Invariant tori in dissipative hyperchaos

Jeremy P. Parker (Lead / Corresponding author), Tobias M. Schneider

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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.
Original languageEnglish
Article number113102
Number of pages8
JournalChaos: An Interdisciplinary Journal of Nonlinear Science
Issue number11
Publication statusPublished - Nov 2022


  • Chaotic dynamics
  • Chaotic systems
  • Periodic-orbit theory
  • Ergodic theory
  • phase space methods
  • Invariant manifold


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