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Characterizing chaos in systems subjected to parameter drift.

Dániel Jánosi1, Tamás Tél2

  • 1Institute for Theoretical Physics, Eötvös Loránd University, Pázmány Péter Sétány 1/A, H-1117 Budapest, Hungary.

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We introduce new methods to study chaos in systems with changing parameters. Following foliations helps identify chaotic behavior and its strength, even when traditional methods fail.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Nonlinear Dynamics
  • Statistical Mechanics

Background:

  • Traditional methods for characterizing chaos often fail in systems with parameter drift.
  • Understanding time-dependent chaos is crucial for many scientific and engineering applications.
  • Existing techniques may not apply when system parameters change over time.

Purpose of the Study:

  • To develop novel qualitative and quantitative approaches for characterizing chaos in systems with parameter drift.
  • To provide alternative methods when traditional chaos detection techniques are insufficient.
  • To analyze the behavior of chaotic sets in both dissipative and Hamiltonian systems under parameter variations.

Main Methods:

  • Qualitative analysis using stable and unstable foliations, approximated numerically.
  • Quantitative analysis using ensemble-averaged pairwise distance related to unstable foliations.
  • Investigation of Smale horseshoe-like patterns in foliation intersections.
  • Examination of snapshot attractors and snapshot chaotic seas in dissipative and Hamiltonian systems.

Main Results:

  • Stable and unstable foliations provide an efficient way to track chaos, even without hyperbolic periodic orbits.
  • Transverse intersections of foliations signify a time-varying chaotic set.
  • In dissipative systems, the chaotic set is not always dense on the snapshot attractor.
  • In Hamiltonian systems, unstable foliations correlate with the snapshot chaotic sea, where chaos is locally dense.
  • Quasiperiodic tori can break up and exhibit chaotic motion over time.

Conclusions:

  • Following foliations offers a robust method for characterizing chaos in drifting parameter systems.
  • The ensemble-averaged pairwise distance effectively quantifies instantaneous time-dependent chaos.
  • The proposed methods extend the analysis of chaotic dynamics to systems previously intractable with traditional approaches.