Intermittent synchronization in non-weakly coupled piecewise-linear expanding map lattice: A geometric-combinatorial...

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Statistical Mechanics

Background:

• Coupled map lattices (CMLs) model spatially extended dynamical systems.
• Prior research on CMLs primarily used the Perron-Frobenius operator, focusing on weak coupling.
• Understanding dynamics beyond weak coupling is crucial for complex systems.

Purpose of the Study:

• To develop a novel method for analyzing CMLs beyond the weak-coupling regime.
• To investigate the dynamical behavior of a two-node CML with identical piecewise-linear expanding maps.
• To derive conditions for unique invariant measures and intermittent synchronization.

Main Methods:

• A novel geometric-combinatorial approach was developed.
• Analysis focused on a two-node coupled map lattice system.
• Piecewise-linear expanding maps were utilized.

Main Results:

• A necessary and sufficient condition was derived for the uniqueness of absolutely continuous invariant measures.
• The condition also guarantees intermittent synchronization, where orbits repeatedly enter and leave a neighborhood of the diagonal.
• The method extends analysis beyond the limitations of the Perron-Frobenius framework for weakly coupled systems.

Conclusions:

• The developed geometric-combinatorial method provides new insights into CML dynamics.
• The findings are significant for understanding synchronization phenomena in complex systems.
• This work opens avenues for studying strongly coupled CMLs.