Chaotic dynamics at the boundary of a basin of attraction via non-transversal intersections for a non-global smooth diffeomorphism

TL;DR

This paper analytically proves the existence of complex chaotic dynamics at the boundary of a basin of attraction in a specific class of non-globally smooth diffeomorphisms, revealing intricate invariant sets and symbolic dynamics.

Contribution

It provides the first analytic proof of transversal homoclinic points and chaotic invariant sets for a family of non-globally smooth diffeomorphisms near a critical cycle.

Findings

Existence of transversal homoclinic points proven analytically.

Boundary of basin contains a Cantor-like invariant set.

Dynamics on the invariant set is conjugate to a full shift.

Abstract

In this paper we give analytic proofs of the existence of transversal homoclinic points for a family of non-globally smooth diffeomorphisms having the origin as a fixed point which come out as a truncated map governing the local dynamics near a critical period three cycle associated to the Secant map. Using Moser's version of Birkhoff-Smale's Theorem, we prove that the boundary of the basin of attraction of the origin contains a Cantor-like invariant subset such that the restricted dynamics to it is conjugate to the full shift of $N$-symbols for any integer $N\ge 2$ or infinity.