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In this paper, we investigate the homoclinic bifurcations from a heteroclinic cycle by using exponential dichotomies. We give a Melnikov—type condition assuring the existence of homoclinic orbits form heteroclinic cycle. We improve some important results.

We consider the n-dimensional differential equations

where is a small parameter, is a parameter. In studying the global bifurcation, we usuaally assume unperturbed differential equations

admits ahyperbolic equilibruim and a homoclinic orbit connecting it. It is the peresistence of homoclinic oribit and heteroclinic that we usually study in global bifurcation, we refer to Wiggins [

If

I = 1, 2

(where we assume ) then we say that is a heteroclinic cycle consisting of, , and. The study of homoclinic bifurcation from a heteroclinic cycle is very important and interest not only from the point of view of bifurcation theory itself but also from the point of view of application, we refer to Kokubu [

The main tool used in this paper is theory of exponential dichotomies. We consider the linear differential equations

where A(t) is a n n continuous bounded matrix on R. We say equation (1.3) admits an exponential dichotomy on interval J if ther exist con stants K, α, a projection P and the fundamental matrix X(t) of equation (1.3) satisfying;

for On the theory of exponential dichotomies, refer to Coppel [

We consider differential equations

where is small parameter, is a parameter. with respect to, where cl\ompact subset, a small interval containing zero, a small interval.

We assume C1. For unperturbed equation

Admits two hyperbolic equilibriums and two heteroclinic orbits connecting respectively (form a heteroclinic cycle), that is,

.

.

We denote the heteroclinic cycle by

.

We want to study under what conditions can a homoclinic orbit bifurcate from the heteroclinic cycle as the second case of Kokubu [

C2. All real parts of the matrix are different from zero; and the number of eigenvalues with positive real parts is

If the conditions C1 and C2 are satisfied then equation

admit an exponential dichotomy on both and, and the sum of dimensions of stable and unstable subspaces is n. If follows from the roughness of exponential dichotomy that (refer to Zeng [

admit an exponential dichotomy on both and, and the sum of dimensions of the stable and unstable subspaces is In the follows, because we want to the exponent of to be greater that 1, without loss of generality, we may assume the constants

Otherwise, we replace by then the exponent of is greater than 1.

C3. The variational equations (2.4) admit a unique (up to a scalar multiple) nontrival bounded solution on R.

Under the conditions C1, C2, C3, we can prove (refer to Zeng [

i = 1, 2 also admit unique (up to a scalar multiple ) nontrival bounded solution, respectively, on R, and an exponential dichotomy on both and, respectively. The constants of the exponential dichotomies are also K, α.

We let

,

.

The main result of this paper is Theorem 1 We assume the conditions C2, C2 and C3 are satisfied, then when sufficiently small equation (2.1) admits a unique hyperbolic equilibrium satisfying .If the 2 × 2 matrix

is invertible, the for sufficiently small there exista a continuous function satisfying

such that the equation

admits a homoclinic orbit connecting in the neighbourhood of the heteroclinic cycle.

Remark If the conditions C1, C2 and C3 are satisfied, uing the standard method (refer to Zeng [

where. If the matrix M is invertible then we can easily prove (refer to Zeng [

and

has two hyperbolic equilibriums, , satisfying and, and two heteroclinic orbits, satisfying

,

.

That is, the heteroclinic cycle persists in the region of parameters

Fiom Theorem 1 of this paper we see that in the region of parameters

a homoclinic orbit connecting bifurcates from the heteroclinic cycle.

Kokubu [

We can also prove that if the conditions C1, C2 and C3 are satisfied then for sufficiently small a homoclinic orbit connecting, bifurcates from the heteroclinic cycle, but the region of parameters of bifurcation is different from.

To prove the main result of this paper, we want to find the bounded solutions of equation (2.1) on and on satisfying

We make a change of variables for equation (2.1)

respectively, and obtain the equations

We write the above equations in the following form

And the boundary value condition in the following form

where is sufficiently large.

. satisfying:

In order to find the bounded solutions of equations (3.1), (3.2) and (3.3), we consider the following boundary value problem

where For any, , we first consider the following boundary value problems for

We let and have the following lemma:

Lemma 1 Assume the conditions C1, C2 and C3 are satisfied.

Then there exists sufficiently small such that for equations (3.9), (3.10) and (3.11) admit a unque continuous except at t = 0 bounded solution satisfying with

Moreover, is differentiable in and with

.

where denotes the left limit of function at t = 0, is a constant independent of, Moreover, if

then, are continuous at t = 0.

Proof Lemma 2 is mainly due to Lin [

Let, be the bounded solutions of equations

Let

then, are the solutions of equations

In the same method as follows, we can show that

.

Now we prove the boundness of. Let

then, are the solutions of equations

From (3.12) we obtain

hence there exsits a constant L > 0 such that

This completes the proof of Lemma 2.

Now we consider equations (3.1)-(3.3). We have the following lemama:

Lemma2 Assume conditions C1, C2 and C3 are satisfied. Then there exist sufficiently small and the constants, L > 0 such that for equations (3.1)-(3.3) admit aunque continuous except at t = 0 bounded solution satisfying

with

Moreover, if

then, are continuous at t = 0.

The proof of Lemma 2 can be proved by contract fixed point theorem and is similar to that of Lin [

From Lemma 2 we see that if we have proved that bifurcative equations (3.16) and (3.17) can be can be solved then we find the continuously bounded solutions of equations (3.1), (3.2) and (3.3)

and

.

Now we mainly solve bifurcative equations (3.16) and (3.17). We make a change of variable for equations (3.16) and (3.17) and obtain the following bifurcative equation

From (3.15) we have

Leting in the above equation, we obtain

(Remark ACTUALLY, is defined only for. but due to the existence of its limit, here we define the vaule of the limit to be the value at. In the sequel, we make the same definition.)

From the property of we have

hence

From the representation of (3.18), (3.19), (3.21) and (3.33) we obtain

In the same way, we can obtain

For convenience, we define a matrix

then we have

We define

Obviously, for equation

And equation

equivalent. Now we want to find the solutions of equation (3.26). We first compute. From (3.18) we have

Now we compute (3.27). Since

we have

and hence is bounded for.

Since

we have

Noting, we can easily prove that

hence

Last, since

we obtain

From (3.28), (3.29) and (3.31) we have

In the same way, we can prove

Hence we have

Let

then we have

From (3.34) we have

Since the matrix M is invertible, it follows from the implicit function theorem that for sufficienly small there exists a continuous function satisfying

Hence for sufficiently small we have

Hence for sufficiently small equations (3.6), (3.7) and (3.8)

So for sufficiently small the equation

has two solutions

satisfying

We construct a solution of equation (3.38) by making use of and

Since, is a continuously bounded solution of equation (3.38).

Now we show is a homoclinic orbit connecting the equilibrium. Since when

Hence for any, there exist and such that when and, we have

Since is hyperbolic, we obtain (refer to [

In the same way, we can prove that

Hence is a homoclinic orbit connecting in the neighbouthood of the heteroclinic cycle.

Theorem 1 discussed the second case of bifurcations of kokubu [

Admits three hyperbolic equilibriums and two heteroclinic orbits, connecting to, to, respectively, that

We denote by.

Theorem 2 We assume the conditions B1, C2 and C3 are satisfied, then when sufficiently sall equation (1.1) admits two hyperbolic equilibrium, satisfying,. If the matrix

Is invertible, then for sufficiently small there exists a continuous function satisfying

Such that the equation

Admits a heteroclinic orbit connecting to in the neighbourhood of the heteroclinic cycle.