Phase transitions in an adaptive network with the global order parameter adaptation.

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Network Science

Background:

• Kuramoto oscillators are fundamental models for synchronization in coupled systems.
• Adaptive networks introduce dynamic changes in network structure, affecting collective behavior.
• Understanding bifurcation phenomena is crucial for characterizing transitions in dynamical systems.

Purpose of the Study:

• To investigate synchronization transitions in adaptive networks of Kuramoto oscillators with dyadic coupling.
• To analyze the role of global order parameter and adaptive coupling strength on network synchronization.
• To explore the influence of the phase-lag parameter on synchronized and bistable states.

Main Methods:

• Analysis of continuous and abrupt synchronization transitions via pitchfork and saddle-node bifurcations.
• Derivation of low-dimensional evolution equations for global order parameters using the Ott-Antonsen ansatz.
• Numerical simulations to validate theoretical bifurcation conditions.

Main Results:

• Identified continuous synchronization via pitchfork bifurcation and abrupt transitions via saddle-node bifurcations, creating bistable regions (R1 and R2).
• Demonstrated transitions from incoherent to bistable states and between bistable states as a function of coupling strength.
• Showed that the phase-lag parameter significantly expands the parameter space for weakly synchronized and bistable states.

Conclusions:

• The adaptive network exhibits complex synchronization behaviors including continuous and abrupt transitions.
• Bifurcation analysis and the Ott-Antonsen ansatz provide accurate predictions for network dynamics.
• The phase-lag parameter offers a mechanism to control and broaden synchronized states in adaptive oscillator networks.