A computer-assisted proof of the existence of Smale horseshoe for the folded-towel map

TL;DR

This paper provides a rigorous, computer-assisted proof of symbolic dynamics chaos and hyperbolicity in the 4th iterate of a generalized Hénon map, confirming a prior conjecture and demonstrating complex chaotic behavior.

Contribution

It offers the first computer-assisted proof of chaos and hyperbolicity in the 4th iterate of the folded-towel map, validating a previous conjecture.

Findings

Existence of symbolic dynamics chaos in the 4th iterate of the map.

Proof of uniform hyperbolicity of the invariant set.

Validation of the conjecture from Li and Yang (2007).

Abstract

The paper contains a rigorous proof of existence of symbolic dynamics chaos in the generalized H\'enon map's 4th iterate $H^4$, which was conjectured in the paper \textit{A 3D Smale Horseshoe in a Hyperchaotic Discrete-Time System} of Li and Yang, 2007. We prove also the uniform hyperbolicity of the invariant set with symbolic dynamics. The proofs are computer-assisted with the use of C++ library \textit{CAPD} for interval arithmetic, differentiation and integration.