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We discovered a phase transition in coupled oscillators driven by random frequencies. A heteroclinic bifurcation explains the shift between contractible and noncontractible limit cycles in collective dynamics.

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Area of Science:

  • Nonlinear dynamics
  • Statistical physics
  • Dynamical systems theory

Background:

  • Coupled oscillator systems exhibit complex collective behavior.
  • Phase transitions in dynamical systems are crucial for understanding emergent properties.
  • Saddle-node on invariant circle bifurcations are common in oscillatory systems.

Purpose of the Study:

  • To investigate the collective dynamics of coupled phase oscillators near a saddle-node on invariant circle bifurcation.
  • To identify the mechanism driving the phase transition in the system's collective behavior.
  • To characterize the topological differences in limit cycles before and after the bifurcation.

Main Methods:

  • Ott-Antonsen reduction for analyzing collective dynamics.
  • Geometric techniques for ordinary differential equations.
  • Analysis of vector fields on a cylinder to identify bifurcations.

Main Results:

  • A heteroclinic bifurcation was identified as the mechanism for the phase transition.
  • This bifurcation separates two distinct families of limit cycles.
  • Limit cycles before the bifurcation are contractible, while those after are noncontractible.
  • Both families of limit cycles were found to be stable.

Conclusions:

  • The study elucidates the role of heteroclinic bifurcation in coupled oscillator dynamics.
  • A clear topological distinction in limit cycles characterizes the phase transition.
  • The findings provide insights into the stability and behavior of complex dynamical systems.