Chimera states on complex networks

Zhu, Yun; Zheng, Zhigang; Yang, Junzhong

doi:10.1103/physreve.89.022914

Cited by 137 publications

(94 citation statements)

References 20 publications

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“…To represent the results, we use snapshots of three attributes (Zhu et al, 2014): (i) the phase profile, (ii) the effective angular velocities of oscillators and (iii) the fluctuation of the instantaneous angular velocity of oscillators. The effective angular velocity of oscillator i is defined as…”

Section: Resultsmentioning

confidence: 99%

“…In this study, we kept the global coupling strength constant and allowed the non-local coupling strength, κ, to vary from one simulation to the next one, similar to what was done in the recent work of Zhu et al (2014).…”

Section: Chimera Statesmentioning

confidence: 99%

“…. , N. Although we have not investigated the influence of the global coupling strength on the steady-state solution of the Kuramoto model, we treat this term as constant, in particular K = 1, based on observations made by Zhu et al (2014). We would like to stress that the model in Eq.…”

Section: Mathematical Model Of the Earthquake Sequencingmentioning

confidence: 99%

“…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.…”

Section: Mathematical Model Of the Earthquake Sequencingmentioning

confidence: 99%

“…In our studies on earthquake sequencing, we consider a directed graph as a representation of an earthquake complex network. The occurrence of chimera states as solutions to non-linear dynamics on both undirected and directed graphs has recently been investigated (Zhu et al, 2014;Vasudevan and Cavers, 2014a). As a precursor to studying earthquake sequencing with real data from the earthquake catalogues, we investigated the Kuramoto model on synthetic networks that mimic Erdös-Rényi random networks, small-world networks, and scale-free networks and directed graphs adapted from them, and examined chimera-state solutions (Vasudevan and Cavers, 2014a).…”

Section: Mathematical Model Of the Earthquake Sequencingmentioning

confidence: 99%

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

Vasudevan

Cavers

Ware

2015

Nonlin. Processes Geophys.

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Abstract. Earthquake sequencing studies allow us to investigate empirical relationships among spatio-temporal parameters describing the complexity of earthquake properties. We have recently studied the relevance of Markov chain models to draw information from global earthquake catalogues. In these studies, we considered directed graphs as graph theoretic representations of the Markov chain model and analyzed their properties. Here, we look at earthquake sequencing itself as a directed graph. In general, earthquakes are occurrences resulting from significant stress interactions among faults. As a result, stress-field fluctuations evolve continuously. We propose that they are akin to the dynamics of the collective behavior of weakly coupled non-linear oscillators. Since mapping of global stress-field fluctuations in real time at all scales is an impossible task, we consider an earthquake zone as a proxy for a collection of weakly coupled oscillators, the dynamics of which would be appropriate for the ubiquitous Kuramoto model. In the present work, we apply the Kuramoto model with phase lag to the nonlinear dynamics on a directed graph of a sequence of earthquakes. For directed graphs with certain properties, the Kuramoto model yields synchronization, and inclusion of nonlocal effects evokes the occurrence of chimera states or the co-existence of synchronous and asynchronous behavior of oscillators. In this paper, we show how we build the directed graphs derived from global seismicity data. Then, we present conditions under which chimera states could occur and, subsequently, point out the role of the Kuramoto model in understanding the evolution of synchronous and asynchronous regions. We surmise that one implication of the emergence of chimera states will lead to investigation of the present and other mathematical models in detail to generate global chimera-state maps similar to global seismicity maps for earthquake forecasting studies.

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Section: Resultsmentioning

confidence: 99%

Section: Chimera Statesmentioning

confidence: 99%

Section: Mathematical Model Of the Earthquake Sequencingmentioning

confidence: 99%