The size of the sync basin revisited

Abstract

In dynamical systems, the full stability of fixed point solutions is determined by their basins of attraction. Characterizing the structure of these basins is, in general, a complicated task, especially in high dimensionality. Recent works have advocated to quantify the non-linear stability of fixed points of dynamical systems through the relative volumes of the associated basins of attraction [Wiley et al., Chaos 16, 015103 (2006) and Menck et al. Nat. Phys. 9, 89 (2013)]. Here, we revisit this issue and propose an efficient numerical method to estimate these volumes. The algorithm first identifies stable fixed points. Second, a set of initial conditions is considered that are randomly distributed at the surface of hypercubes centered on each fixed point. These initial conditions are dynamically evolved. The linear size of each basin of attraction is finally determined by the proportion of initial conditions which converge back to the fixed point. Armed with this algorithm, we revisit the problem considered by Wiley et al. in a seminal paper [Chaos 16, 015103 (2006)] that inspired the title of the present manuscript and consider the equal-frequency Kuramoto model on a cycle. Fixed points of this model are characterized by an integer winding number q and the number n of oscillators. We find that the basin volumes scale as (1-4q/n), contrasting with the Gaussian behavior postulated in the study by Wiley et al.. Finally, we show the applicability of our method to complex models of coupled oscillators with different natural frequencies and on meshed networks.

Order parameter allows classification of planar graphs based on balanced fixed points in the Kuramoto model

“…In order to calculate this balancing ratio of a circulant graph for an arbitrary number of vertices in the graph, we need to derive an expression for the basin stability of the graph's fixed points. The basin stability S B (q c ) in dependence of the winding number q c for the fixed points in the Kuramoto model on a cycle graph is known to follow a Gaussian distribution S B (q) = and different explanations for this scaling have been suggested [22,34]. As we found here, the variance σ 2 C N (N ) of this Gaussian distribution scales to good approximation linearly with the number of vertices N as shown in Fig.…”

Robustness of Synchrony in Complex Networks and Generalized Kirchhoff Indices

“…Optimization of synchronization has been investigated from various angles. The synchronous state can be optimal from the point of view of linear stability [5], the range of parameters that allow synchronization [12][13][14], the value that an order parameter takes at synchrony [15] or the volume of the basin of attraction around a stable synchronous fixed point [16][17][18]. Here we extend these investigations by asking what makes synchronous states more or less fragile against external perturbations.…”

Concurrent formation of nearly synchronous clusters in each intertwined cluster set with parameter mismatches

“…Twisted states of identical oscillators, originally discovered in ring structures [31,32], have been recently reported in square and cubic lattices [25]. In the twisted states of lattices, oscillators in each line are synchronized in square lattices while oscillators in each plane are synchronized in cubic lattices.…”