TRANSVERSE HOMOCLINIC ORBIT BIFURCATED FROM A HOMOCLINIC MANIFOLD BY THE HIGHER ORDER MELNIKOV INTEGRALS∗

Consider an autonomous ordinary differential equation in R that has a d dimensional homoclinic solution manifold W . Suppose the homoclinic manifold can be locally parametrized by (α, θ) ∈ Rd−1 × R. We study the bifurcation of the homoclinic solution manifold W under periodic perturbations. Using exponential dichotomies and Lyapunov-Schmidt reduction, we obtain the higher order Melnikov function. For a fixed (α0, θ0) on W , if the Melnikov function have a simple zeros, then the perturbed system can have transverse homoclinic solutions near W .


Introduction
The problem of the bifurcation of homoclinic orbit under small periodic perturbation is very important in dynamic system because they are related to some complex dynamic behaviors, such as chaotic motions. Homoclinic orbit is a special invariant set of a differential equation. Suppose the equationẋ = f (x) has a solution γ(t), which asymptotic to the hyperbolic equilibrium x = 0 in both forward and backward time direction. The orbit γ corresponding to the solution γ(t) in phase space is called homoclinic orbit [11]. Suppose the variational equation ofẋ = f (x) along γ has d linearly independent bounded solutions. Let W s (0), W u (0) denote the stable and unstable manifolds of the equilibrium 0, respectively. Clearly, the homoclinic orbit γ lies on W s (0) W u (0). From the number of the linearly independent bounded solutions of the variational equation, we know the dimension of the intersection of the tangent space of W s (0) and W u (0) is equal to d. Apparentlẏ γ ∈ T γ(0) W s (0) T γ(0) W u (0), so d ⩾ 1. If d = 1, the homoclinic orbit γ is called nondegenerate; otherwise it is called degenerate [15].
From geometrical viewpoint, in 1963 Melnikov [17] used Poincare map to investigate the persistence of homoclinic orbit in R 2 . In 1980, Chow, Hale and Mallet-Parret [3] used functional methods to study the persistence of homoclinic orbit of Duffing's equation under damping and periodic forcing. Later Palmer [18] extended the work in [3] to N -dimensional system. He assumed the unperturbed system has a nondegenerate homoclinic solution. By the Exponential dichotomies and the methods of the Lyapunove-Schmidt reduction, he obtained the bifurcation function which are Melnikov types integrals. The zeros of the function correspond to the persistence of the homolicnic orbit for the perturbed system. Also by the Shadowing Lemma, Palmer proved that the persistent homoclinic orbit is transversal. Hence the periodic map of the perturbed system exhibits chaotic motion.
In 1984, Hale [12] suggested a further extension of the functional method to a more general case where the unperturbed equation has a degenerate homoclinic orbit. In 1992, Gruendler [8] studied the persistence of the homoclinic orbit for an autonomous ordinary differential equation with an autonomous perturbation in R N . He assumed the autonomous system has a homoclinic solution and the variational equation has d bounded solutions, d ⩾ 1. By using functional methods, he obtained the bifurcation function which deponed on d dimension independent parameters except for the perturbation parameters. The low order term of the bifurcation function are also Melnikov types integrals. He expanded the bifurcation function about those parameters to the second derivative by Taylor's Theorem. And by a final application of the Implicit Function Theorem, he proved the bifurcation function has a zero. Therefore the perturbed system has a homoclinic orbit. In 1995, Gruendler [9] generalized this result to the periodic perturbed ordinary differential equation. In 1996, Gruendler [10] showed that the variational equation alone the perturbed homoclinic orbit has no nozero bounded solution by the exponential dichotomies and Lyapunov-Schmidt reduction. Hence the perturbed system exhibits chaos.
In 1990, Palmer [19] considered the bifurcation of the degenerate homoclinic orbit under the periodic perturbation. He assumed unperturbed equation has a family of homoclinic orbits which depend on two parameter family. In 1992, Battelli and Palmer [2] given us one way of degenerate homoclinic orbit. That is the intersection of the stable and unstable manifold have branches which is a two dimensional homoclinic solution manifold. And this can occur in the integrable Hamiltonian system [16]. Moreover in [20], Zhu and Zhang investigated the bifurcations of a degenerate homoclinic loop in R N . They obtained an invariant manifold of a definite dimension bifurcated from the degenerate homoclinic orbit.
The definition of the degenerate homoclinic orbit is equivalent to require the tangent space of W s (0) and W u (0) at least have a two dimension intersection. In certain integrable Hamiltonian system with more than two degrees of freedom, the corresponding stable and unstable manifold can coincide or intersect in a more than two dimension submanifold. In this paper we suppose W s (0) W u (0) have a branch which is a d dimension homoclinic solution manifold denoted by W H . Let (t, α) ∈ R×R d−1 , d > 1, be the local coordinates on W H . For each α ∈ R d−1 , d > 1, let γ(t, α) are homoclinic orbits which asymptotic to the hyperbolic equilibrium 0. We will using the higher order melnikov types integrals to study the bifurcation of the homoclinic manifold W H under the periodic perturbation. Meantime many authors studied the homoclinic orbit bifurcation and limit cycle bifurcations by high order Melnikov method [1, 4-7, 13, 14]. Consider the following system: we make the following assumptions: (H2) f (0) = 0 and the eigenvalues of Df (0) lie off the imaginary axis.
(H3) Unperturbed equationẋ(t) = f (x(t)) has a homoclinic solution manifold W H which can be parameterized by which uniformly with respect to α ∈ R d−1 , d > 1.
We will investigate the homoclinic bifurcations of (1.1) near the homoclinic manifold γ(t, α) by the higher order melnikov integrals. This paper is organized as following. In section 2, we list some properties of fundamental solutions of the variational equation along the homoclinic manifold γ(t, α). The main result is presented. In section 3, we prove the main result. By using the functional analytic method, we obtained the bifurcation function. And expanded it about d dimensional parameters to the third order derivatives by Taylor's Theorem. We obtained the lower order term of the bifurcation function which denote by M . Take a fixed point (α 0 , θ 0 ) on the homoclinic manifold W H , if M has a simple zero, then the system (1.1) has a homoclinic solutions near γ(t, α 0 ). In section 4, we prove the homoclinic orbit bifurcated from the homoclinic manifold W H is transversal. Hence the periodic map of the system (1.1) can have chaotic motion.

Preliminaries and Main result
For each α ∈ R d−1 , d > 1, since γ(t, α) is a homoclinic orbit which asymptotic to the hyperbolic equilibrium 0. Hence, from [9,18], variational equation (1.2) has exponential dichotomies on J = R ± respectively. Let U (t, α) be the fundamental solution matrix of the system (1.2). In particular, there exist projections to the stable and unstable subspaces, P s + P u = I, and constants m > 0, K 0 ≥ 1 which are uniformly with respect to α ∈ R d−1 , such that Take the same m in (2.1), define the Banach space with the norm z = sup t∈R |z(t)|e m|t| . The linear variational system The domains of (2.2) and (2.3) are the dense subset of Z, defined as From the theory of homoclinic bifurcation [18], γ(t, α) asymptotic to the hyperbolic equilibrium x = 0, so L α : Z → Z be Fredholm operators with index 0, for α ∈ R d−1 . And we have ) be an orthonormal unit basis of N (L α ) and (φ 1 (t, α), ..., φ d (t, α)) be an orthonormal unit basis of N (L * α ). By the defination of u j (t, α), we take derivatives about t and α on both side of (1.2). So we havë So from (2.5) , we have We define some Melnikov types of integrals that will be used in the future. For integers i, j, k, l from the set {1, ..., d}, let where v j (t, α) be as in (2.6).

The proof of the main result
By (H3), system (1.1) with µ = 0 has a homoclinic solution manifold W H which can be parameterized by γ(t, α). In this section, we will find conditions such that (1.1), with small µ = 0, has homoclinic solution γ µ near W H for some α 0 . Let D i h, D ij h or D ijk h denote the derivatives of a multivariate function h with respect to its i-th , i, j-th or the i, j, k-th variables. And suppose d > 1 in the rest of the paper. With the change of variable As in [9], one can prove that g satisfies the following properties: Lemma 3.1. The function g(·, µ, α, θ) : Z → Z satisfies the following properties: (1) g(0, 0, α, θ) = 0, D 11 g(0, 0, α, θ) = D 11 f (γ(t, α)), γ(t, α)).
For any α ∈ R d−1 , i = 1, ..., d, we define the subspace of Z which consists the range of L α by the method in [9,18]. Let Consider a nonhomogeneous equatioṅ If h ∈ Z, using the variation of constants, with some phase condition, there exists an operator K : Z → Z such that Kh is a solution of (3.3). Clearly, the general bounded solution of (3.3) is Recall that L α be a family Fredholm operators with index 0 which independent of α. And we know dim N (L α ) = dim(Z⧸R(L α )) = d. So we suppose φ 1 (t, α), ..., φ d (t, α) be an orthonormal unit basis of Z/R(L α ). Define a map P : As in [18], one can prove that Lemma 3.2. The map P satisfies the following properties: (1) P and I − P are projections; We now use the Lyapunov-Schmidt reduction to solve (3.1). Applying P and (I − P ) on (3.1), we find that (3.1) is equivalent to the following systeṁ
then z = ϕ is a solution of (3.1) and hence the perturbed system (1.1) has a homoclinic orbit x = γ + ϕ, where ϕ is given in (3.9). Let We have the following Lemma.

Lemma 3.4.
For i, j, k, l ∈ {1, ..., d}, the function H i (β, µ, α, θ) has the following properties: where a (i) (α, θ) and c (i) jkl (α) be as in (2.8) and (2.9). Proof. To prove those properties of H i ,we should take derivatives with respect to µ and β in (3.2) by the formula of H i , where z is equal to ϕ which is in (3.9). From Lemma 3.1,Lemma 3.2, so it is easy to check (i), (ii) through direct calculations. Next we prove (iii) and (iv).

The Transversality
In this section we will prove that the homoclinic solutions γ s (t) are transverse for 0 = s ∈ I. From Section three, the solution γ s (t) satisfyinġ From (3.13), through calculations, we have and Dδ j (0)u j (t, α 0 ).
Note that D 1 M (β 0 , µ 0 , α 0 , θ 0 ) is nonsingular. Then there exists a regionÎ,Î ⊂Ĩ such that V (S(s)) is nonsingular when 0 = s ∈Î. And we again take I =Î Then the variational equation along γ s has no nonzero bounded solutions. So γ s is a transverse homoclinic solution of (1.1) and its periodic map exhibits chaotic motion.