Robust chaos in autonomous time-delay system

TL;DR

This paper introduces a modified autonomous time-delay system based on the logistic differential equation that exhibits robust chaotic attractors, including Smale-Williams and Anosov types, with potential implications for stable chaos modeling.

Contribution

It presents a novel autonomous time-delay system with two feedback loops that generates robust chaotic attractors related to hyperbolic dynamics, expanding understanding of stable chaos.

Findings

System exhibits Smale-Williams attractor with double-expanding circle map.

System can produce attractors close to a torus with Anosov dynamics.

Attractors demonstrate robustness with no regularity windows.

Abstract

We consider an autonomous system constructed as modification of the logistic differential equation with delay that generates successive trains of oscillations with phases evolving according to chaotic maps. The system contains two feedback loops characterized by two generally distinct retarding time parameters. In the case of their equality, chaotic dynamics is associated with the Smale-Williams attractor that corresponds to the double-expanding circle map for the phases of the carrier of the oscillatory trains. Alternatively, at appropriately chosen two different delays attractor is close to torus with Anosov dynamics on it as the phases are governed by the Fibonacci map. In both cases the attractors manifest robustness (absence of regularity windows under variation of parameters) and presumably relate to the class of structurally stable hyperbolic attractors.