Diffusion and scaling in escapes from two-degrees-of-freedom Hamiltonian systems

Area of Science:

• Physics
• Dynamical Systems
• Statistical Mechanics

Background:

• Hamiltonian systems with two degrees of freedom can exhibit global stochasticity.
• The behavior of orbits in such systems is influenced by nonintegrable corrections.
• Understanding escape dynamics is crucial for characterizing chaotic behavior.

Purpose of the Study:

• To investigate the statistical properties of escaping orbits in three distinct Hamiltonian systems.
• To analyze the influence of a nonintegrable correction parameter (epsilon) on orbit escape.
• To identify critical parameters and scaling laws governing the escape dynamics.

Main Methods:

• Simulating ensembles of orbits in three different two-degrees-of-freedom Hamiltonian systems.
• Analyzing the escape probability (P) as a function of time and system parameters.
• Investigating the scaling of escape probability and convergence time with system parameters and region size.

Main Results:

• Escape probability exhibits an initial exponential decay, transitioning to power-law decay near a critical parameter value (epsilon(1)).
• A critical parameter epsilon(1) marks an abrupt change in escape behavior.
• Escape probability P(0) and convergence time T scale as power laws of (epsilon - epsilon(1)) and region size r, with critical exponents alpha, beta, and delta.

Conclusions:

• The transitional escape behavior is linked to the breakdown of KAM tori or cantori.
• Late-time power-law escape reflects intrinsic diffusion of chaotic orbits.
• The universality of critical exponents across different potentials suggests fundamental properties of chaotic systems.