Configurational stability for the Kuramoto-Sakaguchi model.

Area of Science:

• Complex systems
• Nonlinear dynamics
• Network science

Background:

• The Kuramoto model describes synchronized behavior in coupled oscillators.
• The Kuramoto-Sakaguchi model introduces a phase lag, breaking the gradient structure and complicating analysis.
• Understanding stability in such networks is crucial for various scientific domains.

Purpose of the Study:

• To analyze the stability of phase-locked configurations in the Kuramoto-Sakaguchi model.
• To develop criteria for determining stability and instability of synchronized states.
• To investigate the impact of the phase-lag parameter on network dynamics.

Main Methods:

• Derivation of analytical conditions for stability and instability.
• Utilizing a topological invariant (modulo 2 count) for the unstable manifold dimension.
• Numerical simulations for small and large oscillator networks.

Main Results:

• A sufficient condition for the stability of phase-locked configurations was established.
• A sufficient condition for instability was derived, linked to the parity of the unstable manifold dimension.
• Numerical results validated the theoretical findings for diverse network sizes.

Conclusions:

• The Kuramoto-Sakaguchi model's dynamics can be analyzed using novel stability criteria.
• The phase-lag parameter significantly impacts network stability, offering new insights into synchronization phenomena.
• The findings contribute to the theoretical understanding of complex oscillatory systems.