By Elhadj Zeraoulia, Julien Clinton Sprott
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Extra resources for 2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach
Assume that there exist n homoclinic points for these invariant manifolds. Let W s (Q) be the stable manifold and W u (Q) be the unstable manifold, and let x be the homoclinic point farthest from the fixed point Q along the unstable manifold W u (Q). Since W s (Q) and W u (Q) are invariant manifolds, f (x) ∈ W s (Q) ∩ W u (Q). Thus f (x) is either a homoclinic point or the fixed point Q. Since f takes a point on the unstable manifold W u (Q) away from the fixed point Q, then f (x) cannot be the fixed point.
1987)]. 22. (a) The invariant manifold of a map f is a set of points X such that f (X) ⊂ X 12 . (b) Given a fixed point Y0 in the invariant manifold, the manifold Y is called stable if ∀y ∈ Y , limn→∞ f n (y) → Y0 . Similarly, a manifold is called unstable if ∀y ∈ Y , limn→∞ f −n (y) → Y0 . (c) A fixed point is hyperbolic if it is the intersection of one or more stable manifolds and one or more unstable manifolds. (c) A homoclinic point is a point x, different from a fixed point, that lies on both a stable manifold and an unstable manifold of the same fixed point Q.
This method has been successfully applied to the H´enon map. See Sec. 6. 2 Homoclinic orbit shadowing In [Coomes et al. (2005)], a new computer-assisted technique with two main components for the rigorous proof of the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in Rn is presented. The computation of a suitable pseudo (approximate) homoclinic orbit to a pseudo-periodic orbit was done using the global Newton’s method introduced in Sec. 1. 24 given below) is applied to prove the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo-orbits of the H´enon map [H´enon (1976)]14 .