Chaos in Hamiltonian systems subjected to parameter drift

Area of Science:

• Nonlinear dynamics
• Hamiltonian systems
• Chaos theory

Background:

• Low-dimensional Hamiltonian systems are fundamental models.
• Parameter drifts introduce time-dependence and complexity.
• Understanding long-term dynamics under perturbations is crucial.

Purpose of the Study:

• To analyze the dynamics of Hamiltonian systems under parameter drifts.
• To identify the mechanisms and conditions for torus breakup.
• To explore the relationship between intact tori and fluid vortices.

Main Methods:

• Following ensembles of initial conditions on tori.
• Analyzing 'snapshot tori' and their evolution.
• Investigating collisions with hyperbolic orbits and chaotic seas.
• Calculating finite-time Lyapunov exponents and using the polar rotation angle method.

Main Results:

• Snapshot tori emerge, exhibiting time-dependent shapes and central elliptic orbits.
• Torus breakup occurs due to collisions with hyperbolic orbits and chaotic seas.
• Intact tori persist, analogous to coherent vortices in fluid dynamics.
• Exponential divergence of initially close points signals torus breakup, characterized by a novel Lyapunov exponent.

Conclusions:

• Parameter drifts lead to complex dynamics and torus breakup in Hamiltonian systems.
• Torus breakup is a critical transition driven by chaotic interactions.
• Intact tori serve as analogs for coherent structures in fluid dynamics.
• The study provides insights into the behavior of perturbed dynamical systems.