Invariant tori in dissipative hyperchaos

TY - JOUR

T1 - Invariant tori in dissipative hyperchaos

AU - Parker, Jeremy P.

AU - Schneider, Tobias M.

PY - 2022/11

Y1 - 2022/11

N2 - 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.

AB - 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.

KW - Chaotic dynamics

KW - Chaotic systems

KW - Periodic-orbit theory

KW - Ergodic theory

KW - phase space methods

KW - Invariant manifold

UR - https://doi.org/10.1063/5.0119642

U2 - 10.1063/5.0119642

DO - 10.1063/5.0119642

M3 - Article

C2 - 36456339

SN - 1089-7682

VL - 32

JO - Chaos: An Interdisciplinary Journal of Nonlinear Science

JF - Chaos: An Interdisciplinary Journal of Nonlinear Science

IS - 11

M1 - 113102

ER -