Some aspects of the synchronization in coupled maps

Area of Science:

• Complex systems
• Nonlinear dynamics
• Chaos theory

Background:

• Coupled map lattices (CMLs) are widely used to model complex spatio-temporal phenomena.
• Synchronization is a fundamental emergent behavior in coupled nonlinear systems.
• Power law interactions introduce long-range correlations, potentially altering synchronization dynamics.

Purpose of the Study:

• To numerically investigate the synchronization behavior of a CML composed of chaotic logistic maps with power law interactions.
• To determine a practical lower bound for lattice size to approximate the thermodynamic limit in numerical simulations.
• To explore the relationship between system complexity (Lyapunov dimension) and synchronization speed.

Main Methods:

• Numerical simulations of a one-dimensional coupled map lattice.
• Utilizing chaotic logistic maps as the base dynamical elements.
• Implementing power law interaction rules for coupling between adjacent maps.
• Calculating Lyapunov dimension and averaged synchronization time.

Main Results:

• A practical lower bound for lattice size was identified for reliable thermodynamic limit approximations.
• A strong positive correlation was observed between the Lyapunov dimension and the averaged synchronization time.
• The system exhibits complex synchronization patterns influenced by power law interactions.

Conclusions:

• The choice of lattice size is critical for accurate numerical studies of CMLs in the thermodynamic limit.
• Higher system complexity, as indicated by Lyapunov dimension, leads to longer synchronization times in this model.
• Power law interactions significantly impact the synchronization dynamics of chaotic map lattices.