Infinite Bifurcations in Thomas system

TL;DR

This paper analyzes the complex dynamics of the Thomas system, revealing rich behaviors including bifurcations, chaos, and infinite fixed points, driven by a single bifurcation parameter and exploring the zero-dissipation limit.

Contribution

It provides both analytical and numerical insights into the bifurcation structure and chaotic behavior of the Thomas system, highlighting the effects of dissipation and the zero-dissipation limit.

Findings

System exhibits stable regimes, limit cycles, and chaos depending on parameter b.

Infinite bifurcations and fixed points occur as b varies.

System behaves like Brownian motion when dissipation approaches zero.

Abstract

In this paper we are going to make an analytical and numerical analysis for the Thomas system. Physically, this system describes a particle, driving by a system of oscillators, dissipated by a dissipation term b > 0. Mathematically, this system is very interesting because it contains rich dynamics in it which is generated by only one bifurcation parameter b. Depending on the value of b, the system is undergo through a stable regime, limit cycles, infinite amount of bifurcations, a series growing to infinity of fixed points, and chaos containing multiple attractors. Another interesting behaviour of the system is in the limit of b goes to zero, which means that there is no dissipation term. The system is then containing an infinite number of fixed points and behaves like a Brownian motion.