Wielding intermittency with cycle expansions.

Area of Science:

• Dynamical Systems Theory
• Statistical Mechanics
• Chaos Theory

Background:

• Periodic orbit theory struggles with computing observable averages in intermittent dynamical systems.
• Intermittent systems exhibit complex behavior near singularities, challenging traditional analysis methods.

Purpose of the Study:

• To develop a novel scheme combining cycle expansion and perturbation theory for precise computation of observable averages in intermittent systems.
• To improve the accuracy of dynamical average calculations for systems exhibiting intermittency.

Main Methods:

• The proposed method integrates periodic orbit theory (cycle expansion) with local perturbation theory.
• It revises intermittent maps to preserve the natural measure, using Taylor expansions near singularities.
• Periodic orbit theory is applied to the non-singular regions of the phase space.

Main Results:

• The combined approach successfully computes observable averages more precisely for intermittent systems.
• The method effectively captures the natural measure near singularities and in other phase space regions.
• Application to one-dimensional intermittent maps with a single singularity yielded improved results.

Conclusions:

• The cooperation between cycle expansion and perturbation theory offers a more accurate method for analyzing intermittent dynamical systems.
• This integrated technique provides a robust framework for calculating dynamical averages, overcoming limitations of periodic orbit theory alone.
• The revised map approach successfully maintains the natural measure, leading to enhanced precision in calculations.