Chaos in conservative discrete-time systems subjected to parameter drift

Area of Science:

• Dynamical Systems
• Chaos Theory
• Nonlinear Dynamics

Background:

• Parameter drifts in dynamical systems can alter system behavior.
• Understanding the evolution of invariant tori under perturbations is crucial.

Purpose of the Study:

• To analyze the dynamics of area-preserving low-dimensional mappings under parameter drifts.
• To identify conditions leading to torus breakup and understand its consequences on system dynamics.

Main Methods:

• Following ensembles of initial conditions on initial tori.
• Analyzing snapshot tori and their evolution.
• Investigating stable and unstable foliations within a chaotic sea.
• Identifying conditions for torus breakup based on map discontinuities and stable manifolds.

Main Results:

• Snapshot tori change location and shape under parameter drifts.
• Torus breakup occurs due to map discontinuities or specific stable manifolds, leading to strong stretching.
• Stable pseudo-foliation can extend beyond the chaotic sea.
• Exponential divergence of nearby points signifies torus breakup.
• Finite-time Lyapunov exponents exhibit splitting into shorter periods.

Conclusions:

• Ensemble dynamics provide insight into systems with parameter drifts.
• Torus breakup is a critical event with predictable visualizable conditions.
• The behavior of Lyapunov exponents is complex and depends on phase space structure.