Symplectic blenders near whiskered tori and persistence of saddle-center homoclinics

TL;DR

This paper demonstrates how symplectic blenders can be constructed near whiskered tori in symplectic diffeomorphisms, ensuring the persistence of saddle-center homoclinic intersections under small perturbations, with implications for dynamical stability.

Contribution

It introduces a method to create symplectic blenders near whiskered tori and proves the persistence of saddle-center homoclinic intersections under perturbations.

Findings

Symplectic blenders can be generated near whiskered tori via small perturbations.

Non-transverse homoclinic intersections are robust under perturbations.

Results extend to continuous-time dynamical systems.

Abstract

A blender is a hyperbolic basic set such that the projection of its stable/unstable set onto some center subspace has a higher topological dimension than the set itself. We prove that, for any $C^s$ symplectic diffeomorphism (where $s=2,\dots\infty,\omega$), if it has a one-dimensional whiskered torus with a homoclinic orbit, then a symplectic blender can be created by an arbitrarily $C^s$-small perturbation. Using this result, we show that the non-transverse homoclinic intersection between the invariant manifolds of a saddle-center periodic point is persistent, in the sense that the original system lies in the $C^s$-closure of a $C^1$-open set of symplectic diffeomorphisms where those having saddle-center homoclinics are dense. Our results also hold in the corresponding continuous-time settings.