Exotic equilibria of Harary graphs and a new minimum degree lower bound for synchronization

Abstract

This work is concerned with stability of equilibria in the homogeneous (equal frequencies) Kuramoto model of weakly coupled oscillators. In 2012 [R. Taylor, J. Phys. A: Math. Theor. 45, 1-15 (2012)], a sufficient condition for almost global synchronization was found in terms of the minimum degree-order ratio of the graph. In this work, a new lower bound for this ratio is given. The improvement is achieved by a concrete infinite sequence of regular graphs. Besides, non standard unstable equilibria of the graphs studied in Wiley et al. [Chaos 16, 015103 (2006)] are shown to exist as conjectured in that work.

The lower bound of the network connectivity guaranteeing in-phase synchronization

“…One of the most important questions is how the tendency of synchronization depends on connectivity, or connection density, of the network [14][15][16][17][18][19][20] . The connectivity µ of a network having N nodes has been defined as the minimum degree of the nodes divided by N − 1, the total number of other nodes.…”

“…Besides the upper bound, a series of studies has also revealed the lower bound of µ c 15,17,19 . In particular, Townsend et al have provided a circulant network whose connectivity is less than 0.6828 • • • and has a stable solution other than the in-phase synchronization 19 , which means that the best known lower bound of µ c is 0.6828…”

Sparsity-driven synchronization in oscillators networks

“…it is the sole network attractor 10,11,21 . Conversely, in sparse networks, the ISS is often sharing the system state-space with other synchronization patterns such as q-twisted states 10,22,23 , traveling waves [23][24][25] , solitary and chimera states [26][27][28][29] . In such a scenario, the ISS possess a domain of attraction, i.e., a finite portion of the network state-space from where all trajectories converge to the ISS.…”

Sufficiently dense Kuramoto networks are globally synchronizing

“…[1][2][3][4][5][6][7] Among the many questions raised by synchronization, one of the most natural asks how network topology can either promote or prevent global synchronization. [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] We say that a network of oscillators globally synchronizes if it converges to a state for which all the oscillators are in phase, starting from all initial conditions except a set of measure zero. Otherwise, we say that the network supports a pattern.…”

“…For example, a network in which any two oscillators are connected by exactly one path (i.e., a tree) can be quite sparse, yet all trees of identical Kuramoto oscillators are known to be globally synchronizing; conversely, there are dense Kuramoto networks that nevertheless support a pattern. 9,15,23,24 In this paper, we study the homogeneous Kuramoto model introduced by Taylor 11 , in which each oscillator has the same frequency ω. By going into a rotating frame at this frequency, we can set ω = 0 without loss of generality.…”