Notes on homoclinic bifurcation theory Ale Jan Homburg KdV Institute for Mathematics University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The Netherlands [email protected] Version 20051027 1 Introduction These notes contain a proof of the disappearance of an isolated periodic orbit in a homoclinic bifurcation with a homoclinic loop to a hyperbolic saddle equilibrium. The notes include a treatment of the implicit function and the stable manifold theorem. 2 Implicit function theorem At several places we will find the need to solve equations on Banach spaces. We include the statement of the implicit function theorem allowing us to do so. Implicit function theorems on Banach spaces were developed by Hildebrandt and Graves [12]. We follow the treatment in [7]. See [25] for an account of the history of implicit functions theorems. 2.1 Contraction theorem The Banach contraction theorem says that a contraction on a complete metric space has a unique fixed point. Let S be a metric space with metric d. Recall that T : S → S is a contraction if for some θ ∈ [0, 1), d(T (x), T (y)) ≤ θd(x, y). (2.1) The constant θ is called a contraction constant. 1 Note that in an equivalent metric d˜, this gives d˜(T (x), T (y)) ≤ Cθnd˜(x, y) Theorem 2.1. If (S,d) is a complete metric space and T : S → S is a contraction, then n there is a unique fixed point x¯ of T in S. Also, for any x0 ∈ S, T x0 → x¯ as n →∞. 2.2 Uniform contraction theorem For a contraction depending on a parameter, also the fixed point will depend on a parameter. A contraction is called uniform if the contraction constant can be chosen independent of the parameter. Theorem 2.2. Let U, V be open sets on Banach spaces X,Y , let U¯ be the closure of U, T : U¯ × V → U¯ a uniform contraction on U¯ and let g(y) be the unique fixed point of T (·,y) in U¯. If T ∈ Ck(U¯ × V,X), 0 ≤ k < ∞ then g(·) ∈ Ck(V,X). Proof. We will show that g is continuous. This follows easily using the fixed point formulas and the definition of contraction. Compute |g(y + h) − g(y)| = |T (g(y + h),y + h) − T (g(y),y)| ≤ |T (g(y + h),y + h) − T (g(y),y + h)| + |T (g(y),y + h) − T (g(y),y)| ≤ θ|g(y + h) − g(y)| + |T (g(y),y + h) − T (g(y),y)|. Therefore 1 |g(y + h) − g(y)| ≤ |T (g(y),y + h) − T (g(y),y)|. 1 − θ This implies continuity of g. Derivatives of g are treated as follows. Start at the fixed point equation T (g(y),y)= g(y), differentiate to obtain equations for derivatives, solve these equations and prove that the solution indeed equals the derivative of g. Formally differentiating T (g(y),y)= g(y) with respect to y gives DxT (g(y),y)Dg(y)+ DyT (g(y),y)= Dg(y). So Dg(y) should be a solution M(y) to DxT (g(y),y)M(y)+ DyT (g(y),y)= M(y). (2.2) This equation can be solved since DxT (g(y),y) gives a contraction. To establish that M(y) equals the derivative of g, we must show that |g(y + h) − g(y) − M(y)h| = o(|h|) as h → 0. Write γ(y)= g(y + h) − g(y). Now γ(y) = T (g(y)+ γ(y),y + h) − T (g(y),y) = DxT (g(y),y)γ(y)+ DyT (g(y),y)h + R, (2.3) 2 where R = T (g(y)+ γ(y),y + h) − T (g(y),y) − DxT (g(y),y)γ(y) − DyT (g(y),y)h. Since T is differentiable, for any ǫ > 0 there is δ > 0 with |R| < ǫ(|γ(y)| + |h|) if |γ(y)|, |h| < δ. Since γ is continuous in h we may further restrict δ so that this estimate holds for |h| < δ. −1 From (2.3) we get γ(y)=(I − DxT (g(y),y)) (DyT (g(y),y)h + R), so that |γ(y)| < C|h| for some C > 0, if |h| < δ. This implies |R| < ǫ(1 + C)|h| for |h| < δ. Equation (2.2) shows that (I − DxT (g(y),y))M(y) = DyT (g(y),y). With (2.3) this yields (I − DxT (g(y),y))(γ(y) − M(y)h)= R. We obtain |γ(y) − M(y)h| < Kǫ|h| for some K > 0, if |h| < δ. This proves that M(y) equals the derivative Dg(y). Higher order derivatives are treated by induction. Assume that T is of class Ck and g has been shown to be Cj, j<k. The above reasoning shows that M(y) is Cj and equals the derivative Dg(y). So g is of class Cj+1. 2.3 Implicit function theorem The implicit function theorem is a corollary of the uniform contraction theorem, by combining it with a Newton type algorithm. Theorem 2.3. Suppose X,Y,Z are Banach spaces, U ⊂ X,V ⊂ Y are open sets, F : U × V → Z is continuously differentiable, (x0,y0) ∈ U × V , F (x0,y0)=0 and DxF (x0,y0) has a bounded inverse. Then there is a neighborhood U1 × V1 ∈ U × V of (x0,y0) and a function f : V1 → U1, f(y0) = x0 such that F (x, y)=0 for (x, y) ∈ U1 × V1 if and only if k k x = f(y). If F ∈ C (U × V,Z), k ≥ 1, then f ∈ C (V1,X). −1 Proof. If L = [DxF (x0,y0)] , G(x, y) = x − LF (x, y), then the fixed points of G are the solutions of F = 0. The map G has the same smoothness properties as F , G(x0,y0) = 0, DxG(x0,y0) = 0. Therefore, |DxG(x, y)| ≤ θ < 1 for all x, y is a neighborhood U1 × V1 of (x0,y0). It is easy to choose the neighborhood so that G : U1 × V1 → U1. The result now follows from Theorem 2.2. Remark 2.4. Note that the above reduction to an application of the uniform contraction theorem works if y 7→ F (x, y) is continuous, as long as x 7→ F (x, y) is continuously differ- entiable and DxF (x0,y0) has bounded inverse. See [5]. 3 Invariant manifolds Consider a flow φt for a differential equationx ˙ = f(x) on a compact manifold M. A submanifold W ⊂M is invariant if φt(W )= W , t ∈ R. It is positive invariant if this holds for t > 0. It is locally invariant if f(x) ∈ TxW for x ∈ W . Invariant manifolds play an important role in the qualitative study of flows. There exists a substantial theory on robust invariant manifolds, for which perturbed flows have perturbed invariant manifolds. 3 We will see a glimpse of the general theory by studying dynamically defined invariant man- ifolds near equilibria. 3.1 Stable manifold theorem Consider a differential equation x˙ = f(x) (3.1) on a smooth compact manifold M. We assume that X is a smooth, that is C∞, map. Write (t, x0) 7→ φt(x0) for the flow of X giving the solution to (3.1) equal to x0 at time t = 0. Standard theory of ordinary differential equations shows that φt(x0) depends smoothly on (t, x0). Suppose (3.1) has an equilibrium p ∈M, f(p) = 0. Suppose further that the origin is a hyperbolic equilibrium, i.e. the spectrum of Df(p) is disjoint from the imaginary axis. s u s u There is a Df(p) invariant splitting E ⊕ E of TpM where Df(p) restricted to E (E ) has spectrum with negative (positive) real part. We refer to Es as the stable subspace of Df(p) and to Eu as the unstable subspace of Df(p). Define the stable set s W (p)= {x ∈M| lim φt(x)= p}. t→∞ u The unstable set W (p) is defined as the stable set for φ−t. There are several proofs of the stable manifold theorem (for a hyperbolic equilibrium) stated below. We give a variant of Perron’s proof [29], see also [19, 20, 21, 32, 16]. A more geometric approach to stable manifolds is with graph transforms. This approach originates with [11]. An extensive use of graph transforms is in [13]. We quote D.S. Anosov [2]: Every five years or so, if not more often, someone discovers the theorem by Hadamard and Perron, proving it by Hadamard’s method of proof or by Perron’s. s s s Proposition 3.1. W (p) is an injectively immersed smooth manifold with TpW (p)= E . Proof. We construct a local stable manifold defined as s Rn Wloc(p)= {x ∈ | d(x, p) < σ, lim φt(x)= p} t→∞ s s for a sufficiently small σ > 0. Then W (p) = ∪t≤0Wloc(p). Using a local coordinate system we may assume that p equals the origin in Rn and that we are working with a differential equationx ˙ = f(x) on Rn. Using a test function we may assume that X is globally close to Df(0). Indeed, let ψ : Rn → R be a test function, i.e. a smooth nonnegative function such that ψ = 0 outside a neighborhood of 0 and ψ = 1 near 0. Consider the rescaled version ψǫ(x) = ψ(x/ǫ). Then ψǫX + (1 − ψǫ)Df(0) converges together with first order derivatives to Df(0) as ǫ → 0. 4 Consider the time 1 map F = φ1.