Stickiness in generic low-dimensional Hamiltonian systems: A recurrence-time statistics approach

Area of Science:

• Physics
• Nonlinear Dynamics
• Statistical Mechanics

Background:

• Generic Hamiltonian systems often exhibit divided phase space, leading to complex dynamics.
• Chaotic diffusion models, like the random model, assume uncorrelated cell visits and exponential recurrence times.
• Understanding 'stickiness' is crucial for characterizing transport in chaotic systems.

Purpose of the Study:

• To analyze the structure and stickiness in chaotic components of Hamiltonian systems with divided phase space.
• To identify sticky regions using recurrence time statistics.
• To develop a quantitative measure of stickiness.

Main Methods:

• Utilized recurrence time statistics of chaotic orbits within phase space cells.
• Performed numerical studies on the Chirikov standard map, Robnik billiards, and lemon billiards.
• Compared cell filling to a random model of chaotic diffusion.

Main Results:

• Introduced a stickiness measure S (ratio of standard deviation to mean recurrence time).
• Found S=1 in the bulk of the chaotic sea and S>1 in sticky regions.
• Observed separation of timescales in recurrence times due to dynamical trapping.

Conclusions:

• Stickiness induces correlations in cell visits, deviating from random models.
• The measure S effectively quantifies and distinguishes sticky regions.
• Results visualized using animated grayscale plots of S.