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Phase and frequency linear response theory for hyperbolic chaotic oscillators.

Ralf Tönjes1, Hiroshi Kori2

  • 1Institute of Physics and Astronomy, Potsdam University, 14476 Potsdam-Golm, Germany.

Chaos (Woodbury, N.Y.)
|April 30, 2022
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We developed a linear theory for phase and frequency response in chaotic systems, generalizing concepts from limit cycle oscillators. This approach uses a shadowing conjecture and co-variant Lyapunov vectors to analyze chaotic dynamics and measure phase response curves.

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Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Statistical Mechanics

Background:

  • Phase response theory is established for autonomous limit cycle oscillators.
  • Extending this theory to hyperbolic chaotic dynamics remains a challenge.

Purpose of the Study:

  • To formulate a linear phase and frequency response theory for hyperbolic flows.
  • To generalize phase response theory to hyperbolic chaotic dynamics.

Main Methods:

  • Based on a shadowing conjecture for perturbed trajectories.
  • Utilizing a unique time isomorphism identified as phase.
  • Solving an adjoint linear equation for the phase sensitivity function.
  • Employing co-variant Lyapunov vectors in tangent space.

Main Results:

  • Developed a generalized phase response theory for chaotic dynamics.
  • Phase sensitivity function estimates average phase velocity changes.
  • Frequency changes are experimentally accessible for measuring phase response curves.
  • Identified limits of the linear response regime.

Conclusions:

  • The new theory provides a framework for understanding linear response in chaotic systems.
  • Phase response curves can be defined and measured for chaotic oscillators.
  • Co-variant Lyapunov vectors are key to constructing shadowing trajectories and phase.