Chaotic polynomial maps

TL;DR

This paper studies polynomial maps in Euclidean spaces, exploring conditions for chaos, hyperbolic sets, and strange attractors, including classical maps like Logistic and Hénon, with theoretical and computational analysis of their chaotic dynamics.

Contribution

It introduces new conditions for chaos and hyperbolicity in polynomial maps, extending classical examples and analyzing high-dimensional cases with computer simulations.

Findings

Existence of Smale horseshoes in polynomial maps under certain conditions

Identification of hyperbolic invariant sets conjugate to full shifts

Discovery of chaotic attractors with positive Lyapunov exponents in 3D maps

Abstract

This paper introduces a class of polynomial maps in Euclidean spaces, investigates the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets, studies the chaotic dynamical behavior and strange attractors, and shows that some maps are chaotic in the sense of Li-Yorke or Devaney. This type of maps includes both the Logistic map and the H\'{e}non map. For some maps in three-dimensional spaces under certain conditions, if the expansion dimension is equal to one or two, it is shown that there exist a Smale horseshoe and a uniformly hyperbolic invariant set on which the system is topologically conjugate to the two-sided fullshift on finite alphabet; if the system is expanding, then it is verified that there is an forward invariant set on which the system is topologically semi-conjugate to the one-sided fullshift on eight symbols. For three types of high-dimensional polynomial maps with degree two, the existence of Smale horseshoe and the uniformly hyperbolic invariant sets are studied, and it is proved that the map is topologically conjugate to the two-sided fullshift on finite alphabet on the invariant set under certain conditions. Some interesting maps with chaotic attractors and positive Lyapunov exponents in three-dimensional spaces are found by using computer simulations. In the end, two examples are provided to illustrate the theoretical results.