Rhythmic states and first-order phase transitions in adaptive coupled three-dimensional limit-cycle oscillators

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Statistical Physics

Background:

• Coupled limit-cycle oscillators are fundamental in modeling diverse phenomena.
• Understanding phase transitions in high-dimensional systems is crucial for complex network analysis.

Purpose of the Study:

• To investigate phase transitions in three-dimensional coupled limit-cycle oscillators with adaptive coupling.
• To explore the influence of natural frequency distribution on emergent rhythmic states and phase transitions.

Main Methods:

• Analysis of coupled three-dimensional limit-cycle oscillators with adaptive coupling.
• Investigation of systems with uniform and Gaussian natural frequency distributions.
• Theoretical analysis of incoherent states and fixed points, verified by numerical simulations.

Main Results:

• Multiple-cluster rhythmic states emerge for uniform frequency distribution (Case I) but not for Gaussian distribution (Case II).
• Two sequential first-order phase transitions occur with increasing coupling strength (K).
• A nonhysteretic transition occurs at K→0⁺, while a hysteretic transition emerges for K>0, dependent on frequency distribution width.

Conclusions:

• The natural frequency distribution critically affects the emergence of complex rhythmic states and phase transition behaviors.
• Adaptive coupling in high-dimensional oscillator systems exhibits rich phase transition dynamics.
• The findings offer insights into the collective dynamics of complex oscillatory networks.