From incoherence to synchronicity in the network Kuramoto model

Abstract

We study the synchronisation properties of the Kuramoto model of coupled phase oscillators on a general network. Here we distinguish the ability of such a system to self-synchronise from the stability of this behaviour. While self-synchronisation is a consequence of genuine non-perturbative dynamics, the stability in dynamical systems is usually accessible by fluctuations about a fixed point, here taken to be the synchronised solution. We examine this problem in terms of modes of the graph Laplacian, by which the absolute Lyapunov stability of the synchronised fixed point is readily demonstrated. Departures from stability are seen to arise at the next order in fluctuations where the dynamical equations resemble those for species population models, the logistic and Lotka-Volterra equations. Methods from these systems are exploited to analytically derive new critical couplings signalling deviation from classical stability. We observe in some cases an intermediate regime of behaviour, between incoherence and synchronisation, where system wide periodic behaviours are exhibited. We discuss these results in light of simulations.

Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model

“…Though it is not the critical coupling, it is a threshold that demarcates different dynamical regimes of behaviour visible in the numerical solutions. This is in contrast to the ordinary Kuramoto model where linearisation gives only a weak indication of the point for change in dynamical behaviour [33].…”

“…(8), exists as a sharp condition in the ordinary Kuramoto model: the best one can do is demand some consistency for the linearisation approximation to hold [33,34]. In this regime, the order parameter r defined via…”

Synchronization in complex networks of phase oscillators: A survey

“…works indicating certain (e.g., degree-dependent) scaling laws (Nishikawa et al, 2003;Restrepo et al, 2005;Gómez-Gardeñes et al, 2007;Moreno and Pacheco, 2004;Kalloniatis, 2010;Skardal et al, 2013).…”

“…Regarding the potential and equilibrium landscape, a few interesting and still unresolved conjectures can be found in Tavora and Smith (1972a); Araposthatis et al (1981); Baillieul and Byrnes (1982); Mehta and Kastner (2011);Korsak (1972) and pertain to the number of (stable) equilibria and topological properties of the equilibrium set. Finally, the complex networks, nonlinear dynamics, and statistical physics communities found various interesting scaling laws in their statistical and numerical analyses of random graph models, such as conditions depending on the spectral ratio λ 2 /λ n of the Laplacian eigenvalues, interesting results for correlations between the degree deg i and the natural frequency ω i , and degree-dependent synchronization conditions (Nishikawa et al, 2003;Moreno and Pacheco, 2004;Restrepo et al, 2005;Boccaletti et al, 2006;Gómez-Gardeñes et al, 2007;Arenas et al, 2008;Kalloniatis, 2010;Skardal et al, 2013). It is unclear which of these results and findings are amenable to an analytic and quantitative investigation.…”

Approximating frustration scores in complex networks via perturbed Laplacian spectra

“…For example, one may ask which link or node removals maximally or minimally impact the algebraic connectivity λ 2 or the eigenratio λ 2 / λ N [57], both being summary measures of dynamic synchronization [5, 58, 59]. One may also examine an individual row of the eigenvector matrix, i.e.…”