Loss of Coherence in Dynamical Networks: Spatial Chaos and Chimera States

Abstract

We discuss the breakdown of spatial coherence in networks of coupled oscillators with nonlocal interaction. By systematically analyzing the dependence of the spatio-temporal dynamics on the range and strength of coupling, we uncover a dynamical bifurcation scenario for the coherence-incoherence transition which starts with the appearance of narrow layers of incoherence occupying eventually the whole space. Our findings for coupled chaotic and periodic maps as well as for time-continuous Rössler systems reveal that intermediate, partially coherent states represent characteristic spatiotemporal patterns at the transition from coherence to incoherence. Understanding the dynamics on networks is at the heart of modern nonlinear science and has a wide applicability to various fields [1,2]. Thus, network science is a vibrant, interdisciplinary research area with strong connections to physics. For example, concepts of theoretical physics like the Turing instability, which is a known paradigm of non-equilibrium self-organization in spacecontinuous systems, have recently been transferred to complex networks [3]. While spatially extended systems show pattern formation mediated by diffusion, i.e., local interactions, a network takes also into account long-range and global interactions yielding more realistic spatial geometries.Network topologies like all-to-all coupling of, for instance, phase oscillators (Kuramoto model) or chaotic maps (Kaneko model) were intensively studied , and numerous characteristic regimes were found [4][5][6]. In particular, for globally coupled chaotic maps they rangefor decreasing coupling strength -from complete chaotic synchronization via clustering and chaotic itineracy to complete desynchronization. The opposite case, i.e., nearest-neighbor coupling, is known as lattice dynamical systems of time-continuous oscillators, or coupled map lattices if the oscillator dynamics is discrete in time. These kinds of networks arise naturally as discrete approximation of systems with diffusion and have also been thoroughly studied. They can demonstrate rich dynamics such as solitons, kinks, etc. up to fully developed spatiotemporal chaos [6][7][8][9][10].The case of networks with nonlocal coupling, however, has been much less studied in spite of numerous applications in different fields. Characteristic examples pertain to neuroscience [11,12], chemical oscillators [13,14], electrochemical systems [15], and Josephson junctions [16]. A new impulse to study such networks was given, in particular, by the discovery of so-called chimera states [17,18]. The main peculiarity of these spatio-temporal patterns is that they have a hybrid spatial structure, partially coherent and partially incoherent, which can develop in networks of identical oscillators without any sign of inhomogeneity.In this Letter we discuss the transition between coherent and incoherent dynamics in networks of nonlocally coupled oscillators. We start with coupled chaotic mapswhere z i are real dynamic variables (i = 1, ..., N , N 1 and the inde...

Chimera patterns in two-dimensional networks of coupled neurons

“…First found in identical and symmetrically coupled phase oscillators [2], chimera states have been the focus of extensive research for over a decade now. Both theoretical and experimental works have shown that this counter-intuitive collective phenomenon may arise in numerous systems including mechanical, chemical, electro-chemical, electrooptical, electronic, and superconducting coupled oscillators [3][4][5][6][7][8][9][10][11][12][13][14].…”

Amplitude and phase dynamics in oscillators with distributed-delay coupling

“…Recent work on laser communication networks has extensively used coupled oscillator models to study the dynamics of in-phase or anti-phase and complete chaos synchronization [6][7][8][9][10]. Chimera states, where a network of oscillators splits into coexisting domains of coherent, phaselocked and incoherent, desynchronized behaviour, have also been observed [11][12][13][14][15]. These studies have shown that coupling between different elements can play a dual role in the dynamics: it can both lead to suppression of oscillations and also facilitate a certain degree of synchronization between different elements.…”

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].…”