The size of the sync basin revisited

Area of Science:

• Dynamical Systems and Chaos Theory
• Nonlinear Dynamics
• Computational Physics

Background:

• Fixed point stability in dynamical systems is crucial for understanding system behavior.
• Basins of attraction define the set of initial conditions leading to a specific fixed point.
• Quantifying basin volumes offers insights into nonlinear stability, but is computationally challenging, especially in high dimensions.

Purpose of the Study:

• To propose an efficient numerical method for estimating basin volumes of fixed points in dynamical systems.
• To analyze the scaling behavior of basin volumes in the equal-frequency Kuramoto model.
• To demonstrate the method's applicability to complex coupled oscillator systems.

Main Methods:

• Identify stable fixed points within the dynamical system.
• Generate initial conditions randomly distributed on hypercubes centered around fixed points.
• Dynamically evolve initial conditions and determine basin size by the convergence proportion.

Main Results:

• Developed an efficient algorithm to estimate basin volumes.
• Found that for the equal-frequency Kuramoto model, basin volumes scale as (1-4q/n)ⁿ, where q is the winding number and n is the number of oscillators.
• This scaling contradicts the previously postulated Gaussian behavior.

Conclusions:

• The proposed numerical method provides an efficient way to quantify nonlinear stability through basin volume estimation.
• The discovered scaling law for the Kuramoto model offers new insights into the structure of its basins of attraction.
• The method is applicable to diverse and complex systems of coupled oscillators, highlighting its broad utility.