Phase order in chaotic maps and in coupled map lattices

Area of Science:

• Nonlinear dynamics
• Chaos theory
• Statistical physics

Background:

• The logistic map is a fundamental model for studying chaotic behavior.
• Understanding phase transitions in dynamical systems is crucial for various scientific fields.

Purpose of the Study:

• To define and investigate the concept of 'direction phase' in the logistic map.
• To analyze the transition of this direction phase as a parameter changes.
• To explore phase synchronization in coupled map lattices.

Main Methods:

• Defining direction phase based on sequential iterations of the logistic map.
• Analyzing the parameter-dependent transition of the net direction phase (M).
• Investigating the order state and synchronization of direction phases in coupled map lattices.

Main Results:

• A transition from zero to a finite value for the net direction phase (M) was observed as the parameter (μ) increased.
• A scaling law M ≈ (μ - μ(0))^α with α = 0.5 was found near the transition point.
• Phase synchronization of direction phases was identified in coupled map lattices, even amidst chaotic behavior.

Conclusions:

• The study reveals a novel transition phenomenon in the logistic map related to direction phases.
• The observed scaling behavior provides quantitative insights into the transition dynamics.
• Phase synchronization can occur in chaotic coupled map lattices, highlighting complex emergent order.