The size of the sync basin

Area of Science:

• Nonlinear dynamics
• Complex systems
• Network science

Background:

• The synchronous state in coupled oscillator networks can be stable but not globally attractive.
• Understanding the conditions and likelihood of synchronization from random initial states is crucial for many real-world applications.
• The size of the basin of attraction for the synchronous state (sync basin) is a key, yet challenging, quantity to determine.

Purpose of the Study:

• To investigate the size of the sync basin in a network of identical phase oscillators with varying coupling ranges.
• To explore the emergence and characteristics of coexisting attractors when synchronization is not globally stable.
• To analyze the statistical distribution of these coexisting states and their dependence on network parameters.

Main Methods:

• Analytical calculations for the sync basin size in a ring network with sinusoidal coupling.
• Numerical simulations of coupled phase oscillators with varying numbers of neighbors (k) and total oscillators (n).
• Characterization of coexisting attractors by their winding number (q) and analysis of their basin sizes.

Main Results:

• For extensive coupling (k/n > ~0.34), the sync basin covers the entire phase space.
• As coupling becomes short-range (k/n < ~0.34), uniformly twisted wave attractors emerge.
• The distribution of winding numbers (q) for these twisted states follows a Gaussian law, with standard deviation scaling with sqrt(n/k).

Conclusions:

• Network connectivity significantly impacts the likelihood of global synchronization.
• Short-range coupling in oscillator networks leads to a rich variety of stable, non-synchronous states.
• The statistical distribution of these complex states presents an open problem for theoretical explanation.