Exponential Dichotomies and Homoclinic Orbits from Heteroclinic Cycles *

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.


Introduction
We consider the n-dimensional differential equations where , n x R   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 [1], Palmer [2,3], Naudot [4] and Meyer and Sell [5].But in studying the pulses solutions of some recation-diffusion equations, we often meet the problem of homoclinic bifurcations from the heteroclinic cycles, refer to Kokubu [6], Chow, Deng and Terman [7], Gambaudo [8] and reference therein.Suppose Equation (1.2) has two hyperbolic equilibriums 1 2 and two homoclinic orbits and two homoclinic orbits .
, p p (where we assume     q t q t  ) then we say q t q t q q      consisting of , , 1 and 2 .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 [6], Chow, Deng and Terman [7].The main purpose of this paper is to invertigate the homoclinic bifurcation from heteroclinic cycles   by making use of exponential dichotomies and Melnikov technique.For convenience, we only discuss the case of heteroclinic cycles with length = 2. Using the theory of exponential dichotomies, Melnikov functions and Slinikov chang of variable, Kokubu [6] investigate the periodic and homoclinic bifurcations from a heteroclinic cycle.In Kokubu [6], he needs to divide the problem into critical and non-critical two cases.Moreover, he needs that the heteroclinic orbits approach the hyperbolic equilibriums along the eignspaces associated with the principal eigenvalues.Chow, Deng and Terman [7] also studied the same problem in the non-critical case by making use of Liapunov-schmidt method and Silnikov's changes of variable and Poincare map and obtain some analytical results.Chow, Deng and Terman [7] also the conditions as in Kokubu [6].Melnikov functions were not obtained in Chow, Deng and Terman [7].Chow, Deng and Terman [9] studied the same problem as this paper, Kokubu [7] did not need to divide the problem into critical and non-critical two cases and unified the two cases and didn't ndde that the heteroclinic orbits approach the hyperbolic equilibriums along the eigenspaces associated with the principal eigenvalues.The results of Chow, Deng and Terman [9] are weaker than those of Kokubu [6] and Chow, Deng and Terman [7] under weaker assumptions because of the topological approachs.The purpose of this paper is to improve the above results by a analystic method (Lin's method [10]).We can also unify the critical and non-critical cases and weak the condi-tions of Kokubu [6], Chow, Deng and Terman [7,9].Moreover, it is also an interesting to provide an analystic method of studying bifurcations of heteroclinic cycles.Many ideas of this paper come from Lin [10], Meyer and Sell [5], Kokubu [6] and Palmer [2,3].But it should note that the results of this paper cannot be followed directly from these papers, much technique has been made.Let us finally mention the related results on the bifurcations of heteroclinic cycles.Sandstede [11] investigated the forced symmetry breaking of heteroclinic cycles.Guckenheimer and Holmes [12] discussed the spontaneous symmetry breaking of heteroclinic cycle.Krupa and Melbourne [13] studiecd the stability of heteroclinic cycle.On the other related results on heteroclinic cycles, we refer to the references of the above mentioned papers and good survey of Krupa [14].The paper is organized as following.In Section 2, we give the main result; in Section 3, the proof of the main result is given.
The main tool used in this paper is theory of exponential dichotomies.We consider the linear differential equations where , 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 [15], Sacker and Sell [16] and Meyer and Sell [17].On the relations between exponential dichotomies and homoclinic, heteroclinic bifurcations, we refer to Palmer [18] and Meyer and Sell [16]., t s J  . 

Main Result
We consider differential equations  , , where , with respect to We assume C1.For 0, 0, v Admits two hyperbolic equilibriums 1 and two heteroclinic orbits , q t q t 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 [6] C2.All real parts of the matrix are different from zero; and the number of eigenvalues with positive real parts is i If the conditions C1 and C2 are satisfied then equation ,0,0 , 1, 2.
x i admit an exponential dichotomy on both R  and R  , and the sum of dimensions of stable and unstable subspaces is n.If follows from the roughness of exponential dichotomy that (refer to Zeng [12], Sacker and Sell [16], Coppel [15]) that the variational equations along admit an exponential dichotomy on both R  and R  , and the sum of dimensions of the stable and unstable subspaces is Under the conditions C1, C2, C3, we can prove (refer to Zeng [12] that the adjoint equations of equations of (2.3), (2.4) also admit unique (up to a scalar multiple ) nontrival bounded solution t , respectively, on R, and an exponential dichotomy on both and R  R  , 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 , v  sufficiently small Equation (2.1) admits a unique hyperbolic equilibrium sufficiently small there exista a continuous function admits a homoclinic orbit connecting Remark If the conditions C1, C2 and C3 are satisfied, uing the standard method (refer to Zeng [19]), we can obtain the bifurcative equations of persistence of the two heteroclinic orbits and where .If the matrix M is invertible then we can easily prove (refer to Zeng [19]) that for , and two heteroclinic orbits 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 We can also prove that if the conditions C1, C2 and C3 are satisfied then for 0     sufficiently small a homoclinic orbit connecting 2 , bifurcates from the heteroclinic cycle , but the region of parameters of bifurcation is different from

The Proof of the Main Result
To prove the main result of this paper, we want to find the bounded solutions of Equation (2.1) We make a change of variables for Equation (2.1) , .
We write the above equations in the following form , , , , .
And the boundary value condition in the following form where  is sufficiently large.
  For any , we first consider the following boundary value problems for   2 z t are continuous at t = 0. Proof Lemma 2 is mainly due to Lin [10].For the proof of the first part of existences of the solutions   , i z t  satisfying (3.12), (3.13) and (3.14), we refer to Lin [10] and omit the proof.We now want to prove the second part that ( ),ln .ln ln ln ln where denotes the left limit of function , l ,ln .
In the same method as follows, we can show 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 0 0   and the constants 2 , L > 0 such that for The proof of Lemma 2 can be proved by contract fixed point theorem and is similar to that of Lin [10].
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) Now we mainly solve bifurcative Equations (3.16) and (3.17).We make a change of variable for Equations (3.16) and (3.17) ṽ v  and obtain the following bifurcative equation , , , , , , , d ln ln , , 0 From (3.15) we have Leting 0   in the above equation, we obtain   is defined only for 0   .but due to the existence of its limit, here we define the vaule of the limit to be the value at 0   .In the sequel, we make the same definition.)From the property of we have From the representation of (3.18), (3.19), (3.21) and (3.33) we obtain , 0 , , 0, 0 , 0, 0 d lim ln ln , , , ,0,0,0 d In the same way, we can obtain For convenience, we define a matrix Obviously, for 0 equivalent.Now we want to find the solutions of Equation (3.26).We first compute .From (3.18) we have , , , Noting , we can easily prove that In the same way, we can prove ,0 ,0 ,0 ,0 ,0,0 ,0,0 d ,0,0 ,0,0 d ,0,0 d ,0,0 d Since the matrix M is invertible, it follows from the implicit function theorem that for 0   sufficienly small there exists a continuous function We construct a solution of Equation (3.38) by making use of ln , , 0 , ln , , 0 ,  q t q t q q p       .
e assume the conditions B1, Theorem 2 W e satisfied, then when ,   sufficiently sall Equation variational Equations (2.4) admit a unique (up to a scalar multiple) nontrival bounded solution   i t  on R.

  2 ,Lemma 1 
z t  be the bounded solutions, which are continous except at t = 0 and satisfy (3.12), (3.13) and (Assume the conditions C1, C2 and C3 are satisfied.Then there exists sufficiently small 0 0 9), (3.10) and (3.11) admit a unque continuous except at t = 0 bounded solution i z t  satisfying with second case of bifurcations of Kokubu[6].Acutally, we slso investigate the first case of bifurcation as in Figure2in the same way and have the following result.We assume B1