Hierarchical structure in sharply divided phase space for the piecewise linear map

Area of Science:

• * Dynamical Systems
• * Hamiltonian Systems
• * Chaos Theory

Background:

• * Hamiltonian systems exhibit complex dynamics with stable and chaotic regions.
• * Stable regions in phase space can possess intricate hierarchical structures.
• * Understanding slow dynamics is crucial for characterizing system behavior.

Purpose of the Study:

• * To investigate the influence of hierarchical stable regions on slow dynamics in Hamiltonian systems.
• * To analyze the role of stable region structure in chaotic dynamics.
• * To determine the factors governing recurrence time distributions.

Main Methods:

• * Studied a two-dimensional piecewise linear map.
• * Employed symbolic representation for stable regions.
• * Developed a procedure to compute polygon edges of stable regions.
• * Analyzed symbol sequences of recurrence trajectories.

Main Results:

• * Identified infinitely many polygonal stable regions hierarchically distributed in phase space.
• * Demonstrated that the cumulative distribution of recurrence time follows a power law (~t^{-2}).
• * Showed that the hierarchical structure does not significantly affect the power-law exponent.

Conclusions:

• * Marginal instability at the edges of stable regions is the primary determinant of the power-law exponent.
• * The hierarchical organization of stable regions has a limited impact on the observed power-law dynamics.
• * Findings offer insights into generic chaotic systems with complex stable region structures.