristriction of the tangent bundle Tan to

M. Then
11 _ 1.
(12.4) (TIR IM)x — TxM 0 TxM’ xcM

where TxM and T;M are respectively the tangent Space and the

normal space to M at x.

72

Now by the tubular neighbourhood theorem, there exist a
neighborhood U of T in Txnn‘d, a neighbourhood v of M in
T‘LM 'and a linear map N(0): Rn"d -» R” for each 0cT such that
the vector bundle map F: U C Tan—d -» V C T‘M defined by

(12.5) F(0,z) = 11(9) + cN(0)z

is a diffeomorphism. Here N depends on 0 smoothly. Clearly we

have
(12.6) (Du(0))* 11(0) = o
and

(12.7) N*(0) N(0) = 1‘"d

where (Du(0))* and N*(0) are transposes of Du(0) and N(0)

respectively. By using P, we transform the vector field f + cg on

a.

a.
R“ to the vector field F(f+cg) on Twin—d where F is the
pullback by F. We set

«- d
(12.8) F(f+cg) :02
J:

n—d a

.0
A] + E B .
1 as; (=1‘52

That is,

73

(12.9) f(u(0) + 6N(0)z) + cg(u(0) + 6N(0)z)
= [1311(0) + c g, N(0)z] A1 + (11(0) 131
A h

d n—d

By using (12.6) and (12.7) we have

(12.10) A; = (nu(o))*1(u(o)) + 6(Du(0))* [- g7, (N(0)z)Du(0)f(u(0))
5 -Df(u(o))N(o)z - g(uwm + om
Ad

 

5 w + €9(0,z) + 0(62).

Here we assumed that that (Du(0))*f(u(0)) = w, w = (wl,...,wd) are

rationally independent. Under this assumption we also have

(12.11) 1211 = N*(o)[n1(u(0))N(o)z+g(u(o)) + 0(a) - g, N(0)z A1 1

l.3n-d * * Ad
= N (0)Df(u(0))N(0)z — N (0)639 N(0)z)w
+ N*(o)g(u(o)) + om

a A(0)z + N*(0)g(u(0)) + 0(6)-

Note that A(0) is a linear mapping for each 0.
To obtain the differential equations in TxIRIHl, we need a scale change
in time. This is because no will be changed after perturbation. Thus

we set

74

(12.12) x((1+£fl)t+ca) = F( 0(t),z(t)),

where 36 R and as R. By using (12.12) we have the following system
in szn‘d.

(12.13) 0 = (l+cfl)[w+69(0,z) + 0(3)] = w + c(fluH-6(0,z)) + 0(2)
2: (1+e0)[A(o)z+N*(o)g(u(o)) +0001
= A(0)z+N*(0)g(u(0)) + 0(5).

1 <<—1 so that 7(t1) is enoughly close to M.

Now we choose t
Then from (12.5), there exist unique 01 s a1(tl)cT and

w1 = “11(11): Rn‘d such that

(12.14) 7(t1) = u(al) + cN(al)w1.

l

1andw.

Hereafter we use a and w instead of 0
Consider system (12.13) and let z = E(0,t) where 2 is a
bounded function for t£(-oo,t1] which will be determined later. Then

the '0—equation' in (12.13) becomes

(12.15) 0 = w + c(0o+o(0,2(0,t))) + 0(62).

Let 0(t) 5 0(t;t1,o; E) be the solution of (12.15) with 0(t1) = a. By

using 0(t) for 0, the 'Z-equation' in (12.13) becomes

75

(12.16) i = A(Mz + 11120211660)» + 0(a)

with z(tl) = E(a,tl). Since M is normally hyperbolic, the linear

system 2 = A(0(t))z has an exponential dichotomy on (w,tl]. This

Rn—m _. Rn—m

means that there exist a projection Q: , constants K 2 l

and L > 0 such that

(12.17) |<I>(t,t1)Q<I>(tl,S)| s Ke_L(t—S), s s t s 11,

and

(12.18) |<I>(t,t1)(I—Q)<I>(t1,s)| 5 Ke_L(S_t), 1 g s s 11,

where <I>(t,s) is the transition matrix of 2 = A(-é(t))z. Thus
z(t;t1,2(a,tl)) is a bounded solution of (12.16) on (-oo,t1] if and only
if
(1219) 2021.229» = 20050422211)
t * - -
#2015042) {1 201.9011) («showman

t * - -
+ 0(e)}ds + 20,150 1 201.com («show»)

+ 0(6)}(18.

By letting t = t1, we have

76

l _ _
(12.20) 2w?) = n“ + Q It o(11,s)(N*(o(s;.1,a, 2))g(u(o(s;tl.a», 2»)

+ 0(6)}ds

where ”u c $(I—Q). We can show, by the contraction mapping
principle, that equation (12.20) has a unique solution 2(nu)(a,t1) for
each "u which is enoughly small. Thus (12.20) gives a local expression

of the unstable manifold of system (12.13) near MT in the space
Tan—d.

 

(12.20) has the folliwing expression in the 'original space' Rn.

(We use i = 1 this time).

(12.21) {11 E u1(al) + 6N1(al)Zl(al,tl)
= 111011)
1 _ _
+ 61 0“+N1(a1)q 1t 2(tl,s){N1*(ol(s;tl,al;21) g(u1(01(s;tl,a1;21>))

-oo

+ 0(6)}ds]

where in = N1(ozl)17u c 52(N1(a1)(I—Q)).

Next we choose t2 >> 1 so that 7(t2) is enoughly close to
M2. Then there exist unique 02 = a2(t2)c T2 and W2 = w2(t2) c

n—d2
R such that

(12.22) 7(12) = u2(a2) + (N2(a2)w2.

In exactly the same fashion as before we have the following local

expression of the stable manifold of system (12.13) near u2(a2) c M2

in the 'original Space' Rn.

77

(12.23) {3 a 12(a2) + 1N2(o2)22(o2,12)

= u21a2)

t2
+ 1[ 0 —S+N 22(o )(1 —P) 1 02(123) (N2(22(s 12 11,222))g(u2 (22(112 112,22)»

0(6)}dS]

where 775 c (N2(a2)P) and P is a projection of the exponential
dichotomy on [t2,+oo).

To measure the distance between the sections of the stable and
unstable manifolds given in (12.21) and (12.23), we shall 'carry' the
section of the unstable manifold given in (12.21) to the hyperplane
7(t2) + 52(N2(02)) by the flow of (12.1). Hence we shall need the
following expressions of the unstable and stable manifolds. From (12.14)

and (12.22), we have

(12.24) {11 = 7(11) + 6N1(al)[—w1+nu
t1
+o) 01 (11 ,ls){N( *(01(s;o1,1 1;z1) )g(u 1(01(s;,o11 1 ,21)))

+0(c)}ds]
101) + Mo“)

and

78

(12.25) 5‘1 = 7(12) + 6N2(a2)[-w2+7ls

2

1 ... - _

+(I—P) 1 <I>2(t2,s){N2 (02(312,o2,z2 )g(u2((02112,o2;z2)))
Q

+ 0(6)}d8]
5 7(12) + 1M3(r,3).

Now we consider system (12.2) along the heteroclinic orbit 7. Let

B(t) and a(t) be bump functions such that

(12.26) 5(1) 2 s1 for 1

t1 and a(t)5a1 for t
for t 2

t a2 for t

IV IA
IV IA
6"

and let

(12.27) x((1+cfl(t))1 + 121(1)) = 7(t) + cy(t).

Then system (12.2) becomes

(12-28) 5' = fl(t)y + g(7(t)) + W ()+ a(t+) +(Bt) (t))t}f(7(t) + 0( )
where 13(1) = Df(7(t)). Now let X(t,s) be the transition matrix of

y = B(t)y and let y(17u)(t) be the solution of (12.28) with the initial
condition y(nu)(tl) = MUM“). Also define 72(7)“) so that

(1229) y(1“)(r2(n“)) «2 1112) + 21112012».

79

We note that r 211(7) )= t2 + 0(5) because of the continuous

dependence of solutions on initial conditions. Thus

(12.30) y11“)<r2(n“)) = x1r211“),11)M“(1“)
720711) 2 ll .
+ {1 xv (n ),s){g(7(8)) + (21s) + a(s)

+ 13(8)8)f(7(8)) + 0(6)}dS
2

t
- x02.t1)M“(n“) + 11 X(t2.s){g(1(s))
+ (5(8) + a( s() + B(s)s)f ())7(s )}ds + 0(6).

To measure the distance between y(r)“)(1'2(nu)) and Ms(1)s), we

use linearly independent bounded solutions on R of the adjoint system

(12.31) 10 + 3*(1)¢ =

Let {¢1,...,¢m} be a complete set of linearly independent bounded

solutions on R of system (12.31) which satisfy

2 2 2 1 ._
(12.32) ¢i(t )c{$ (P) n 3(N (a ))} , 1—1,...,m.
Define

(12.33) Mi = 0:02) iy(n“)(r2(n“)) + 111121111» - 102)) - M3113»,
i=1,...,m

80

By using (12.24), (12.25) and (12.30), Mi is computed as follows.

112.34) 12112) 1111“)1r21n“)) 15111021111» - 1112)»
2

* t
= 1,112) (11121511111111) + {1 x112.s)1g111s))

+ 1(1) + 1(1) + 13100111111012 + 141021111»

- 1112)) + 0101
2

:1: t :1:
=¢,1t1)M“11“)+ {1 140111110) +1010 + 1(1)

+ fi(S)S)f(7(8))}d8 + 0(1)
2

:1: t :1:
= 0,111)M“11“)+ {1 1,1s)g111s))ds + 010

= 0:111)N11a1)1—w1 + n“
1
+ Q It 21111.10{N1*17I11s))g1u11211s))
2
t :1:
+ 0(6)}dsl + {1 0,1s)g111s))ds + 010

t1 2

all :1: t :1:
=1 21111011711s))g1u11v11s)))ds + {, 0,1s)g111s))ds

00
+ 0(6)
1 2

:1:

t a:
¢i1s)g1u117i11s)))ds + {1 0,1s)g111s))ds + 01:)
1 :1: t2 :1:
0,1s)g111s»ds + {1 ¢,1s)g111s)))ds + 0(a)
2
¢§1s)g111s))ds + 011).

-oo

1
I
1
I
1
I

-oo

-oo

81

Here we let ¢i(t) = \P%(t)Nl*(71(t)), t 5 t1.

112.35) 02112) M3113)
= 13121112113142 + n,
2 .. .—
+ (1.1») It 22112.1.){N2*1021s))g1u21121s))> + 011)} as

2
t a: :1: - -
1 221W1021s))g1u21021s)))ds + 01:)

t2

1 ¢§1s)g1u21521s)))ds + 011)
00 t2 *
1 ¢,1s)g171s))ds + 010.

00

Here we let ¢i(t) = \II?(t)N2*(-02(t))i t 2 t2.

Thus we have

and we have proved the following

Theorem 12.1. The linear Melnikov vector M = (M Mm) for

1""’
system (12.2) is given by

(12.37) 111. = ¢:(t)g(7(t))dt, i=1,...,m,

l

82
where m is the number of linearly independent bounded solutions of
° *
¢ + [Df(7(t))l ¢ = 0. 0

Remark 12.2. Since the heteroclinic orbit 7 is contained both in
the unstable manifold Wu(M1) of M1 and the stable manifold
w2(M2) of M2, it is interesting to consider how 1hese manifolds
intersect each other along the heteroclinic orbit 7. Consider the case of

a time—independent perturbation and define
_ . u l s 2
(12.38) k — d1m[T7(t)W (M ) n Tymw (M )]
where T 7(t)Ww(M1) and T 7(t)WS(M2) are tangent spaces at 7(t)
to W“(M1) and W3(M2) respectively. Then k and the dimension
m of the Melnikov vector have the following relation.
_ . u 1 . s 2
(12.39) m—n—[dlm W(M)+d1mW(M)—k],
where
(12.40) dim w‘1(M1) = dim Sea—Q1) + 01

and

(12.41) dim w3(M2) = dim 51(P2) + d2.

83

Note that dim w“(M1) = n - [dim WS(M1) - d1].
If we define the splitting index 6(7) of 7 by

(12.42) 0(7) = dim WS(M1) — dim WS(M2),

we have, from (12.40), the following relation

553:. ‘52-. I

77 Jan

I..-
J

(12.43) m = k + 6(7) + (11
which is a generalization of (7.12).

Now we go into a special case to which Theorem 12.1 can easily be
applied. consider a system with a quasi periodic perturbation

(12.44) 2 = f(z) + cg(z,wlt,...,wdt), 211211

where g is periodic in each 't' argument and wl,...,wd are rationally
independent, see Meyer and Sell [13]. We assume that the unperturbed
system 2 = f(z) has a homoclinic or heteroclinic orbit 7 to
hyperbolic critical point(s). System (12.44) is equivallent to the following

system on the torus Td.

(12.45) 2 = f(z) + cg(z,0)

0:0

84

where a = (01,...,0‘1), w = (wl,...,wd) and (2,0): Rnde. This is a
special case of system (12.2) in the sense that the 'z—dynamics' of the
unperturbed system of (12.45) is globally defined in the normal bundle of
Td. By using the homoclinic orbit 7, the homoclinic orbit 7 of

system (12.45) to the torus Td is given by

If"

(12.46) "7(1) = (7(1), 1011 + 01,...,wdt + 0d)

’3. .-_
1

where 0140,22), i=1,...,d. This is because the 'z—dynamics' and
'O—dynamics' of the unperturbed system of (12.45) are completely
decoupled. By Theorem 12.1, we have the following corollary in this

case.

rollar 12.2. The linear Melnikov vector M(01,...,0d) =
(Ml(0 ,...,0d),...,Mm(01,...,0d)) for system (12.44) is given

A

00 :1:
(12.47) Mi(01,...,0d) = 1m ¢i(t)g(7(t),w1t + 01,...,wdt + 0d)dt,

i=1,...,m. Here {¢1,...,¢m} is a set of linearly independent bounded
solutions of 113 + [Df(7(t))]*¢ = 0. 1:1

As a Special case, we shall prove the following pr0position for

two—dimensional systems. See also Meyer and Sell [13] and Wiggins [21].

Proposition 12.3. Consider system (12.44) with the same

assumption as before and let n = 2 and d 2 2. Then the stable and

85

unstable manifolds of system (12.44) intersect transversally if and only if

for the linear Melnikov function M(0l,...,0d) defined in (12.47) (i = 1

in this case), there exist (0 ,...,0d) such that

(12.43) 14(1—9 ,...,0d) = o

and

\ 4‘“:me

E

(12.49) (xwhi)(3l,...,0d) 4 0

where w = (w1,...,wd) and 221111 is the Lie derivative of 114 with

respect to w.

m. Let 0i(a) = 0i - wia, i = 1,...,d, 06R and define

13(5) = M(0l(a),...,0d(a)).

Then we have

(12.50) D(a) = ¢*(t)g(7(t),w1(t-a) + 01,...,od(1-o) + 5d)d1

¢*(t+a)g(7(t+a),wlt + 5,...,ed1 + 1d)d1

2‘8 3‘8

and

86

(12.51) D(O) = 14(31,...,0d) = 0.

From (12.50) 0: works as a 'sweeping' parameter along 7. See Figure
11. Since D(a) changes along 7(a), adR and since the difference of
the true distance of the stable and unstable manifolds and D(s) is of

order 6, the implicit mapping theorem implies, using (12.51), that the
transversal intersection exists if and only if D(O) = 0 and D'(0) ,1 0.
Finally

(12-52) D'(0) =i§l 37: (910-9d) a— (0 )
—d 6M - _ _
= ‘E “’1 3‘9- (911 d) "(JM)( 9(1) U

 

 

TN)

 

>1
é/

 

 

 

 

 

 

Figure 11
Remark 12.4. We note that the case of two—dimensional systems

with periodic perturbations in §10 is a special case of this pmposition.

That is, g; M (00) in Corollary 10.2 is generalized to (.ZwM)(01,...,0d).

87

§13. THREE EXAMPLES

In this section we apply the methods deve10ped in previous sections
to three examples. We shall exame (1) a two—dimensional system
which has transversal intersections, (2) a four—dimensional system which
has both of transversal and tangential intersections and (3) a system for
which condition (ii) in Theorem 7.1 is not satisfied but transversal

intersection exists.

Example 1 (Chow, Hale and Mallet—Paret [4])
We consider the following second order equation
3

(13.1) 35 — x + f x2 = 1 cost,

where c is sufficiently small. That is,

(13.2) g: [’y‘] =f(x ,y) + (g(t)

=:-3[ 243+] [cogt] '

We notice that the unperturbed system (i.e. (=0) has a homoclinic orbit,

sech2(t/2)
(13.3) 1(1) = [113(1)] 5 [Asech2(t/2)tanh(t/2)]

to the origin. See the figure below.

88

'o- 1:
V

fl
CS

211) = a“,
\ 1311:)

 

 

I
0
Figure 12
The linearized system along 7 is
(13.4) 2 = A(t)z
where
_ _ 0 1
(13.5) A(t) .. n1(7(1)) .. [1-300) ].
The adjoint system (is + A*(t)¢, = 0 has only one linearly independent
bounded solution

(13.0) 0(1) = [‘38]

and hence the (linear) Melnikov function is

.‘.¢ 6):!

89

8

A

(13.7) M(a) =

a:

45 (t)g(t-a)dt

p(t)cos(t—a)dt

I
.00
on
I
.00
=-csina

where c > 0 is a constant.

Since
(13 8) ‘1 M(n7r) = (—1)“+1c n = 0 41 42
° 35 a a a 7'",

the perturbed stable and unstable manifolds always intersect transversally

and so tangential intersections never occur.

Example 2 (Gruendler [6])

In this example we consider the following system of two second order

equations.
.. 2 2 .
(13.9) x1 = x1 - 2xl(xl+x2) + c{—3plx1-p2xl
2p 4p
3 2 2 4
+ —2- (3x +x )coswt + —-2 x coswt}
1+1.) 1 2 1+0) 1x2
.. -x —2x(x2+ 2)+ c{— — - 51 +4.“:3 xxcoswt
x2'2 21112 ”1221122 —21+w 12
2p
4 2 2
+ j (x +3x )coswt}.
1+0) 1 2

Here pl, 112. p3 and p4 are parameters, and c is assumed to be

sufficiently small. We consider first the unperturbed system (c = 0).

90

As easily seen, this unperturbed system is a Hamiltonian system. Let
5:1 = x3 and $12 = x4, and let x = (x1, x2, x3, x4). Then the
unperturbed system becomes

(13.10) 51 = XH(X)
where the Hamiltonian function H(x) is given by

(13.11) H(x1,x2,x 3,x4 x4) = 2 (x2 + x2) )+ g(x2 +x2).

Furthermore system ( 13.10) has one more first integral

(13.12) F(x1,x2x3x4) = xlx4 — x2113

which results from the conservation of the angular momentum.

Since

(13.13) {F,H}(x) = dF(x)XH(x)

= [114-113-2221] 1‘3
:4
x1—2x1(x1+x2)

x2"2"2(1‘1 +1‘2),

 

 

and since dF(x) and dH(x) are linearly independent for any

de4\{0}, unperturbed system (13.10) is a completely integrable system

91

in R4\{0}. So we shall utilize this special structure (see Proposition
11.3(ii) and Theorem 11.4) to derive the Melnikov vector even through
the perturbed system is not a Hamiltonian system.

Next we notice that the unperturbed system (13.10) has a

homoclinic orbit 7(t,o) to the origin.

(13-14) 7(t,0) = (D(t), 0, 13(0, 0)

where p(t) = sech t.

In terms of the complete integrability, we know that the stable and the
unstable manifolds of system (13.10), both of which have dimension two,
must coinside along 7(t,o) and in fact, by using a symmetry property

of XH, this 'homoclinic manifold' can be expressed as

(13.15) 7(t,v) = (p(t)oosu, p(t)sinV, p(t)cosu, p(t)sinu),
uc[o,21r), th.

That is, system (13.10) has a family of homoclinic orbits parametrized by
u. Thus system (13.10) is an example to case (ii) in Section 10.

Now we go back to the original perturbed system (13.8).
2n

3 2 2
Let g(tlx,p) = (0,0,—3p1x1—p2x3 + W (3x1-l-x2 )coswt
4114
+ -—42- ooswt,
1+w :152

—plx2—p2x4+ 7x1xzcoswt + —-Q (x1+3x2)coswt),

where ,u = (p1,p2,p3,p4). Then system (13.9) has the form

92

(13.16) 5: = XH(x) + cg(t,x,p).

As we mentioned in case (ii) in Section 10, the linear Melnikov vector
can be used to detect a point of transversal or tangential intersection of
the perturbed stable and unstable manifolds of system (13.16).
Furthermore by virtue of Proposition 11.3 (ii), bounded solutions of the
adjoint system of the linearized system of system (13.10) along an orbit

7(t,u) are given by

(13.17) dH(7(t,u)) = (-p(t)cosu + 2p(t)3cosu, —p(t)sinu
+ 2p(t)3sinu, p(t)cosu, p(t)sinu)

and
(13.18) dF(7(t,1/)) = (p(t)sinu, — p(t)cosu, — p(t)sinu, p(t)cosu).

It is easily shown that dH(7(t,V)) and dF(7(t,u)) are linearly
independent.
We can now compute the linear Melnikov vector

may) = (1911mm, M2(a,u)) as follows.

(13.19) 111W) = I°° dH(7(t,V))g(t-a,7(t,V),u)dt

= J00 {1118008211 + sin2v)p(t)f)(t) _ 2f)(t)2
+ 1:‘r—wlzhcgcosu + #43inV)P(t)2I5(t)cosw(t—a)}dt

= — :2; p2 — xw sech(£‘§’)(—p3sinu + p4cosu)coswa.

93

Let c = 1r sech(1‘2‘—’). Then the Melnikov vector becomes

A

(13.21) M(a,u,p) = - :2; 112 - cw(u3cosu + p4sinu) sinwa

2plsin2u + c(—p3sinu + V4cosv)ooswa .

’

To find points of intersection, consider, for example, the case u = 0.

.I.v‘
. H

In this case the Melnikov vector becomes

(13.22) M(a,o;,u) = - 33-112 — cwp3sinwa . I if.“
Cp4coswa

Solving NI(a,o;p) = 0, we have the following bifurcation set S in the

parameter space (p1,p2,p3,p4).

(mm) S=AUBUQ

where A = {(ul,n2,u3,u4)= #2 = * % CW3, #4 ¢ 0, #1312611},
B = {(#lmzm3m4): |u2l < lgcwp3l, #4 = 0, #141342},
C = {("1’”2’”3”‘4)‘ ”2 = * gcwl‘s’ ”4 = 0’ ”1’”3‘R}'

See Figure 13.
Next we examine the transversality and the tangency of intersection

in the case u = 0. The derivatives of M are given by

(13.24) ~35 M(a,0;p) = —cw2p3coswa
—cwu4sinwa

94

and

(13.25) gamma”) = ”“43”“ .
4p1-Cp3coswa

Since, in this example, the stable and the unstable manifolds of the
unperturbed system coincide and constitute a two dimensional manifold in

R4, rank [35- Nf(a,0;p) ‘3; NI(a,0;u)] = 2 implies a transversal

. ‘hw‘giflfiA-ufl

intersection. (see Proposition 10.3.)

(i) Let pcA. Since
6 ‘ a ‘ 0 t c

rank [35 M(a,0,#) a; M(a.o.p1 = rank [, CW4 4,?‘1‘41 = 2,

intersection is always transversal.
(ii) Let MB. In this case we have

6 . 6 . -cw2p3coswa 0

[6'6 M(0,0,fl) a; M(a,0,fl)] = 0 4fl1_cfl3008wa
(ii—1) If pl ,1 71; cp3coswa, then the rank of the above matrix is 2

and so we have a transversal intersection.

(ii-2) If p1 = i- Cp3coswa, then '3; NI(0:,0,,u) = 0. By computing
the rank of (10.18), we have a tangential intersection.

(iii) Let ch. In this case

3 ' 3 ' 0 0
M 0,07” M 0,01” = .
[‘55 ( ) '5; ( l [0 4l‘1 ]

By computing the rank of (10.18), we have a tangencial intersection if

 

 

 

\\\\

 

 

 

 

 

 

 

96

Example 3

The aim of this example is to give an example in which condition
(ii) of Theorem 7.1 is not satisfied but the stable and unstable manifolds
intersect transversally. To this end we modify the system in Example 2

slightly and consider the following system.

(13.26) x1 = x3
x2 = x4
- 2 2 21‘3 2 2
x3 = x1-2x1(x1+x2) + e{—3plx1—,u2x3+ m (3x1+x2)(coswt
+ film“ wt}
x cos
1+w 1x2
5: — —2 (x2+ 2) + c{— — x + “‘3 xxcoswt
4—X2x21x2 ”1‘2”“ —21+w 12

2
+ 1%? (xf+3x§)cosw1}.

y=y+ccoswt.

Notice that the unperturbed system (5:0) has the following stable and

unstable manifolds.
(13.27) W11 = (p(a)cosu, p(a)sinu, p(a)cosu, p(a)sinu,y),
(13.28) W8 = (p(a)cosu, p(a)sinu, p(a)cosu, p(a)sinu,0)

where p(t) = sech t and a,ydR, uc[0,27r].

 

1

97

Note that dim W11 = 3 and dim W8 = 2, and the 'homoclinic

manifold' is W‘1 n W3.

From (13.17) and (13.18), it is clear that the adjoint system of the
linearized system of the unperturbed system has two linearly independent

bounded solutions on R which are given by

”‘3

(13.29) ¢:(t) = (—p(t)cosu + 2p(t)3cosu, -p(t)sinu _
+ 2p(t)3sinu, p(t)cosu, o(1)3mu, 0)

and
(13.30) ¢;(t) = (mono, —p(t)cosu, —p(t)sinV, p(t)cosu, 0).

Hence the Melnikov vector for system (13.26) is precisely the same as

before. Consider the case 11 = 0. From now on we assume that
(13.31) [160 and ”1 at 0.

Then we know that 35 M(a,0;p) = 0 and g; M(a,0;p) ,1 0. Thus
condition (ii) in Theorem 7.1 is not satisfied. However we shall show
that there exists the transversal intersection of the stable and unstable
manifolds of system (13.26). First recall, by using the notation in (7.3)

and (7.4), that we have the following situation.

98

 

 

 

 

 

 

 

 

 

204.2(3) aw ‘1 3mm)
#3 mi(a,V)
5302(0)) 30-13(01))
( m§(a,v,u‘2’) 11130341!)
(13.32) DP“ = ’1 o o ‘
o 1 o
o o 1
a u a u a
35 m2 3; m2 3? m2 1
”2
(13.33) DFS = r1 0 ‘
o 1
6 s a s
‘65 m1 w "‘2

 

 

Therefore if 35 mi at 0, we have the transversal intersection even
though 6% NI(a,0:p) = 0 which means that 35 m3 = 33 m3. Note
that the linearized system of the unperturbed system of (12.26) has an
unbounded solution (0,0,0,0,cet) on [a,oo) where C is a nonzero

constant. Let

* -t
(13.34) ¢3(t) = (0,0,0,0,Ce ).
Then

(13.35) m§(o) = ¢;(a)ms(a).

99

Referring (6.1) and the proof of Lemma 4.2, we have

a
(13.36) mi(a) = I ce_t cosw(t—a)dt = —C-2 e_a.

co 1+0)
So we have
(1 s _ C —a
(13.37) 35 m1(a) — ——2- e at 0 for any 0.
1+0.)

Thus in this example, there exists the transversal intersection but the
condition by the Melnikov vector can not be used to show it.

Let us summarize these analysis. In these examples the
Hamiltonian nature of the unperturbed systems is effectively used even
though the perturbations are not Hamiltonian. See Proposition 11.3 and
Theorem 11.4. Example 1 is a standard two—dimensional case which
gives the simplest case to which the linear Melnikov function can be
easily applied. We note that the linear Melnikov function gives a
necessary and sufficient condition of the transversal intersection and hence
it also can be used to show the tangential intersection. See Pr0position
10.2. Example 2 gives a higher dimensional case to which the linear
Melnikov vector can be used to detect the transversal and tangential
intersection. See Pr0positions 10.3 and 10.7. In Example 3, we consider
a case to which the Melnikov vector can not give a complete information
about the transversal intersection. This limitation of the Melnikov vector
in higher dimensional cases comes from the fact that the Melnikov vector

is the projection of the real distance between the stable and unstable

100

manifolds to the space of the completely unbounded solutions, i.e., the
complement subspace of $(P(a)) + $(I—Q(a)). Therefore the Melnikov
vector dr0ps the information about the projection of the real distance to

other subSpaces. See the decomposition in (6.9).

101

§14. EXPONENTIALLY SMALL SPITTING OF STABLE AND
UNSTABLE MANIFOLDS

In this section we examine an example for which the Melnikov
function can not be applied to detect the intersection of the stable and

unstable manifolds. Before doing this, we recall Example 1 in §13.
(14.1) x — x + gXZ = 6 cost.

The linear Melnikov function of this system was

A

(14.2) M(a) = -c sina

where c - 0 is a constant. The reason for that 131(0) can be used
to detect the intersection of the stable and unstable manifolds of this
system is that the distance d between the stable and unstable
manifolds is expressed as

A

(14.3) {1 = c(M(a) + 0(3)).

That is, the linear Melnikov function 121(0) constitutes the leading
term.

Now we consider the following rapidly forced system.

(14.4) x — x + gx2 = 6 cos (t—)
‘1

102

where c << 1 and ‘1 << 1. In this case the linear Melnikov
function takes the form

A

(14.5) M(a,c) = — '25 cosech (’2') sin (3).

Hence NI(a,c) can not be the leading term of the expansion of d in
terms of c. See also Holmes, Marsden and Scheule [l 1]. This is a
serious limitation of the perturbation method used in the theory of the
Melnikov function we deve10ped before and in fact it relates to one of
the fundamental problems in dynamics since the time of Poincare.

Resolution of this difficulty has to wait for future study.

[1]
[2]
[3]
[4]
[5]
[6]
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[8]

[9]

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103

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