Chaos in conservative discrete-time systems subjected to parameter drift.

Area of Science:

• Dynamical Systems and Chaos Theory
• Nonlinear Dynamics

Background:

• Understanding parameter drifts in dynamical systems is crucial for predicting system behavior.
• Low-dimensional mappings provide a tractable model for studying complex dynamics.

Purpose of the Study:

• To analyze the dynamics of area-preserving low-dimensional mappings under parameter drifts.
• To identify conditions for torus breakup and understand the resulting chaotic behavior.

Main Methods:

• Following ensembles of initial conditions corresponding to initial tori.
• Analyzing snapshot tori, chaotic seas, and foliations.
• Investigating torus breakup conditions related to map discontinuities and stable manifolds.

Main Results:

• Snapshot tori change location and shape within a time-dependent chaotic sea.
• Two conditions for torus breakup were identified: map discontinuity and stable manifold interaction.
• Exponential divergence of nearby points occurs upon torus breakup, with regimes characterized by different finite-time Lyapunov exponents.

Conclusions:

• The study provides insights into the breakup of invariant tori in low-dimensional maps under parameter drift.
• Stable pseudo-foliation offers a generalized approach to understanding chaotic dynamics beyond the instantaneous chaotic sea.
• Averaged Lyapunov exponents can differ significantly from stationary map values in scenarios with divided phase space.