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Size of the sync basin resolved.

Pablo Groisman1,2,3, Cecilia De Vita1,2, Julián Fernández Bonder1,4

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Researchers studied Kuramoto oscillators and found their basin sizes follow a Gaussian scaling law. This discovery explains the dynamics of complex systems by analyzing winding numbers and timescale separation.

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Area of Science:

  • Complex Systems
  • Nonlinear Dynamics
  • Statistical Physics

Background:

  • Kuramoto oscillators exhibit multistable dynamics, crucial for understanding complex systems.
  • Attractors in these systems are known as twisted states with specific phase winding properties.

Purpose of the Study:

  • To investigate the scaling of basin sizes for twisted states in Kuramoto oscillators.
  • To provide numerical and analytical evidence for a 2006 conjecture on basin size scaling.

Main Methods:

  • Numerical simulations of Kuramoto oscillators on cycle graphs.
  • Analytical derivations utilizing timescale separation and Central Limit Theorem.

Main Results:

  • Confirmed the conjectured Gaussian scaling of basin sizes (e^{-kq^{2}}) with winding number (q).
  • Identified rapid winding number stabilization (t∝logn) as the key dynamical mechanism.
  • Demonstrated that winding number can be treated as a sum of weakly dependent random variables.

Conclusions:

  • The study provides a robust theoretical framework for understanding attractor basin scaling in multistable dynamical systems.
  • The findings have implications for predicting the behavior of large networks of coupled oscillators.