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Invariant tori in dissipative hyperchaos.

Jeremy P Parker1, Tobias M Schneider1

  • 1Emergent Complexity in Physical Systems Laboratory (ECPS), École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland.

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Unstable 2-tori, rarely studied, are found in chaotic systems. These, along with periodic orbits and fixed points, offer a complete dynamical description of chaos.

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Area of Science:

  • Nonlinear dynamics
  • Chaos theory
  • Dynamical systems

Background:

  • Chaotic dynamics in nonlinear dissipative systems are complex.
  • Unstable fixed points and periodic orbits are key to understanding time-periodic dynamics in chaos.
  • Higher-dimensional invariant tori for quasiperiodic dynamics are less understood.

Purpose of the Study:

  • To investigate the presence and significance of unstable 2-tori in hyperchaotic attractors.
  • To establish a complete set of invariant solutions for describing chaotic dynamics.

Main Methods:

  • Numerical identification of unstable 2-tori through bifurcations of unstable periodic orbits.
  • Parametric continuation and stability analysis of these tori.

Main Results:

  • Unstable 2-tori are generically embedded in hyperchaotic attractors of dissipative systems.
  • These tori can be numerically identified and analyzed.
  • 2-tori, periodic orbits, and equilibria form a complete set of invariant solutions.

Conclusions:

  • Unstable 2-tori are crucial for a comprehensive dynamical description of chaos.
  • The findings extend the understanding of invariant solutions in chaotic systems.
  • This work provides a foundation for analyzing quasiperiodic dynamics within chaos.