Bifurcation from a homoclinic orbit in partial functional differential equations, Discrete Contin

We 
consider a family of partial functional differential equations 
which has a homoclinic orbit asymptotic to an isolated equilibrium 
point at a critical value of the parameter. Under some technical 
assumptions, we show that a unique stable periodic orbit 
bifurcates from the homoclinic orbit. Our approach follows the 
ideas of Sil'nikov for ordinary differential equations and of 
Chow and Deng for semilinear parabolic equations and retarded 
functional differential equations.


Introduction.
For an ordinary differential equatioṅ (1.1) where x ∈ R n , ∈ R is a parameter and g is a smooth function, it is known that if x = 0 is a hyperbolic equilibrium for = 0 and the Jacobian matrix D x f (0, 0) = A has a unique eigenvalue λ > 0 which is simple and the real parts of all other eigenvalues are strictly less than −λ, then under certain additional transversality conditions, a unique stable periodic orbit bifurcates from the homoclinic orbit as the parameter changes. See, for example, Andronov et al. [AL73], Chow and Hale [CH86] and Kuznetsov [Ku95]. One of the approaches to the above bifurcation problem, originated in the work of Neimark andŠil'nikov [NS65] andŠil'nikov [Si68] for ordinary differential equations in R n with n ≥ 3, is to reduce the bifurcation problem to a problem of the continuation of fixed points for a one-parameter map in a small neighborhood of the hyperbolic equilibrium. This map resembles the well-known Poincaré map but the points on the stable manifold do not return. In what follows we shall call this map theŠil'nikov map and we refer to Kuznetsov [Ku95] for a detailed description of Sil'nikov's results and techniques. The above result has been generalized to other kinds of equations, while several other methods have also been developed. These include the work of Blázquez [Bl86] for semilinear parabolic equations, and of Walther [Wa90] for retarded functional differential equations. We should particularly mention the work of Chow and Deng [CD89] for some infinite dimensional dynamical systems including semilinear parabolic differential equations and retarded functional differential equations, where they obtained some subtle estimates related to linear variational equations along semiorbits of the nonlinear equations and established the smoothness and the existence of a fixed point of theŠil'nikov map.
In this paper, we consider the following one-parameter family of partial functional differential equations:u (t) = Au(t) + L(u t ) + g(u t , ), (1.2) where A is the generator of an analytic semigroup, L is a linear operator and g is a smooth nonlinear functional. g depends on not only the current but also the historic status of u. More specific descriptions will be given in next section. This kind of equations is motivated by reaction-diffusion equations where the reaction terms may involve time delay and have been studied by many researchers, see, for example, Faria [Fa99,Fa01], Faria et al. [FHW02], Hale [Ha86], Hale and Ladeira [HL93], He [He90], Martin and Smith [MS90], Memory [Me91], Travis and Webb [TW74,TW78], etc. For an introduction of the fundamental theory of such equations and some related references, we refer to the monograph by Wu [Wu96]. The purpose of this paper is to generalizeŠil'nikov's theorem and Chow and Deng's techniques to the above partial functional differential equations. In section 2, we introduce the notations and present the main results. The differentiability of solutions of equation (1.2) with respect to the initial values and parameters and the smoothness of the stable and unstable manifolds are proved in section 3. The local analysis of equation (1.2) near the equilibrium is given in section 4. In section 5, we construct theŠil'nilov map and discuss some of its properties. The proof of the main theorem is presented in section 6.

The Main Results
. Let X denote a Banach space over R = (−∞, ∞) and B(X, X) the Banach space of bounded linear operators from X to X equipped with the operator norm. Let r > 0 be a given constant and C = C([−r, 0]; X) the Banach space of continuous X-valued functions on [−r, 0] with the supremum norm | · |. For any real numbers a ≤ b, t ∈ [a, b] and any continuous mapping u : [a − r, b] → X, u t denotes the element of C given by u t (θ) = u(t + θ) for θ ∈ [−r, 0].
Consider the following family of partial functional differential equationṡ where ∈ (− 0 , 0 ) is a parameter, 0 is a given positive constant, A, L and g satisfy the following assumptions: (H1) A is the infinitesimal generator of an analytic compact semigroup {S(t)} t≥0 on X.
It is known that for each φ ∈ C, the initial value problem has a unique solution defined for t ≥ −r. Denote this solution by then {T (t)} t≥0 is a strongly continuous semigroup of bounded linear operators on C with the generator denoted by A T . Also, for each φ ∈ C and ∈ (− 0 , 0 ), there exists τ (φ, ) > 0 and a unique continuous map u = u(φ, ) : [−r, τ (φ, )) → X such that we have the following variation of constants formula (see He [He90], Memory [Me91] and Wu [Wu96]) for u(φ, ) on [0, τ(φ, )). By assumption (H4), C can be decomposed as C = C s ⊕C u , where C u is the one-dimensional eigenspace of A T associated with {λ} and C s is the generalized eigenspace associated with the remaining spectrum. Let φ λ = φ λ (0)e λ· be the eigenvector of A 0 associated with λ and φ * λ be the eigenvector corresponding to {λ} of the formal adjoint operator associated with the bilinear pairing (see Travis and Webb [Tw74]) where ψ ∈ C * = C([0, r]; X * ), X * is the dual space of X and φ ∈ C. Then Note that for φ ∈ C, Let P s and P u be the projections of C onto C s and C u , respectively, i.e. C s = P s C, C u = P u C. It is shown that P s and P u can be applied to the elements X 0 w with w ∈ X. Define X s 0 and X u 0 by Note that if w ∈ X, then T (t)X u 0 w ∈ C u for all t ∈ R and T (t)X s 0 w ∈ C s for t ≥ r. Moreover, there exist constants K 1 and µ > λ > 0 such that (2.7) Then we have the following variation of constants formula (see He [He90], Memory [Me91] or Wu [Wu96]): Since g is C 3 −smooth, by the differentiability of the solution with respect to initial values and parameters (see Theorem 3.1 in section 3), By assumptions (H3) and (H4) there exist δ 1 > 0 and 1 ∈ (0, 0 ) such that the local stable and unstable manifolds W s loc ( ) and W u loc ( ) exist and are subsets of where h s and h u are C 3 −smooth (see Theorem 3.2 in section 3) and h u is defined by for some positive constant K 2 independent of (φ u , ). In order to state the main theorem, we need one additional assumption: (H5) When = 0, equation (2.1) has a homoclinic orbit Γ 0 asymptotic to the equilibrium 0.
For a fixed ∈ (− 0 , 0 ), let W u + ( ) be the orbit of equation (2.1) through a given Without loss of generality, we assume that the homoclinic orbit Γ 0 = W u + (0). Here, a homoclinic orbit Γ 0 asymptotic to 0 is a continuous mapping u : R → X satisfying for t, s ∈ R with t ≥ s, and lim t→±∞ u(t) = 0. Now we can state our main theorem on homoclinic bifurcation of (2.1), which is a generalization of the results ofŠil'nikov [Si68] and Chow and Deng [CD89] to abstract semilinear functional differential equations.

Preliminaries.
In this section, we prove the differentiability of solutions of equation (2.1) with respect to initial values and parameters and the smoothness of the stable and unstable manifolds.

Differentiability with Respect to Initial Values and Parameters. Let
V be a neighborhood of 0 in C, (a, b) be an open interval in R and F ∈ C k (V × (a, b); X). Consider (3.1) In the proof, we shall use Lemma 4.2 and the argument for Theorem 4.1 in Hale and Verduyn Lunel [HV93].
Proof. Fix ξ ∈ V and α 0 ∈ (a, b). There exist constants M > 0, δ > 0 and N > 0 such that Now choose η ∈ (0, 1) and ν ∈ (0, 1) so that Clearly K(η, ν) is a closed subset of the Banach space This shows that Since ν < 1, we conclude that is a uniform contraction. By Lemma 4.2 of Hale and Verduyn Lunel [HV93], for Standard continuation argument then leads to the C k −smoothness of u(φ, α) with respect to (φ, α) for t in any compact subset of the domain of the definition of u(φ, α).

Smoothness of the Stable and Unstable Manifolds.
In this subsection, we study the C k −smoothness of the stable and unstable manifolds of equation (2.1) (Chow and Lu [CL88a,CL88b]). First, we modify assumption (H3) as follows: ). Note that u(0, )(t) = 0 for all t ≥ 0 is always a solution of (2.1) with u 0 = 0. Therefore, for a fixed τ 1 > r, by the basic theory of (2.1) (see Wu [Wu96]) it follows that there exists an open neighborhood N 0 of 0 in C and Thenf is completely continuous, C k −smooth and D φf (0, 0) = T (τ 1 ) (defined in section 2). Let Σ s = {λ ∈ C; λ is a characteristic value of (2.2) with Reλ < 0}, Σ c = {λ ∈ C; λ is a characteristic value of (2.2) with Reλ = 0}, λ is a characteristic value of (2.2) with Reλ > 0} and assume that For each λ ∈ Σ s ∪Σ u , let M λ be the realized generalized eigenspace of A T associated with λ and denote Then we know that dimC u < ∞, C u and C s are closed subspaces of C such that Theorem 3.2. We have the following results on the smoothness of the stable and unstable manifolds.
Hence, E = C × R has the following decomposition By Theorem II.1 of Krisztin, Walther and Wu [KWW99], there exist open neighbor- Then Therefore, Consequently, u t (φ, ) ∈W . In other words, Asφ s = u s t (φ, ) ∈ N s and ∈ (− s , s ), we must have That is, u t (φ, ) ∈ W. This proves (i).
(ii) Using Theorem III.1 of Krisztin, Walther and Wu [KWW99], the smoothness of the unstable manifold can be proved similarly.

Local Analysis.
Under hypothesis (H3), we may assume, without loss of generality, that the constant K 2 defined in (2.13) is positive and that δ 1 > 0 is chosen so that for |φ s | < δ 1 , |φ u | ≤ δ 1 and ∈ [− 1 , 1 ], we have and for |φ u | < δ 1 and | | ≤ 1 . By (2.4), the definition of X 0 and the fact that (4.7) Thus, (2.12) can be rewritten as follows By the smoothness of the local unstable manifold W u loc ( ), differentiation of h u (θ) with respect to θ ∈ [−r, 0] leads to Similar to the proof of Proposition 3.2 in Chow and Deng [CD89], we have the following lemma. (4.10) Moreover, there exists a constant K 3 > 0 depending on δ 1 , 1 , K 1 and K 2 such that By the smoothness of the stable and unstable manifolds (Theorem 3.2) and following the argument in the proof of Proposition 3.4 in Chow and Deng [CD89], we have the following lemma.
Applying the Implicit Function Theorem to it follows that τ (φ, ) is C 2 . Moreover, by Lemma 4.5, we have Therefore, Similarly we can show that Letφ ∈ Ω + ∪ Ω − . Differentiating (4.24) and using the Chain Rule, we have Therefore, by (4.26) and Lemma 4.4, we obtain This completes the proof.
is the intersection point of Σ(δ 4 /2, ) and the solution orbit of (2.1) with the initial value φ and parameter .
is continuous in (φ, ) and is Lipschitzian in φ for each fixed .
(iii) For every ∈ [− 8 , 8 ], 6. The Proof of the Main Results. To prove Theorem 2.1, we need the following lemmas.