On the dynamics of rotating rank-one strange attractors families

TL;DR

This paper investigates a two-parameter family of rotating rank-one maps on a three-dimensional space, revealing how strange attractors and hyperchaos emerge through resonance phenomena and coupling mechanisms, supported by rigorous analysis and simulations.

Contribution

It provides a rigorous analysis of the dynamics of rotating rank-one strange attractors, linking their existence to resonance tongues in Arnold circle maps, and explores the transition to hyperchaos.

Findings

Existence of strange attractors in resonance tongues.

Unique physical measure determines typical system behavior.

Transition from strange attractors to hyperchaos with positive Lyapunov exponents.

Abstract

In this article, we study a two-parameter family of rotating rank-one maps defined on $\textbf{S}^1\times [1, 1+b]\times \textbf{S}^1$, with $b\gtrsim 0$, whose dynamics is characterised by a coupling of a family of planar maps exhibiting rank-one strange attractors and an Arnold family of circle maps. The main result is about the dynamics on the skew-product, which is governed by the existence and prevalence of strange attractors in the corresponding resonance tongues of the Arnold family. The strange attractors carry the unique physical measure of the system, which determines the behaviour of Lebesgue-almost all initial conditions. This phenomenon can be considered as the transition dynamics from a strange attractor with one positive Lyapunov exponent to hyperchaos. Besides an analytical rigorous proof, we illustrate the main results with numerical simulations. We also conjecture how persistent hyperchaos can be obtained.