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This study introduces generalized m-splay states in coupled phase oscillator networks. We provide simple, observable-based linear stability conditions applicable to large networks, including those with inertia.

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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.