Invariant tori in dissipative hyperchaos

TL;DR

This paper demonstrates that unstable 2-tori are generically embedded in hyperchaotic attractors of dissipative systems, and shows how to numerically identify and analyze their stability, expanding understanding of invariant solutions in chaos.

Contribution

It provides the first demonstration of unstable 2-tori in hyperchaotic systems and introduces methods for their numerical identification and stability analysis.

Findings

Unstable 2-tori are generically embedded in hyperchaotic attractors.

Bifurcations of unstable periodic orbits can be used to identify tori.

Parametric continuation allows characterization of tori stability.

Abstract

One approach to understand the chaotic dynamics of nonlinear dissipative systems is the study of non-chaotic yet dynamically unstable invariant solutions embedded in the system's chaotic attractor. The significance of zero-dimensional unstable fixed points and one-dimensional unstable periodic orbits capturing time-periodic dynamics is widely accepted for high-dimensional chaotic systems including fluid turbulence, while higher-dimensional invariant tori representing quasi-periodic dynamics have rarely been considered. We demonstrate that unstable 2-tori are generically embedded in the hyperchaotic attractor of a dissipative system of ordinary differential equations; that tori can be numerically identified via bifurcations of unstable periodic orbits and that their parametric continuation and characterization of stability properties is feasible. As higher-dimensional tori are expected to be structurally unstable, 2-tori together with periodic orbits and equilibria form a complete set of relevant invariant solutions on which to base a dynamical description of chaos.