Existence of in nitely many connecting orbits in a singularly perturbed ordinary

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Existence of in nitely many connecting orbits in a singularly perturbed ordinary

dierential equation

Hiroshi Kokubu

^y

Department of Mathematics Kyoto University

Kyoto 606-01, Japan [email protected]

Konstantin Mischaikow

^z

School of Mathematics Georgia Institute of Technology

Atlanta, GA 30332, U.S.A.

[email protected] and

Hiroe Oka

^x

Department of Applied Mathematics and Informatics Faculty of Science and Technology

Ryukoku University Seta, Otsu 520-21, Japan

[email protected] April 3, 1996

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We consider a one-parameter family of two-dimensional ordinary dierential equations with a slow parameter drift. Our equation as- sumes that when there is no parameter drift, there are two invariant curves consisting of equilibria, one of which is normally hyperbolic and whose stable and unstable manifolds intersect transversely. The slow parameter drift is introduced in a way that it creates two hyperbolic equilibria in the invariant normally hyperbolic curve that is persistent under perturbation. In this situation, we prove that the number of distinct orbits which connects these two equilibria changes from - nite to innite depending on the direction of the slow parameter drift.

The proof uses the Conley index theory. The relation to a singular boundary value problem studied by W. Kath is, also, discussed.

Research was supportedin part by NSF Grant INT-9315117 and JSPS US-Japan

CooperativeResearchMPCR-294.

y

ResearchwassupportedinpartbyGrant-in-AidforScienti cResearch(No. 06740150

and07740150),MinistryofEducation,ScienceandCulture,Japan.

z

ResearchwassupportedinpartbyNSFGrantDMS-9101412andDMS-9505116.

x

ResearchwassupportedinpartbyScienceandTechnologyFundforResearchGrants

ofRyukokuUniversity.

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1 Introduction

In this paper, we consider a one-parameter family of two-dimensional ordinary dierential equations with a slow parameter drift, which is given by the following form:

u_ = f(u )

_ = g() u²IR² ²IR ²^;1 1] (1) where g : ^;1 1]^!IR is C¹ and satises

g()>0 for ^;1< <1 and g(1) = 0:

Such a family can be viewed as a singularly perturbed autonomous ordinary dierential equation (a vector eld) or its time-rescaled version. The purpose of this paper is to present a topological method which allows one to prove existence of innitely many connecting orbits in this type of equations under some conditions that are detailed below. Before giving a precise statement of the result, we shall explain some motivation and background of the study.

This work is stimulated by a paper due to Kath6], who studied a slowly varying singular boundary value problem of the form:

²d²y

dx² ⁺f(y x) = 0 y(x=1) = 0

where >0 is a small perturbation parameter and the slowly varying poten- tialF(y x) =^Z f(y x)dyis assumed (as a typical case) to have one minimum and one maximum when x= 0, so that the corresponding autonomous prob-

lem ²d²y

dx² ⁺f(y 0) = 0

possesses a center and a saddle with a homoclinic orbit in its phase plane.

Kath approached this problem from qualitative point of view by working in the phase plane instead of using analytical techniques such as matched asymptotic expansions, and he observed that, in some cases, the stable and unstable manifolds of the saddle (which persist under the presence of slowly varying variable x in the nonlinear term f(y x)) can have innitely many (transverse) intersections in the phase plane. The existence of innitely

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many intersections that correspond to innitely many distinct solutions to the problem heavily depends on the mutual position of the stable and unstable manifolds of the saddle, which were measured by a Melnikov-like function or the \energy dierence" in 6] (notice that when x = 0 the problem gives a Hamiltonian equation.) See 6] for more detail.

Unfortunately, the argument given in 6], although very interesting, did not allow us to understand the essential mechanism for creating innitely many such dierent solutions to this problem. Motivated by 6], we shall try to give a complete mathematical proof for this type of result. In this paper, we consider an initial value problem instead of a boundary value problem, as Kath essentially did the same by using the phase plane analysis, and hence we view the parameter in the above equation as one of the phase variables and similarly = ¹ as its equilibrium points. In this way, the problem becomes one of nding innitely many orbits that connect two equilibrium points corresponding to = ¹. Our idea is the use of the Conley index theory which provides us with a topological method for detecting connecting orbits. In order to focus only on the essential features of the problem and since, as explained in 6], the mutual position of the stable and unstable manifolds should be the most important feature in this problem, we shall modify the equation by removing the slowly varying variable and replacing it by a friction term which makes the manner of splitting of the stable and unstable manifolds less degenerate. This modication can be justied by using a result in 15].

A typical form of the problem could therefore be given by x_ = y

y_ = y^;x(x^;a)

_ = (1^;²) x y a ²IR (2) where y is the friction term and the equilibrium points are transformed to (x y ) = (0 0 1). When ⁶= 0, there are exactly four equilibria, B = (0 0 1) andA = (a 0 1). Our main result of this paper will show that for < 0 there exist exactly two distinct heteroclinic orbits B⁺ ^!B^;, i.e. solutions (x(t) y(t) (t)) to (2) such that

tlim^!1(x(t) y(t) (t)) = (0 0 ^;1) = B^;

t^!;1lim (x(t) y(t) (t)) = (0 0 1) = B⁺ ⁽³⁾ 4

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while for >0 there exists innitely many distinct heteroclinic orbits B^; ^! B⁺. To describe the assumptions and to state the results rigorously requires some notation which we now introduce.

First, for technical reasons we wish to extend the set of parameter values at least to a set = ^;1^; 1 +] for some arbitrary > 0. In some applications it is natural to assume that g() and f( ) are dened over IR, and hence, . However, in the case of singular boundary value problems one is lead to consider equations of the form

u_ = f(u )

_ = u²IR² ²IR ²IR (4) where it is assumed that

lim

! 1

f(u ) =F (u )

is well dened. In this case one can use the change of variables = 2^tan^;1

to transfer (4) to

u_ = f(u tan

2 )

_ = ² ^cos²⁽ 2)

which is in the form of (1). Of course, in this case the natural values of are in the interval ^;1 1]. Thus, in this setting, to extend to ² one can dene

f(u ) =

8

>

<

>

:

F⁺(u ) if 1 1 + f(u ) if ^;1< <1 F^;(u ) if ^;1^;^;1

and choose g such that for ⁶= 1, (1^;²)g() > 0. This results in the system which will be studied

u_ = f(u )

_ = g() u²IR² ²IR ²IR (5) 5

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under the assumption that (1^;²)g() >0 when ⁶=1.

Observe that when= 0, the system reduces to a continuous 1-parameter family of equations

u_ =f(u 0): (6)

Let' : IRIR²^!IR² be the ow generated by (6). Let : IRIR²^!IR² be the parameterized ow over the parameter space , i.e.

(t u ) = ('(t u) ):

Recall that a compact set N is an isolating neighborhood for a ow if the maximal invariant set of N is contained in the interior of N, i.e.,

Inv(N ) :=^fu^j(IR u) N^g intN:

We shall make the following assumptions.

A1

There exists an isolating neighborhoodN of such thatN is homeomorphic to 0 1]³ and N := N ^\(IR² ^f^g) is homeomorphic to 0 1]² for all ².

As will be seen, N serves to dene the region of phase space on which the dynamics of interest occurs. In particular, any heteroclinic orbits of : IRIR³ ^! IR³, the ow generated by (5), which do not lie entirely in N will be ignored. In 10] it is shown that if N is an isolating neighborhood for then there exists > 0 such that for ² ^; ], N is an isolating neighborhood of .

A2

There exists A ² IR² and a continuous function B : ^; ] ^! IR² such that

f(A ) = 0 f(B( ) ) = 0

for all ² and ^j^j . Furthermore, for each ² ^;1 1], A and B( 0) are the only equilibria of ' in N.

A3

Let i() and i(), i = 1 2, denote the eigenvalues of Df(A 0) and Df(B( 0) 0) respectively. For ^;1 1,

¹()<0< ²() ²^;1 1]

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Rei()>0 for =1 and Imi()⁶= 0 when Rei() = 0:

Observe that this forcesB( 0) to be a saddle point whileAis a repelling (attracting) xed point for' when >0( <0). The assumptions

A2

and

A3

are local in nature. The next two assumptions contain global information.

A4

Inv(N¹ '¹) consists of the critical points A and B(1 0) and a unique heteroclinic orbit from A to B(1 0). Inv(N^;1 '^;1) consists of the critical points A and B(^;1 0) and a unique heteroclinic orbit from B(^;1 0) to A.

Let B⁺ = (B(1 ) 1) ²N and B^; = (B(^;1 ) ^;1)² N. Observe that B are equilibria of . Let the set of connecting orbits from B⁺ to B^; in N under be denoted by C(B⁺ B^;N ), i.e.,

C(B⁺ B^;N ) :=^f(u )²N ^j(IR u ) N

tlim^!1(t u ) =B^; _tlim

!;1

(t u ) =B⁺^g:

C(B^; B⁺ N ) is dened similarly. Let #C(B^; B⁺N ) denote the number of components ofC(B^; B⁺N ).

The following subset of N will be used to obtain a useful collection of isolated connections. Let

D=Nⁿ(^fA^g^;1^; 1 +]):

Observe that ^fA^g^;1 1] is an invariant set for ". Therefore if" denotes

a B_"⁺ ^! B_"^; connecting orbit, then _"^\ ^fA^g^;1 1] = and _" D.

Therefore,C(B_"⁺ B_"^;D ") =C(B_"⁺ B_"^;N ").

By

A1

, D is homotopic to S¹, and hence ¹(D)¹(S¹) ^Z^Z where ¹ denotes the fundamental group. This suggests the following denition.

Denition 1.1

Let B" = ^S

;1¹B( 0)^S ^S

0j^jj"^jB(1 ). Let " denote

a B_"⁺ ^!B_"^; connecting orbit. Then "B" generates an element of ¹(D).

The winding number of " is dened to be

(") := "B"]²^Z^Z:

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Using the notion of the winding number, we make the nal assumption as follows:

A5

Let "ⁿ be a B_"⁺ⁿ ^!B^;_"ⁿ connecting orbit with "n <0 and ("ⁿ) 0. If

"ⁿ converges to a set;in the Hausdor topology on compact sets as"n ^!0,

then ; = B := B⁰.

This assumption can be veried by looking at the invariant set of the parametrized ow. For instance, it is easy to see that the model equation (2) satises

A5

. In fact, since the limit set ; is a connected invariant set of the parametrized ow which contains B⁺ and B^;, it must consist of the line segment B and, possibly, some of the homoclinic or periodic orbit of the Hamiltonian ow

x_ = y

y_ = ^;x(x^;a)

in the plane given by = 0. But if it contains a homoclinic or periodic orbit, it must turn at least once around the set ^fA^gIR, and hence the winding number cannot be less than or equal to 0, since all the periodic and the homoclinic orbits rotate in the same direction.

The goal of this paper is to prove the following theorem.

Theorem 1.2

Given assumptions

A1

^-

A5

^and ^j^j: (a) if <0, then#C(B⁺ B^;N ) is bounded and 2 (b) if >0, then#C(B^; B⁺ N ) =¹.

Moreover, these connecting orbits are distinguished by the winding numbers.

Let us give a heuristic proof for the theorem in order to clarify the geometric idea behind it and to motivate the rigorous argument presented in the subsequent sections. Our goal is to nd connecting orbits between B_"^;

and B_"⁺ for suciently small " ⁶= 0. Note that the set of equilibria B( 0)

for the fast dynamics forms an invariant curve in the parameterized ow and from

A3

it is normally hyperbolic, hence it has a stable and an unstable manifold. The assumptions

A3

and

A4

imply that these manifolds wrap around the line ^fA^g^;1^; 1 +] and let us assume that they intersect transversely as indicated in Figure 1. It then follows that there exists at least one connection between B_"^; and B_"⁺ besides the trivial connection.

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R× { }A

B(λ ε, ) Bε

B + ε

−

Figure 1: The transverse intersection of the stable and unstable manifolds for the curve of hyperbolic xed points B( 0).

For our purpose, it is easier to work on the universal covering space.

To be more precise, we consider the set IR² ⁿ^fA^g ^;1 ^; 1 + ] and take its universal covering. The picture we obtain by lifting Figure 1 to the universal covering becomes either like Figure 2 or like Figure 3, depending on the direction of the slow ow which is determined by the sign of ". In particular, B_"n denotes the equilibria given by the lift of B_", respectively.

The direction of the ow is chosen in such a way that one turn around the set^fA^g^;1^; 1+] in the base space corresponds to going one oor up in the covering space. For instance, a connection fromB_"^; toB⁺_" in the base space corresponds to a connection from B_"^;⁰ to B_"k⁺ for some k 0 in the universal covering space. Note that the connection between B_"k^; and B_"k⁺ for any k ²^Z^Z indicates the lift of the invariant curve close to B( 0) for all ²^;1^; 1 +].

Let us rst consider the case " < 0 corresponding to Figure 2. In this case, we are looking for B_"⁺⁰ ^! B_"k^; connections for k > 0. Clearly there exists one such connection since there exists at least one connection from

B_"⁺ to B^;_" in the base dynamics due to the transverse intersection of the

stable and unstable manifolds. For simplicity we think of this connection in the universal covering space as a B_"⁺⁰ ^! B_"^;¹ connection. By the deck transformation, we also have a B⁺_"k ^! B_"k^;⁺¹ connection for each k ² ^Z^Z.

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×

B_ε⁺_,0 B_ε⁺_,1 B_ε⁺_,2 B_ε⁺_,3

B_ε⁻_,0 B_ε⁻,1

B_ε⁻_,2 B_ε⁻_,3

Figure 2: The intersections of the stable and unstable manifolds of the points

B_" for " <0.

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Besides these connections as well as trivialB_"k⁺ ^!B_"k^; connections, we hope to convince the reader that there is no way of nding other connections in Figure 2. Suppose there could exist aB⁺_"⁰ ^!B_"^;²connection. It then must lie in the unstable manifold ofB_"⁺⁰ and the stable manifold ofB^;_"², and hence the connecting orbit must go two oors up in the gure. However there already exists a B⁺_"⁰ ^! B_"^;¹ connection. If the desired B_"⁺⁰ ^!B^;_"² connection goes lower than the B_"⁺⁰ ^!B_"^;¹ connection in the unstable manifold of B_"⁺⁰, then it fails to get on and goes underneath the stable manifold of B_"⁺¹ and hence it cannot reach B_"^;². If on the other hand, the B_"⁺⁰ ^! B_"^;² connection goes above the B_"⁺⁰ ^! B_"^;¹ connection, then it runs over the stable manifold of

B_"^;¹ and hence it again cannot go across to B_"^;². Therefore in either case,

there could not exist B⁺_"⁰ ^! B_"^;² connections. Similarly one can imagine that there also exist no B_"⁺⁰ ^!B_"k^; connections for any k 2.

If the direction of the slow dynamics changes, namely if " >0, then the situation changes completely as indicated in Figure 3. In this case there exist

B_"^;⁰ ^!B_"k⁺ connections for anyk 0. Indeed, if you go slightly higher than

the a priori existing B_"^;⁰ ^!B_"⁺¹ connection in the unstable manifold ofB_"^;⁰, then you rst go underneath the stable manifold of B_"⁺¹, cross the trivial

B_"^;¹ ^! B_"⁺¹ connection, and nally are able to get on the stable manifold

of B_"⁺². Such a connection can clearly exist between B_"^;⁰ and B_"k⁺ for any

k 2, and therefore, going back to the base space, we have innitely many connecting orbits as desired.

This argument is meant to give a geometric basis for Theorem 1.2. How- ever, our proof is algebraic and makes use of the Conley index theory rather than explicit assumptions concerning the transversal intersections of stable and unstable manifolds. In particular, we compute the Conley indices of the connecting orbits of "when" <0, directly, and then use this information to compute the Conley index for the connecting orbits when" >0. It is worth remarking that in the context of the Conley index theory this is a singular transformation. Recall that the Conley index is dened for isolated invariant sets. However, as one passes from " < 0 to " > 0, one passes through the parameterized ow ⁰ at which point all relevant isolation is lost. The key to our being able to make this transformation involves a new result due to C. McCord and the second author 9] which allows us to associate algebraic properties to the transition matrices used in our computations. This result and the relevant background for transition matrices is presented in the next section.

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B_ε⁻_,0 B_ε⁻,1

B_ε⁻_,2 B_ε⁻_,3

B_ε⁺_,0 B_ε⁺_,1 B_ε⁺_,2 B_ε⁺_,3

Figure 3: The intersections of the stable and unstable manifolds of the points

B_" for " >0.

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In Section 3, Theorem 1.2(a) is proven. This proof follows directly from the algebraic properties of Conley's connection matrix. Part (b) of Theo- rem 1.2 is proven in Section 4 and makes use of the above mentioned result concerning transition matrices.

2 Conley Index Theory

As was mentioned in the introduction our proof of Theorem 1.2 makes use of the Conley index theory. Thus, we begin with a brief review of the relevent portions of the theory. However, the primary purpose of this section is to establish the notation and the reader is referred to 1, 3, 12, 14, 17, 18, 19]

for details.

Recall that a compact set N X is an isolating neighborhood for a ow : IRX ^!X if

S = Inv(N ) :=^fx² N ^j(IR x) N^g intN:

In this case the maximal invariant set S of N is called an isolated invariant set.

Given an isolated invariant set S, one can nd a pair of compact sets (N L) which form a regular index pair 18], in which case the (homological) Conley index of S is given by

CH(S)H(N L):

Remark 2.1

To simplify the calculations we shall always use homology with

ZZ

2 coecients, i.e.

CH(S)H(N L^Z^Z²):

There are various ways of decomposing an isolated invariant set which are natural with respect to the index theory. The most important of these is the following. A pair of compact invariant subsets (A R) of an isolated invariant set, S, forms anattractor-repeller pair decomposition of S if

1. A is an attractor in S, i.e. there is a neighborhood U of A such that

!(U ^\S) =A.

2. Ris the dual repeller to A inS, i.e. R=Sⁿ^fx^j!(x) A^g 13

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whereand!denote the alpha and omega limit sets. The set ofconnections fromR toA within S is dened by

C(R AS) =^fx ²S^j(x) R !(x) A^g: An obvious, but crucially important observation, is that

S =RC(R AS)A:

When there is no confusion concerning the invariant set S we shall simplify the notation and merely writeC(R A) to denote the set of connecting orbits.

The natural generalization of an attractor-repeller pair decomposition is a Morse decomposition. To be precise, a Morse decompositionof an isolated invariant set S is a collection of disjoint compact invariant subsets ofS,

M(S) =^fM(p)^jp²^P^g

indexed by a nite set ^P, for which one can nd a partial order > such that if x ⁶² S (^S_p^2PM(p)), then there exists p q ² ^P such that p > q, (x) M(p), and!(x) M(q). Any partial order which satises the above condition is admissible. These individual invariant subsets, M(p) are called Morse sets, and the remaining portion,SⁿM(p), is referred to as the set of connecting orbits. In particular, given two Morse sets M(p) and M(q), the set of connecting orbits from M(p) to M(q) is denoted, as before, by

C(M(p) M(q)S) :=^fx²S ^j!(x) M(q) (x) M(p)^g:

If ^M(S) = ^fM(p) ^j p ² ^P^g is a Morse decomposition of S, then each M(p) is an isolated invariant set. S contains other isolated invariant sets, some of which can be produced by the partial order on^P as follows. A subset I ^P is an interval in^P if r ²I whenever p < r < q and p q²I. Disjoint intervals are adjacent if IJ = IJ is also an interval (i.e. if no element of

P lies \between"I and J). If I is an interval, let M(I) :=

i²IM(i)

!

0

@

ij²IC(M(j) M(i))

1

A:

A regular index triple for an attractor{repeller pair (A R) inS is a triple of compact spaces (N² N¹ N⁰) such that (N² N⁰) is a regular index pair for

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S, (N² N¹) is a regular index pair forRand (N¹ N⁰) is a regular index pair for A. Since the Conley index of an isolated invariant set is dened in terms of regular index pairs, regular index triples provides for the following relation between the Conley indices of the invariant sets. The inclusion induced exact sequence

@

!Hn(N¹ N⁰)^!Hn(N² N⁰)^!Hn(N² N¹)^!^@ Hn^;1(N¹N⁰)^! can be re-written as an exact sequence

@⁽RA⁾

! CHn(R)^!CHn(S)^!CHn(A)^@⁽^!^RA⁾CHn^;1(R)^!

which is referred to as the homology attractor-repeller pair sequence. The boundary map@(R A) is called theconnecting homomorphism, as@(R A)⁶= 0 implies that connections between Rand A exist.

Just as one can decompose isolated invariant sets, one can decompose sets of connecting orbits. Given an attractor repeller pair decomposition (A R) of an isolated invariant set S and an indexing set J a separation of C(R A) is a collection ^fCj(R A)^j j ² J^g of open invariant subsets of C(R A) such

that C(R A) =

j²JCj(R A):

It is easy to check that this forces J to be a nite set. Furthermore, if N is an isolating neighborhood of S, then for each j ²J there exists, Nj N such that N_j is an isolating neighborhood for ARC_j(R A). For this reason, the sets Cj(R A) are called sets of isolated connections. Let

Sj = InvNj

then (A R) is an attractor repeller pair decomposition ofS_j. This of course, implies that there is an associated index triple (Nj² Nj¹ Nj⁰) and the corresponding connection map @(R Aj).

Theorem 2.2

(C. McCord, 7, Theorem 2.5]) For any separation ofC(R A),

@(R A) =^X

j²J@(R Aj):

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Connection matrices are the generalization of the connecting homomor- phisms for an attractor{repeller pair to Morse decompositions. For our pur- poses it is sucient to recall that given a Morse decomposition, a corresponding connection matrix is a linear map dened on the graded vector spaces made up of the sum of the Conley indices of Morse sets in a Morse decomposition, that is

# : ^M

p^2PCH(M(p))^! ^M

p^2PCH(M(p)):

We shall write this as a matrix

# = #(p q)]_qp^2P where

#(p q) :CH(M(q))^!CH(M(p)):

Furthermore, connection matrices satisfy the following conditions.

1. They are upper triangular, i.e. if p⁶> q then #(p q) = 0.

2. They are boundary operators, i.e. they are degree^;1 maps

#(p q)CHn(M(q)) CHn^;1(M(p)) and they square to zero, ## = 0.

3. If p and q are adjacent in the ow dened order then the connection matrix entry #(p q) equals the connecting homomorphism for the attractor repeller pair (M(p) M(q)) of M(q p), i.e.

#(p q) =@(M(q) M(p)):

4. Recall that for any interval I in^P, the setM(I) is an isolated invariant set, and hence,CH(M(I)) is dened. The relation between the indices of the Morse sets in M(I), and CH(M(I)) is

CH(M(I)) Ker#(I) Image#(I) where

#(I) = #(p q)]_qp²_I: 16

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The following theorem, due to Franzosa 3], is fundamental.

Theorem 2.3

Given a Morse decomposition, there exists at least one connection matrix.

Returning now to the realm of dierential equations, consider the 1- parameter family of equations

x_ =F(x ) x²IRⁿ ²IR which we assume generates a continuous family of ows

: IRIRⁿ ^!IRⁿ:

Let = ^;1^; 1 +]. The parameterized ow over will be denoted by

$ : IRIRⁿ ^!IRⁿ where

$(t x ) = ((t x) ):

We will make the following assumptions on the parameterized ow.

H1

N is an isolating neighborhood for $.

H2

A Morse decomposition of K = Inv(N $) is given by

M(K) =^fM(p)^jp= 1 ::: P^g with an admissible ordering

P > P ^;1> ::: >1:

Furthermore, for each p = 1 ::: P, the homotopy Conley index of each Morse set is

h(M(p))%^m for some m:

As before we shall adopt the notation that N =N ^\(IRⁿ^f^g) and M(p) =M(p)^\(IRⁿ^f^g):

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H3

K ¹ = ^P

p⁼¹M ¹(p):

Observe that

H3

implies that there are no connecting orbits between the Morse sets under the ows ¹.

Now consider a system of equations of the form x_ = f(x )

_ = g() (7)

where g : IR ^! IR is continuous and satises (1^;²)g() > 0 for ⁶= 1.

Though, formally, equations (7) resemble those of (5) they are not the same since the Morse decompositions at = 1 dier. In the application of Theorem 2.4 (stated below) to the proof of Theorem 1.2 the equations (7) are obtained by choosing a covering of the isolating neighborhood on which (5) are dened.

We shall denote the ow generated by (7) by

$: IRIRⁿ⁺¹ ^!IRⁿ⁺¹:

Let K = Inv(N $). Observe that under $ the Morse decomposition consists of 2P Morse sets

M(K) =^fM(p )^jp= 1 :::P^g (8) where M(p ) := M ¹(p) ^f1^g IRⁿ . As was shown in 16], for 0<^j^j <, the connection matrix for the Morse decomposition ^M(K") has the form

#" =

"

0 T"

0 0

#

: (9)

The submatrixTwill be called thesingular transition matrixassociated with

$". Observe that:

if <0 then T :^M^P

p⁼¹CH(M(p⁺))^!^M^P

p⁼¹CH(M(p^;))

if >0 then T :^M^P

p⁼¹CH(M(p^;))^!^M^P

p⁼¹CH(M(p⁺)):

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Hence, nonzero entries in the transition matrices correspond to M(q⁺) ^! M(p^;) or M(q^;)^!M(p⁺) orbits for somep q.

For the purpose of the following discussion let < 0 (the case > 0 is analogous). Let

T = T(p q)]_qp⁼¹_:::P where

T(p q) :CH(M(q⁺))^!CH(M(p^;)): Since connection matrices are degree ^;1 operators

T(p q) :CHn⁺¹(M(q⁺))^!CHn(M(p^;)): However, by 16, Theorem 5.4], or 11, Theorem 2.10]

CHn⁺¹(M(q⁺))H~n⁺¹(%h(M¹(q)))CHn(M¹(q)) (10) where %h(M¹(q)) denotes the suspension of the homotopy Conley index of M¹(p) under the ow ¹ and

CHn(M^;1(q))CHn(M(q^;)): We shall indicate these isomorphism by

%n(q⁺) : CH_n(M¹(q))^!CH_n⁺¹(M(q⁺)): and n(q^;) :CHn(M^;1(q))^!CHn(M(q^;)): Similarly if >0 then there exist isomorphisms

%n(q^;) :CHn(M^;1(q))^!CHn⁺¹(M(q^;)) and _n(q⁺) : CH_n(M¹(q))^!CH_n(M(q⁺)): Let % :=^M^P

q⁼¹%(q ) :^M^P

q⁼¹CH(M ¹(q))^!^M^P

q⁼¹CH(M(q ))

and :=^M^P

q⁼¹(q ) : ^M^P

q⁼¹CH(M ¹(q))^!^M^P

q⁼¹CH(M(q ))

An important result which is proven in 9] is the following.

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Theorem 2.4

(^;)^;1T^;%⁺ = (%^;)^;1T^;1 ⁺:

Furthermore, as is, also, shown in 9] one can choose bases for all the spaces such that on the level of matrices this equality can be written as

T^;=T^;1:

We shall from now on assume that this has been done.

3

^"

for

^" ^< ⁰

.

This section contains the proof of Theorem 1.2(a), i.e., that for " <0

#C(B_"⁺ B_"^;N ")2

and addresses the question of the structure of these connection orbits. For

j"^j<", let S":= Inv(N ").

Lemma 3.1

For " <0,

M(S") =^fB_" A ^g

is a Morse decomposition of S" with an admissible ordering A⁺ > B⁺_" > B_"^; > A^;:

Proof. The planes =1 are invariant under " and the ows on these planes are given by ' ¹. Since" <0 and (1^;²)g() >0

(Inv(N^;1 '^;1) Inv(N¹ '¹))

forms an attractor-repeller pair decomposition ofS". Thus, by

A4

,^fB_" A ^g is a Morse decomposition of ". The admissibility of the ordering is guaran- teed by

A4

," <0, and (1^;²)g() >0. ²

Let #":CH(A^;)CH(B^;_")CH(B_"⁺)CH(A⁺)

!CH(A^;)CH(B_"^;)CH(B_"⁺)CH(A⁺)

denote a connection matrix for ^M(S"). Let #"(Y X) :CH(X)^! CH(Y) denote the corresponding submatrix of #".

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Lemma 3.2

^For ^;"" <0

#" =

2

6

4

0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0

3

7

5

:

Proof. Under the ow ", the Conley indices for the equilibria are as follows:

CH(A^;) = (^Z^Z² 0 0 :::)

CH(B_"^;) = (0 ^Z^Z² 0 :::)

CH(B_"⁺) = (0 0 ^Z^Z² 0 :::)

CH(A⁺) = (0 0 0 ^Z^Z² 0 :::):

Since connection matrices are degree ^;1 operators, the only possible nonzero entries are #"(A^; B_"^;), #"(B_"^; B_"⁺) and #"(B_"⁺ A⁺). By

A4

and 16],

#"(A^; B_"^;) = 1 and #"(B_"⁺ A⁺) = 1. Since #"#" = 0, #"(B_"^; B_"⁺) = 0.

2

The proof of Theorem 1.2(a) will follow from the fact that #"(B_"^; B_"⁺) = 0. Let

;"(j) :=^f"^j(") =j^g:

Lemma 3.3

For "²^;" 0), B_"⁺B_"^;;"(j) is an isolated invariant set.

Proof. Suppose that for some"²^;" 0) and somej 0,B_"⁺B^;_" ;"(j) is not compact in D.

Let ^fn ^j n = 0 1 2 :::^g ;"(j) and let n = B_"⁺ B_"^; n. Since

" ² ^;" 0), N is an isolating neighborhood and hence n N. Since N is compact, there exists a subsequence ^fnⁱ^g such that nⁱ ^! as i ^! ¹ where N but ⁶D. The latter implies that ^\^fA^g^;1^; 1+]⁶=

. However, since each nⁱ is a compact connected invariant set, so is . Therefore ^\^fA^g^;1^; 1 +] consists of A⁺, A^;, or ^fA^g^;1 1].

Assume A⁺ ² ^\^fA^g^;1^; 1 +]. Since " < 0, B⁺ ² , and is connected, there exists a connecting orbit from B⁺ to A⁺ which contradicts

A3

. A similar argument implies that A^; ⁶²^fA^g^;1^; 1+]. Therefore ^\^fA^g^;1^; 1 +] = which is a contradiction. ²

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Let D"(j) D be an isolating neighborhood for " such that

Inv(D_"(j)) =B_"⁺B_"^;;"(j):

and let @(B⁺ B^; j) denote the connecting homomorphism for the index triple associated with the attractor-repeller pair (B_"^; B_"⁺) decomposition of

InvD"(j).

Since we are using ^Z^Z² coecients for the Conley index @(B⁺ B^;j) = 0 or 1. A standard index theory result is the following.

Lemma 3.4

@(B⁺ B^;j) = 1^,CH(InvD"(j))0⁾;"(j)⁶=.

@(B⁺ B^;j) = 0^,CH(InvD"(j))CH(B_"^;)CH(B_"⁺).

Lemma 3.5

For ²^; 0),

@(B⁺ B^;j) =@(B^;⁺ B^;^;j) i.e., it is independent of .

Proof. By Lemma 3.4, @(B_"⁺ B_"^;j) is determined by CH(InvD"(j)).

Clearly, the Conley index of InvD(j) can change only if for some"²^;" 0),

InvD"(j) is not isolated. Since N is an isolating neighborhood for ",

" ² ^;" 0), InvD"(j) intN. Since the orbits of ;"(j) and ;"(k) gener- ate dierent homotopy classes in D, the only way InvD"(j) could cease to be isolated would be for a connecting orbit to approachA^;1^; 1 +].

However, as was demonstrated in Lemma 3.3, this is impossible for" <0. ²

Lemma 3.6

There exists a positive integer J such that if ^jj^j > J, then

@(B_"⁺ B_"^;j) = 0.

Proof. By Lemma 3.4, if ;"(j) =then@(B_"⁺ B_"^;j) = 0. By Lemma 3.5, it is sucient to show that ;"(j) =. ;"(j) is bounded away fromA^;1 1], hence if"denotes aB⁺_" ^!B^;_" connecting orbit, then there exists a maximal number of times that " can wind around the line A^;1 + 1^;] for a xed > 0. By choosing suciently small, we can approximate the orbit

of " over the regions ^;1 ^;1 + and 1^; 1 via the ow

'^;1 and '¹ respectively. By

A4

, _" cannot wind around A in N ¹. Thus, the winding number of any "is bounded above. Therefore, forJ suciently large, if ^jj^j> J then ;"(j) =. ²

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Lemma 3.7

@(B_"⁺ B_"^;0) = 1 for any" ²^;"0).

Proof. By Lemma 3.5 it is sucient to show that @(B_"⁺ B_"^;0) = 1 for some value of"²^;" 0). With this in mind, letQdenote a-neighborhood

of ^S

;1;¹⁺B( 0). By

A3

and 10], there exists "⁰ > 0 such that if 0 < ^j"^j < "⁰ then Q is an isolating neighborhood for ". Furthermore, CH(Inv(Q "))0. Observe that if ;"(0) Q then

CH(Inv(Q ")) =CH(InvD"(0)) and hence by Lemma 3.4, @(B_"⁺ B_"^;0) = 1.

Thus, it remains to be shown that ;"(0) Q for "suciently small. By 16], in the Hausdor metric

lim"^!0;"(0) = ;

where ; is a compact invariant set for ⁰. Therefore, from

A5

, we have

; = B(= ^S

;1¹B( 0)), and hence ;"(0) Q for " suciently small.

Since @(B_"⁺ B_"^;0) = 1 is proven by the index computation as above, it remains unchanged as long as the isolation of ;"(0) is preserved. Thus, from Lemma 3.3, @(B_"⁺ B_"^;0) = 1 must hold for all"²^;" 0). ² Proof of Theorem 1.2(a). By Lemma 3.2 and Theorem 2.2, we have

0 = #"(B^;_" B_"⁺) = ^X

j^2Z^Z@(B⁺ B^;j)

for any "²^;" 0).

The assumption

A5

implies that any B_"⁺ ^!B_"^; connecting orbit with a negative winding number cannot exist for suciently small ". Namely, there exists a sequence of negative real numbers ^f"k^g with "k ^! 0 such that for any "² "k 0), no B_"⁺^!B_"^; connecting orbit" with^;k (")<0 exists.

Therefore for any " ²"k 0), we have

X

j^2Z^Z@(B⁺ B^; j) = ^X

j^;k^;1@(B⁺ B^;j) + 1 +^X

j¹@(B⁺ B^;j):

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Here we have used Lemma 3.7 as well. From Lemma 3.5, @(B⁺ B^; j) does not depend on " for " ² ^;" 0), and hence, the above formula in fact holds for all "²^;" 0). Since " does not depend on n, we conclude that

0 = 1 +^X^J

j⁼¹@(B⁺ B^;j): (11)

Therefore, there exists at least onej ²^f1 ::: J^gsuch that@(B⁺ B^;j) = 1.

2

Remark 3.8

Observe that there exist an odd number of positive winding numbers j 1 for which @(B⁺ B^;j) = 1. These numbers will be denoted by 1j¹< j² << j²r⁺¹:

4 The Covering Flow

In the previous section the proof of Theorem 1.2(a) was reduced to equation (11) where the summation was indexed by elements of the fundamental group of D. Thus, to further exploit this equality, it is natural to consider covering spaces of the isolating neighborhoods. This is the focus of this section and will lead to the proof of Theorem 1.2(b).

Throughout this section, D will be considered to be the pointed space

(D B_"⁺). LetD^fdenote the universal covering space of D with covering map

:D^f^!D :

There exists a unique local ow ^e" dened onD^fwhich satises the relation

(^e"(t v)) = "(t (v)): (12)

Since ¹(D)^Z^Z, " has innitely many xed points which will be denoted

by ^fB_"j ^j j ²^Z^Zg where (B_j) = B . D^fwill be considered as the pointed

space (D B^f _"⁺⁰). Furthermore, the notation is chosen such that if B_"j is connected to B_"⁰ by a path , then

()B"] =j ²^Z^Z¹(D):

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