Chaotic diffusion on periodic orbits: the perturbed Arnold cat map.

Area of Science:

• Dynamical systems theory
• Statistical mechanics
• Chaos theory

Background:

• Chaotic diffusion is a key phenomenon in nonlinear dynamics.
• Periodic orbits (POs) are fundamental structures in understanding chaotic systems.
• The Arnold cat map serves as a model for studying chaotic behavior.

Purpose of the Study:

• To calculate the diffusion coefficient for chaotic diffusion on POs in a perturbed Arnold cat map.
• To compare the accuracy of three different methods for calculating the diffusion coefficient based on POs.
• To analyze the conditions under which different calculation methods yield accurate results.

Main Methods:

• Utilizing the curvature expansion of the Ruelle zeta function to calculate diffusion coefficients.
• Employing the weighted average of the PO winding-number squared (w(2)) with a stability factor.
• Using the uniform (nonweighted) average of w(2).

Main Results:

• Formulas based on the Ruelle zeta function curvature expansion and weighted average of w(2) show excellent agreement with standard methods.
• The uniform average of w(2) provides reasonably accurate results for small perturbation parameters and non-uniform hyperbolicity.
• Accuracy of the uniform average is linked to uniformity sum rules of PO Lyapunov eigenvalues.

Conclusions:

• The curvature expansion of the Ruelle zeta function and stability-weighted average of w(2) are reliable methods for calculating diffusion coefficients.
• The uniform average of w(2) is a viable, simpler approximation under specific conditions of small perturbations and non-uniform hyperbolicity.
• The study highlights the importance of POs and their properties in understanding chaotic diffusion.