Chimeras in random non-complete networks of phase oscillators

Abstract

We consider the simplest network of coupled non-identical phase oscillators capable of displaying a "chimera" state (namely, two subnetworks with strong coupling within the subnetworks and weaker coupling between them) and systematically investigate the effects of gradually removing connections within the network, in a random but systematically specified way. We average over ensembles of networks with the same random connectivity but different intrinsic oscillator frequencies and derive ordinary differential equations (ODEs), whose fixed points describe a typical chimera state in a representative network of phase oscillators. Following these fixed points as parameters are varied we find that chimera states are quite sensitive to such random removals of connections, and that oscillations of chimera states can be either created or suppressed in apparent bifurcation points, depending on exactly how the connections are gradually removed.

Chimera states in population dynamics: Networks with fragmented and hierarchical connectivities

“…Typical models that have been numerically investigated include the Kuramoto phase oscillator [7][8][9], periodic and chaotic maps [10,11], the Stuart-Landau model [13,14], the Van der Pol oscillator [12] as well as models addressing neuron dynamics such the FitzHughNagumo oscillator [15], the Hindmarsh-Rose model [16], the so-called SNIPER model of excitability type-I [17], or the Hodgkin-Huxley model [18]. Moreover, chimera states have been reported in populations of coupled pendula [19], in autonomous Boolean networks [20], in one-Following the theoretical predictions, chimera states were experimentally verified for the first time in populations of coupled chemical oscillators [23] and in optical coupled-map lattices realized by liquid-crystal light modulators [24].…”

Chimera patterns: influence of time delay and noise**This work was supported by DFG in the framework of SFB 910.

“…In particular, they exist in ensembles of coupled heterogeneous elements Laing (2010) and networks with irregular coupling topologies Ko and Ermentrout (2008); Shanahan (2010); Laing et al (2012); Yao et al (2013); Zhu et al (2014). In a recent study both of these inhomogeneity types are considered Omelchenko et al (2015).…”

“…In the latter case chimeras represent a chaotic saddle state. Moreover, the lifetime of a chimera statethe time till it transforms into the completely coherent state -grows exponentially with the system size Wolfrum Chimera states have been shown to be robust to various kinds of perturbations Laing et al (2012). In particular, they exist in ensembles of coupled heterogeneous elements Laing (2010) and networks with irregular coupling topologies Ko and Ermentrout (2008) Zhu et al (2014).…”

“…Chimera states have been shown to be robust to various kinds of perturbations Laing et al (2012). In particular, they exist in ensembles of coupled heterogeneous elements Laing (2010) and networks with irregular coupling topologies Ko and Ermentrout (2008); Shanahan (2010); Laing et al (2012); Yao et al (2013); Zhu et al (2014).…”

Earthquake sequencing: chimera states with Kuramoto model dynamics on directed graphs

“…One proven modification is the inclusion of non-local effects of the geometry of the system that has been shown to lead to a co-existence of partially synchronized and partially asynchronized states of oscillators as a steady-state solution. Such states, addressed as chimera states, are the subject of recent theoretical and experimental studies (Kuramoto and Battogtokh, 2002;Abrams and Strogatz, 2004;Abrams et al, 2008;Ko and Ermentrout, 2008;Omel'chenko et al, 2008;Sethia et al, 2008;Sheeba et al, 2009;Laing, 2009a, b;Laing et al, 2012;Martens et al, 2013;Yao et al, 2013;Rothkegel and Lehnertz, 2014;Kapitaniak et al, 2014;Pazó and Montbrió, 2014;Panaggio and Abrams, 2014;Zhu et al, 2014;Gupta et al, 2014;Vasudevan and Cavers, 2014a, b). We focus our present study on defining a Kuramoto model with a phase lag that would accommodate the existence of chimera states.…”