When Nonlocal Coupling between Oscillators Becomes Stronger: Patched Synchrony or Multichimera States

Abstract

Systems of nonlocally coupled oscillators can exhibit complex spatio-temporal patterns, called chimera states, which consist of coexisting domains of spatially coherent (synchronized) and incoherent dynamics. We report on a novel form of these states, found in a widely used model of a limit-cycle oscillator if one goes beyond the limit of weak coupling typical for phase oscillators. Then patches of synchronized dynamics appear within the incoherent domain giving rise to a multichimera state. We find that, depending on the coupling strength and range, different multi-chimeras arise in a transition from classical chimera states. The additional spatial modulation is due to strong coupling interaction and thus cannot be observed in simple phase-oscillator models. During recent times the investigation of coupled systems has led to joint research efforts bridging between diverse fields such as nonlinear dynamics, network science, and statistical physics, with a plethora of applications, e.g., in physics, biology, and technology. As the numerical resources have developed at fast pace, analysis and simulations of large networks with more and more sophisticated coupling schemes have come into reach giving rise to an abundance of new dynamical scenarios. Among these a very peculiar type of dynamics was first reported for the well-known model of phase oscillators. Such a network exhibits a hybrid nature combining both coherent and incoherent parts [1][2][3][4], hence the name chimera states. The most surprising aspect of this discovery was that these states exist in a system of identical oscillators coupled in a symmetric ring topology with a symmetric interaction function. Recent works have shown that chimeras are not limited to phase oscillators, but can in fact be found in a large variety of different systems. These include time-discrete and time-continuous chaotic models [5,6] and are not restricted to one spatial dimension. Also two-dimensional configurations allow for chimera states [7,8]. Furthermore, similar scenarios exist for time-delayed coupling [9], and their dynamical properties and symmetries were subject to theoretical studies as well [6,10,11]. It was only in the very recent past that chimeras were realized in experiments on chemical oscillators [12] and electro-optical coupled-map lattices [13]. The nonlocality of the coupling -a crucial ingredient for chimera states -also suggests an interesting connection to material science, see, for instance, magnetic Janus particles that undergo a synchronization-induced structural transition in a rotating magnetic field [14,15]. Nonlo- * corresponding author: schoell@physik.tu-berlin.de cality is of great importance not only for chimera states, but also for other dynamical phenomena such as turbulent intermittency [16]. Hybrid states were also reported in the context of neuroscience under the notion of bump states [17]. They were later confirmed for nonlocally coupled Hodgkin-Huxley models [18] and may account for experimental observation of partial synchrony in neu...

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

“…As we move away from the bifurcation point, the oscillators are organized in one coherent and one incoherent region, thus, forming a 1-chimera state The mean phase velocity profile develops a small dip, which increases and eventually reaches the level of the coherent part as the parameter p 1 increases. Similar scenarios have been observed for nonlocally coupled FitzHughNagumo systems [15].…”

“…Impact of coupling range R Chimera states with multiple domains of incoherence and coherence have been reported in several works and are referred to as clustered chimera or multichimera states. It is known that they may be achieved through time delay [62] or by manipulating the range of the coupling between oscillators [15][16][17]63]. The range R of the coupling reflects the migration range of the different species in the system.…”

“…This introduces nonlocality in the form of a ring topology as proposed in Ref. [15,61]. There, the authors considered a rotational coupling matrix in order to achieve both direct and crosscoupling between the system variables.…”

“…Initial conditions in all simulations herein are taken to be randomly distributed on an ellipsoid enclosed by the triangle with corners on the fixed points Q 1 , Q 2 , and Q 3 . As a measure indicating the existence of a chimera state we employ the mean phase velocity of each oscillator ω k = 2πM k /∆T , where M k is the number of periods of the kth oscillator during a time interval ∆T [5,15]. This quantity is depicted in Fig.…”

“…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].…”