Mechanism of synchronization in a random dynamical system.

Area of Science:

• Dynamical systems and statistical physics, focusing on chaotic dynamics and synchronization phenomena.

Background:

• The Zaslavsky map is a model for deterministic chaos with a unique phase space structure.
• Understanding synchronization mechanisms is crucial in various physical systems.

Purpose of the Study:

• To investigate the synchronization mechanism within the random Zaslavsky map.
• To analyze the phase space structure and identify key bifurcations.
• To reveal the nature of on-off intermittency in particle ensembles.

Main Methods:

• Analysis of error dynamics for two particles to understand phase space structure.
• Identification of a transcritical bifurcation between saddle and stable fixed points.
• Verification of on-off intermittency using a biased random walk model.
• Application of a modified snapshot attractor size definition for particle ensembles.

Main Results:

• A transcritical bifurcation was identified, characterizing the system's dynamic transitions.
• On-off intermittency was confirmed and linked to biased random walk behavior.
• The structure of on-off intermittency in ensembles of particles was explicitly revealed near the transition point.

Conclusions:

• The study elucidates the synchronization mechanism in the random Zaslavsky map.
• On-off intermittency is a key feature, explainable via random walk dynamics.
• The findings provide insights into chaotic system behavior and transitions in particle ensembles.