Generalized splay states in phase oscillator networks.

Area of Science:

• Complex systems
• Nonlinear dynamics
• Network science

Background:

• Coupled phase oscillators are crucial for understanding emergent collective phenomena.
• Phase-locked states are common, but their stability analysis can be complex.
• Generalized m-splay states represent a specific subclass of phase-locked states.

Purpose of the Study:

• To introduce and analyze generalized m-splay states in networks of coupled phase oscillators.
• To derive simple and broadly applicable linear stability conditions for these states.
• To extend the analysis to oscillators with inertia and adaptive coupling.

Main Methods:

• Definition of generalized m-splay states characterized by a vanishing mth order parameter.
• Derivation of explicit linear stability conditions based on network observables.
• Application and exemplification using the Kuramoto-Sakaguchi model.
• Generalization to include inertial and adaptively coupled phase oscillators.

Main Results:

• Generalized m-splay states exhibit typically incoherent dynamics and form high-dimensional solution families (splay manifolds).
• Explicit linear stability conditions are derived, expressed using simple observables like the order parameter and Jacobian trace.
• The derived conditions are independent of network size and applicable to arbitrary network sizes.
• The findings are successfully extended to phase oscillators with inertia and adaptive coupling.

Conclusions:

• The introduced stability conditions offer a simplified and powerful tool for analyzing complex oscillator networks.
• The results are broadly applicable across various models of coupled phase oscillators.
• This work provides fundamental insights into the dynamics and stability of collective phenomena in complex systems.