Bifurcation of bounded solutions of ordinary differential equations

(1)

BIFURCATION OF BOUNDED SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

JAMES ROBERT WARD, JR.

ABSTRACT. Conley index is used to study bifurcation from equi- libria of full bounded solutions to parameter dependent families of ordinary differential equations of the form _~

=

1.

Introduction

The concepts and results of the Leray-Schauder degree and index the- ories have been very effectively applied to prove global continuation and bifurcation of solutions to nonlinear equations in a Banach space. The homotopy invariance of degree under certain conditions is a fundamen- tal property generally used to establish continuation or bifurcation of solutions. The Conley index also has invariance properties that make it a useful tool to study some bifurcation phenomena in dynamical sys- tems. In this paper we study non-autonomous ordinary differential sys- tems, and use the Conley index to prove the existence of continua of full bounded solutions bifurcating from the trivial solution of these sys- tems. More precisely, we study parameter dependent families of ordinary differential equations of the form

dx dt = cF(x, t,

where

is a continuous function of (x, t,

EA

n x JR x JR, n c JRm is an open set and c,

JR are parameters. Bya (full) bounded solution

Received December 22, 1999.

2000 Mathematics Subject Classification: 34C, 37B, 37G.

Key words and phrases: Conley index, bifurcation, bounded solution, almost pe- riodic, ordinary differential equations.

Research supported in part by NSF grant DMS-9501497.

of (1) we mean a solution x

x(t) satisfying (1) for all t

JR and such that

Ilxll

sup Ix(t)1 <

tElR

Throughout this paper, we assume F(O, t, f.l) = 0 for all (t, f.l)

JR x R, so that x = 0 is an equilibrium value for all f.l and c. The parameter f.l will generally be a bifurcation parameter, while c will generally be a real number of small magnitude. We will also assume that F(x, t, f.l) is almost periodic in t

JR, uniformly for (x, f.l) in compact sets (see [3] or [10]). Let

Fo(x, f.l)

lim T 1

fT F(x, t, f.l) dt.

T-oo

lo

We will study the bifurcation of full bounded solutions of (1) making use of the averaged equations

(2) dx dt = Fo(x, f.l).

Proofs are based upon applying Conley index theory to a family of skew-product flows associated with (1), and obtaining indices via homo- topy to the associated averaged equations. Notice that the Conley index cannot be applied directly to nonautonomous differential equations, since the solutions of such equations do not define a flow (dynamical system) on the space of initial values. For this reason we use the associated skew product flows. For the properties of the homotopy index we refer the reader to [2] or [9], and for skew product flows to [10] or [5]; the use of these in the present context is discussed in [11], [12], and [13].

We will make use of a weak topology on our families of differential equations. This topology was studied by Artstein [1] in much greater depth and generality than will be needed here. Let n c JRm be an open set and f : n x JR

JRm. Suppose f satisfies the CaratModory conditions:

for each x

n the function t

! (x, t) is Lebesgue measurable, and for almost all t E JR (in the sense of Lebesgue measure) the function

x

f(x, t) is continuous on n. ^Suppose

(Cl) For every compact set A c n there exist two locally L

functions m

(t) and

(t) such that if x,

and

JR then:

. (1) If(x, t)\ :S mA(t),

(2) If(x, t) - !(y, t)1 :S kA(t) Ix - y[,

(3) for every

> 0 there exists a

=

(€) > 0 such that if E c JR

is measurable, contained in an interval It, t+ 1], and with measure less than

then fE mA(t) dt :s;

and

(4) there exists a number

A ^{such that} ftt+1 kA(s) ds:S;

A ^{for all}

t E R

Given a function f satisfying (Cl) one can define an associated set of functions 9 on

x JR. that contains the time translates of f, defined for T

JR. by f7(X, t)

f(x, T+t) for all (x, t)

x R With an appropriate topology on 9, one can then define skew-product flows on

x 9.

DEFINITION

1([1]). Let f satisfy (Cl) and for every compact set A c

[2

and

> 0 let N A and 6A(E) be given by (Cl). The family 9 = G(j) consists of all CaratModory functions

JR.

JR.m satisfying : For every compact A c

there exist two locally L 1 _functions MA,g ^and KA,g such that if x,

and

JR. then

(1) Ig(x, t)1 :s; MA,g(t),

(2) Ig(x, t) - g(y, t)1 :s; KA,g(t) Ix - YI, and the functions MA,g and KA,g

(3) if

c It, t+ 1] and the Lebesgue measure of

is less than

A(E) then fE MA,g(s)ds :::;

and

(4)

ftt+1 KA,g(s)ds :s;

A ^{for all} t

R

If 9 E

9(j) then so is g7 ^{for any}

R Moreover, for each Xo

the initial value problem

dx dt = g(x, t), x(O) = Xo

has a unique solution x(t; xo, g) defined on a maximal interval of exis- tence I(xo,g) = (a(xo,g),;3(xo, g)). Artstein in [IJ gives the space 9 a weak metrizable topology which we will impose. This is given by

DEFINITION

2([1]). Let {gk} be a sequence in 9. We say {gk} con- verges (weakly) to

9 provided for every x

and

JR. the sequence {f; gk(X, s) ds} converges in JR.m to f; g(x, s) ds.

Convergence in 9 is induced by a metric, which is explicitly given

in [1]. The topological space 9 is compact and is closed under time

translations. A local flow

can be defined on

x 9 by 1f(xo, g, t) =

(x(t;xo,g),gt) ^for t

I(xo,g). The flow (dynamical system)

is askew

product flow (see [10] or [5]).

We are not interested in flows on all of D

9 however. Instead we take the closure in 9 of the time translates of f, the (weak) hull of f,which we denote by Hw(f). Since 9 is compact and Hw(f) is closed, it follows that Hw(f) is compact. The flow 1r is locally invariant on D x Hw(f) and we restrict our attention entirely to this local flow, which we also denote by 1r.

f(x, t) is a continuous and uniformly almost periodic in t

then the usual hull of f is the closure of the time translates of f in the topology of uniform convergence on

sets. We will call this the strong hull of f, and denote it by Hs(f). Now if {ftJ is a sequence of translates of the uniformly almost periodic function f (x, t) converging in the weak topology to

every subsequence of that sequence has in turn a subsequence converging uniformly on compact sets to some h in the strong hull; thus this subsequence converges to h in the weak topology also. Hence for each

D,

t) =

t) a.e., and in fact {ft

con- verges to h in the strong topology. Thus in the uniform almost periodic case the topologies are equivalent if we identify the equivalence classes of functions in Hw(f) with their continuous representatives. Moreover, uniform convergence on compact sets of a sequence of uniformly almost periodic functions is equivalent to its uniform convergence on sets of the form

x

compact [3]. Thus our weak topology is equivalent in this case to this latter topology. Nevertheless, the weak topology will be useful to us in the study of parameter dependence. Since the hull of f

is essentially independent of these topologies, we will simply denote it by H (f). Recall that if 9

H (f) then 9 is also uniformly almost periodic and f

H(g). The space D x H(f) is a locally compact metric space in any of these topologies.

We will apply the following abstract continuation result regarding the Conley index; see [2] or [9]. Recall that if 1r is a flow on a metric spa~e X, N c X is said to be a compact isolating neighborhood for 1r if N is compact and the maximal1r invariant set contained in

is contained in the interior of N.

compact set I is called a compact isolated invariant set if I has a compact neighborhood in which I is the maximal invariant set.

THEOREM

3. Let X be a locally compact metric space and suppose for each A

that 1r>.. is a local flow on X. Suppose: (a) The map A

1r>..

is continuous in the sense that

{An}

{x n} eX, and {tn}

are sequences with An

A,

n

tn

t, and

t) is defined, then

n,tn) is defined for all large nand

n,tn)

^1r>..

t) as n

(b) There is a compact set N in X such that, for each >.

JR, N is an isolating neighborhood for 7f,\. Let 1(>')

{x

7f,\(x, t)

N 'for all t

JR}. Then the Conley (homotopy) index h(7f,\, 1(>')) is defined and its value is independent of >.

JR.

Recall that the Conley homotopy index is a homotopy class of com- pact pointed spaces, and that if the index of a compact isolated invariant set 1

is not the homotopy class of 0, the one-point pointed space, then 1

# 0. We will write h(7f, 1) for the Conley index of a compact isolated invariant set

in a flow 7f.

We also must explain our use of the term bifurcation. Let

be a locally compact metric space with distance function

Let J = [a,

be a compact interval of real numbers containing the number >'0 in its interior. Let 7f,\, >.

be a continuous family of (local) flows on

A solution through x for a flow 7f,\ is a function a mapping a real interval I

0 into X such that a(t) = 7f,\(x, t) for all tEl. A full solution is one with

A full bounded solution is a full solution a = a(t) such that

closure{a(t) : t

JR} is compact.

We assume there is a compact set

c

which is invariant for all 7f,\, that is, for all >.

J, x

and t

JR, 7f,\(x, t)

Since each 7f,\(', t) is a homeomorphism, 7f,\(A, t) = A for all >.

and t

We define bifurcation from

DEFINITION

4. We will say that >'0

is a bifurcation value from A if for every

> 0 there is a pair (>', x)

J x

with x

and such that there is a full 7f,\-solution a

a(t) through x satisfying

d(a(t), A) + I>' - >'01 <

for all

E JR.

Since we assume X is locally c;ompact, it follows that a is a full bounded solution. Notice that if a(t) is a full solution through x ^and a(to) =

then p(t) = a(t + to) is a full solution through

the 7f,\

form a family of semiflows, the definition may be modified to fit that situation. In this paper we study ordinary differential equations only and have no need for semiflows.

Let

be a locally compact metric space and let 7f,\, >.

be a

continuous family of (local) flows on

Let

be a compact set in

invariant for all1r>.. Define the set S as follows:

S' = {( >.., x) : there is a full bounded solution for 1r>. through x}

So = {( >.., x)

S' :

A } and let

S=So.

THEOREM 5. Assume: (1) {1r>. : >.. E J} is a continuous family of local flows on X. (2) There is

compact connected set A which is invariant for all >.. E

(3) There is a >"0 in the interior of J such that for all >.. E J\ {Ao},

is

isolated invariant set for 1r>.. (4) The Conley index h(1r>., A) is constant on the intervals [a, >"0) and (>"0, b]. (5) h(1r

A) :f: h(1rb' A).

Then (cl): For all

> 0, there exists (>.., x) E J

X, x

A, and a fu111r>.-solution eT through x satisfying d(eT(t) , A) + lA - >"01 < c for all t E JR; that is, >"0 is a bifurcation point from A. (c2): Let C denote the component of

containing {AO}

then either

is unbounded or else C meets {a,b} xX.

This theorem extends one given in [14]; the proof will be given else- where.

2. Bifurcation in Almost Periodic Systems

Let F : A = JRm x JR x JR

JRm be continuous. We assume there are continuous functions

h, and

such that

F(x, t, J.L) = g(x, J.L) + /'i,(J.L)h(x, t), and F(O, t, J.L) = °

for all (t, J.L)

JR

with g(x, J.L) and h(x, t) locally Lipshitz continuous in x

JRm, uniformly with respect to t and J.L in compact sets. Consider the parameterized family of differential equations

(3) dx dt = cg(x, J.L) + c/'i,(J.L)h(x, t).

By our assumptions, initial value problems for (3) have unique solutions,

which vary continuously with the parameters c and J.L. We also assume

that h(x, t) is almost periodic in t

JR , uniformly with respect to x in

compact sets. We also assume that for each compact subset

JRm,

F(x, t, J.L) is continuous in J.L

JR, uniformly with respect to (x, t)

K x

R Thus, given

> 0 and compact

c

there exists 6 > 0 such that IF(x, t,

F(x, t, P2)! <

whenever (x, t)

K x

and IPI - P2! < 6.

For T

define the translated function hT by hT(x, t) = h(x, t + T) for all (x, t). We let

H

Hull(h)

closure{hT :

where the closure is in the topology of uniform convergence on compact sets.

is known that since h is uniformly almost periodic in t, then H is compact. Moreover, if h*

H then Hull(h*) = Hull(h). We may define a flow "( on H by "((h*, t) = h;. For any h*

H, the positive limit set w(h*) and the negative limit set o.(h*) are each equal to H itself. Let ho(x) denote the average of h(x, t), so that

liT

ho(x) = lim - h(x, t) dt.

T-ooT

°

We relate (3) to the averaged equation

(4) dx

dt

g(x, p) + K,(p)ho(x)

Fo(x, /1).

It

follows from our assumptions that initial value problems for (4) have unique solutions. We will denote by 13)1 the flow on

generated by initial value problems for (4). Thus

13/l(xo,t) =u(t;xo,Fo(·,/1)), where u(t) satisfies

du dt = Fo(u, /1), u(O; xo, Fo(., p)) = Xo·

We will represent a generic member of Has h* (x, t). Consider the family of differential equations

(5)

for h* EH.

Recall "( denotes the translation flow on

"((h*, t)

h;. Let

denote the homotopy type of the topological space H;

is compact and

connected. Let H* denote the Conley homotopy index of H under the

flow,. Then H* = [lBI,p] where the distinguished point p

lHI, since the flow on H has an empty exit set.

Fix

We wish to define a family of skew product flows associated with (5). Let u(t; xo, ch*) be the solution of (5) with u(O; xo, ch*) = Xo.

We define 1rj.L,E: as a flow on lR

H by

1rj.L,E:(xo, h*, t) = (u(t; Xo, ch*), h;) for (xo, h*)

lR

A solution eT through (xo, h*)

lR

x

is a function eT = eT(t) on an interval I into lR

H such that eT(O) = (xo, h*) and eT(t) = 1fj.LAxo, h*, t) for tEl. A full solution is one with I =

A full bounded solution is a full solution with first component bounded:

lu(t; Xo, ch*) I <

Since

is compact, the second component must be bounded in any case.

Define the set SE: as follows:

S; = {(p" Xo, h*)

[a, b] x lR

x H :

there is a full bounded solution for 1fj.L,E: through (xo, h*)}

SE:,O = {(p" x)

S; : x

A } and let

THEOREM

6. Let J = [a, b] be

compact interval. Suppose: (i) There is a P,o

a < P,o < b, such that {O} is an isolated invariant set for (3j.L,

J\{/Lo}. Moreover, for a::;

<

{O} is asymptotically stable for (3j.L' and for P,o <

b, {O} is not asymptotically stable; the Conley index h((3j.L' {O}) is non-zero and constant

the subintervals [a, P,o) and

b]. (ii) h((3a, {O}) # h((3b, {O}).

Then: There is an co > 0 such that for 0 < c < co: (cl) P,o is a bifurcation point from {O} x H for the skew product Bows 1rj.L,E: (for each fixed 0 < c < co). (c2) Let CE: denote the component of SE: containing

{O} x H); then either CE: is unbounded

else CE: meets {a, b} x lR

x

If, in addition to conditions (i) and (ii), {O} is a repeller for P,o ::;

b then there is a continuum of bounded solutions of (3) bifurcating from (P,o,O) inlRx lR

Proof. There are two parts to the proof. In the first part, we show that there is an co > 0 such that for each 0 < c < co and

J\ {P,o}

the invariant set {O} x H in the skew product flow 1fJI.,E: associated with

(5) has non-trivial Conley index, and this index changes as p crosses Po.

This will prove bifurcation in the skew product flows. We then consider bifurcation in the original family of differential equations, and almost periodicity.

Part

1: Let p

l\{po}. First notice that if x = x(t) is a solution of (5) for some E > 0 then y(t) = x(t/c) is a solution of

(6) dy

dt

g(y, p) + K,(p)h*(y, tic).

We will homotopy (6) for fixed small c > 0 to the equation averaged at c

O. Let 0 < c < 1 and consider the homotopy

(7) dy dt = (1 - A)Fo(Y) + AF*(y, tE-

p)

for F*(y, t, p) = g(y, p)+K,(p)h*(y, tic), and Fo(Y) = g(y, p)+K,(p)ho(y).

Let B(r) be an isolating neighborhood of {O} in the flow

generated by (4). The family of equations (7) has B(r)

{x

lR

Ixl

r} as an isolating neighborhood.

this is not the case, then for each positive integer

there is a bounded function Yn = Yn (t), for all t

JR, and numbers An

[0,1],

(0, n-

Sn

lR , and an hn

such that for all t, Yn(t) satisfies the differential equation

(8) d~n ^{= (1 - An)Fo(Yn) + An [g(Yn, p) +} K,(p)hn(Yn, tIEn)]

and Yn(Sn)

aB(r). Let zn(t) = _Yn(t+sn); then Zn satisfies (8) translated by Sn' Now both Zn and dznldt are uniformly bounded on 1R indepen- dently of n

N, so there is a subsequence of {zn} uniformly conver- gent on compact subsets of lR to some Z = z(t) with z(t)

B(r) and z(O)

aB(r) We can assume An

AO

[0,1]. Now the key here is that the sequence of functions {h

'c;;-l)} converges in our weak topology to h o(')' ([13], Lemma 4.1). From all of this it follows that z satisfies

dz dt = (1 - Ao)Fo(z(t)) + AoFo(z(t))

Fo(z(t))

for all t

R Since z(O)

aB(r) and z(t)

B(r) for all t, this contradicts

the hypothesis that B (r) is an isolating neighborhood for (4), and proves

the existence of 0 < Eo

1 satisfying the claim.

Now fix

(0,

well

f.-l

J\{f.-lo}. We return to the equations (5) or (7) with x

x(t)

y(ct), for our fixed ° ^<

^:S

and consider the homotopy

(9) ~~ ^{= (1 - A)cFo(x) +} AcF*(x, t, f.-l)

for A

[0,1], where, as before, F*(x, t, f.-l) = g(x, f.-l) + K,(f.-l)h*(x, t). We define a family of skew product flows

on JRm

associated with (9) by

1r~(xo,

P*, t) = (x(t; Xo, ACK,(f.-l)h*), hn

where x(t; Xo, ACK,(J.l)h*) denotes the solution to (9) with x(o) = xo.

(Here we are holding J.l fixed, but later will let it vary).

It

follows, essentially from results of Artstein [1], that the family of flows {1rn is continuous in the sense of Theorem 1. Moreover we have just shown that B(r) x H is a compact isolating neighborhood for each

1r~,

A

[0,1]. Thus the Conley index of the maximal invariant set in B(r) x H for

is independent of A

[0,1]. Let 1(0) denote this set for 1rg. Now in this case, A = ° so (9) becomes the equation

(10) dx dt = cPo(x, f.-l).

The flow determined by (10) is the same

that of (4) except for a change of independent variable

We denote this flow by

By hypothesis, h({3p., {o}) i= O. Let, denote the flow on H given by ,(h*, t) = h;. Then

is a product flow on JRm

H given by

1ro =

and

1(0) = {o} x

It follows (see [1] or [9]) that

h(1rg, 1(0))

h({3p. x" {O} x H)

h({3p., {O})

h(r,H),

where

denotes the smash product of pointed topological spaces (ho- motopy types). By hypothesis, h({3p., {o}) of 0, and h(r, H) = [lHI*, p]

is of the form of a compact connected topological space with separated

distinguished point (see above; recall

has an empty exit set under the flow,). It follows from a result in [12J that h(7ri{", 1(0)) ::f. 0. Hence

h(7ri, 1(1)) = h(7rb' 1(0)) ::f. 0,

where 7rl is the skew product flow on IR

x

generated by the initial value problems

~~ ^{= cF*(x, t; f.L), x(O) =}

where h* EH. Now 1(1) = {O} x H also.

The idea now is that the index h(7ri, 1(1)) = h(7ri, {O} x H) actually depends upon f.L. For

f.L < f.Lo, h(7ri, {O} x H) = h(f3Jl, {O})

[ IBI*,pJ = 2:0 ^A[IBI*,pJ = [1HI*,plo since {O} is asymptotically stable for (10). Moreover, [JR[*,

is not a connected topological space. On the other hand, for f.Lo < f.L

b, h(7ri, {O} x H) = _h(f3Jl, {O})

[IBI*,plo but in this case our hypothesis is that h(f3Jl , {O}) is connected.

is now easy to show that h(f3Jl,{O})

[1HI*,p] = (h(f3Jl,{O}) x [1HI*,pJ)j(h(f3Jl,{O})

[1HI*, pJ) is also connected. (By h(f3Jl, {O})

[IBI*, plo we mean, as usual, the (homotopy type of) space obtained by joining the two spaces at their respective distinguished points). Thus the index changes as f.L crosses

It follows from

2 that there is a continuum of bounded solutions in the skew product flows bifurcating in IR

x

from {O} x

This proves the first part of the theorem.

Part 2: We now attend to the question of bifurcating solutions to the original differential equation, and the question of almost periodic solutions. These are separate from the question of bifurcation in the skew product flows. This is because if o-(t)

7ri(xo, h*, t) is a bifurcating solution in the skew product flow then u(t)

u(t; xo, cK({..l)h*), with h*

is a full bounded solution to

(11) ~~ ^{= cg(x, f.L) + cK(M)h*(x, t),}

but this does not insure the existence of a full bounded solution to our original differential equation

(12)

dt = cg(x, f.L) + cK(f.L)h(x, t).

This is because, while h

Hull(h*), so there exists a sequence

with

t) = h*(x, t + an)

h(x, t) as n

the corresponding

sequence {u

may converge to u

0, instead of converging to a nonzero

solution of (12). The additional hypotheses in the second part of the theorem are designed to guarantee that this does not happen.

Thus we now assume in addition to conditions (i) and (ii), {O} is a repeller for

:S

:S b. We claim there is a continuum of bounded solutions of (3) bifurcating from

0) in lRx lR

and these solutions are almost periodic near the bifurcation point. Let (p, xo, h*)

Cc; then either

<

or

2::

The cases are similar, and we treat only the first; so we suppose p < Po. Now let U(t) = u(t; xo, cf\,(p)h*) , so U(t) is a full bounded solution of (11), so

U'(t) = cg(U(t), p) + cf\,(p)h*(U(t), t)

for all t

lR. Since

:S p < J.lo, the zero solution of (11) is asymp- totically stable. Let A = a(xo,h*), i.e., A is the negative limit set of (U(t), hn in lR

x H. Since U(t) is a full bounded solution and h(x, t) is uniformly almost periodic in t, A is compact. Moreover, ({O} x H) nA =

o since the zero solution is asymptotically stable. Now there exists a sequence {an} with an

as n

and h* (x, t + an)

h(x, t) uniformly on compact sets. There is a subsequence of {an}, which we re- label as {an}, such that U(an)

Po

A. We have that (U(an),

(Po, H)

A. The solution of (12) passing through Po at t = 0 is the required full bounded solution.

Since the case of

:S J.l :S b is almost identical to the one considered,

this completes the proof of the theorem. 0

3. An application

Let

C(lR, lR) be continuous almost periodic functions. We con- sider the following example:

d?x dx

(13) dt

+ c

(J.lp(t) - x

dt + c

q(t)x = 0 With c > 0 and p

lR. This is equivalent to the system

~~ =

(14) d~

-c(J.lp(t) - x

)y - cq(t)x.

The averaged system is

dx

=y

(15) il~ (- 2) -

dt

= -

where we assume the mean values of

and

are positive:

liT liT

p= lim T p(s)ds> 0, and q= lim T q(s)ds>O.

T-+oo 0 T-+oo 0

When

0 the origin is a repeller for the averaged equation (15), and hence has Conley index h(,B/L' {O})

2:

However, when

> 0, the origin is asymptotically stable, and h(f3J.L' {O}) = 2:

follows from

THEOREM

3 that for all sufficiently small c > 0, the system (14) has a bifurcating continuum of full bounded solutions in

(x, y)) space.

References

[1] Z. Artstein, Topological dynamics of an ordinary differential equation,J.of Dif- ferential Equations 23 (1977), 216-223.

[2] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS 38, American Mathematical Society, Providence, R. I. 1978.

[3] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathe- matics, Springer-Verlag, Berlin 377 (1974).

[4] R.Gaines andJ.Mawhin, Coincidence Degree and Nonlinear Differential Equa- tions, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 568 (1977).

[5] J.K.Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence,R.1. 1988.

[6] M. Krasnosel'skii and P. Zabreiko, Geometrical Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 1984.

[7] J. Mawhin, Degre topologique et solutions periodiques des systemes differentiels non lineaires, Bull. Soc. Ray. Sci. Liege 38 (1969),308-398.

[8] , Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS 40, American Mathematical Society, Providence, R. I.1979.

[9] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, 1987.

[10] G.R.Sell, Topological Dynamics and Ordinary Differential Equations, Van Nos- trand, Rheinhold, London, 1971.

[11] J. R. Ward, Jr., Conley index and non-autonomous differential equations, Re- sults in Mathematics 14 (1988), 192-209.

[12] ,A topological method for bounded solutions of nonautonomous ordinary differential equations, Trans. Amer. Math. Soc. 333 (1992), 709-720.

[13] , Homotopy and bounded solutions of ordinary differential equations, J.

of Differential Equations 107 (1994), 428-445.

[14] , Bifurcating continua in infinite dimensional dynamical systems and ap- plications to differential equations,J.of Differential Equations 125 (1996), 117- 132.

Department of Mathematics

University of Alabama at Birmingham Birmingham, AL 35294

E-mail: [email protected]