One-dimensional dynamics for traveling fronts in coupled map lattices

Area of Science:

• Complex systems
• Nonlinear dynamics
• Statistical physics

Background:

• Multistable coupled map lattices commonly exhibit traveling fronts that delineate distinct stable phases.
• Understanding the dynamics and velocity of these fronts is crucial for characterizing system behavior.

Purpose of the Study:

• To investigate the relationship between front profiles and propagation velocity in multistable coupled map lattices.
• To analyze the phenomenon of mode locking and its dependence on system parameters and stability boundaries.

Main Methods:

• Reduction of infinite-dimensional lattice dynamics to a one-dimensional circle homeomorphism using an invariant function.
• Analysis of the rotation number of the homeomorphism to determine front propagation velocity.
• Study of front behavior near parametric stability boundaries and in the continuum limit.

Main Results:

• The existence of an invariant function enables a simplified description of front dynamics.
• The rotation number of the resulting circle homeomorphism directly yields the front propagation velocity.
• Mode locking of velocity with system parameters is observed, but tends to vanish near the continuum limit.

Conclusions:

• The invariant function approach provides a powerful tool for understanding traveling fronts in these systems.
• Velocity and mode locking phenomena are intrinsically linked to the topological properties of the reduced dynamics.
• The transition to the continuum limit alters the nature of mode locking, highlighting the importance of lattice discreteness.