Dynamics of coupled maps with a conservation law.

Area of Science:

• Complex Systems
• Statistical Physics
• Chaos Theory

Background:

• Coupled map lattices are fundamental models for studying spatiotemporal chaos.
• Conservation laws significantly influence the behavior of dynamical systems.

Purpose of the Study:

• To investigate the phase structure and phase transitions of a simple coupled map model with a local conservation law.
• To determine the robustness of the phase diagram to violations of the conservation law.
• To calculate critical exponents and explore potential new universality classes.

Main Methods:

• Analysis of a simplified coupled map lattice model.
• Determination of phase structure and phase transition types.
• Calculation of critical exponents for order parameters.
• Numerical investigation of singularities in Lyapunov exponents and Fourier modes.
• Discussion of Lyapunov dimension for spatiotemporal chaos.

Main Results:

• The phase structure and types of phase transitions were identified.
• The phase diagram exhibits robustness against mild violations of the conservation law.
• Critical exponents suggest a possible new universality class.
• A singularity in Lyapunov exponents is linked to a Van Hove singularity in Fourier modes, disappearing when the conservation law is broken.

Conclusions:

• The studied model provides insights into the behavior of systems with conservation laws.
• The robustness of the phase diagram is a key finding.
• The connection between Lyapunov exponents and Fourier modes highlights the importance of conservation laws in characterizing spatiotemporal chaos.