Flexible periodic points

TL;DR

This paper introduces the concept of ε-flexible periodic points in dynamical systems, showing how small perturbations can alter their stability properties and demonstrating their prevalence in systems with non-hyperbolic center-stable bundles.

Contribution

It defines ε-flexible periodic points and proves their perturbability to change stable indices, revealing their generic presence in certain non-hyperbolic systems.

Findings

ε-perturbations can transform stable index two points into index one points

Flexible points are common in systems with non-hyperbolic two-dimensional center-stable bundles

The stable manifold of the perturbed point can be arbitrarily shaped as a C^1-curve

Abstract

We define the notion of $\varepsilon$-flexible periodic point: it is a periodic point with stable index equal to two whose dynamics restricted to the stable direction admits $\varepsilon$-perturbations both to a homothety and a saddle having an eigenvalue equal to one. We show that $\varepsilon$-perturbation to an $\varepsilon$-flexible point allows to change it in a stable index one periodic point whose (one dimensional) stable manifold is an arbitrarily chosen $C^1$ -curve. We also show that the existence of flexible point is a general phenomenon among systems with a robustly non-hyperbolic two dimensional center-stable bundle.