Invariant tori in dissipative hyperchaos

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 quasiperiodic 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; tori can be numerically identified via bifurcations of unstable periodic orbits and their parameteric continuation and characterization of stability properties are 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.

Automated summary

One method to comprehend the chaotic dynamics of nonlinear dissipative systems is through the exploration of non-chaotic yet dynamically unstable invariant solutions, including fixed points, one-dimensional unstable periodic orbits, and two-dimensional unstable invariant tori. This study demonstrates the generic existence of unstable 2-tori in a dissipative system of ordinary differential equations, which can be numerically identified and characterized. These higher-dimensional tori, together with periodic orbits and equilibria, form a complete set of relevant invariant solutions for understanding chaos.