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...

Networks of nonlocally coupled phase oscillators 1 can support chimera states in which identical oscillators evolve into distinct groups that exhibit coexisting synchronous and incoherent behaviours despite homogeneous coupling 2-6 . Similar nonlocal coupling topologies implemented in networks of chaotic iterated maps also yield dynamical states exhibiting coexisting spatial domains of coherence and incoherence 7,8 . In these discrete-time systems, the phase is not a continuous variable, so these states are generalized chimeras with respect to a broader notion of incoherence. Chimeras continue to be the subject of intense theoretical investigation, but have yet to be realized experimentally 6,9-16 . Here we show that these chimeras can be realized in experiments using a liquid-crystal spatial light modulator to achieve optical nonlinearity in a spatially extended iterated map system. We study the coherenceincoherence transition that gives rise to these chimera states through experiment, theory and simulation.Our system is an experimental realization of a coupled-map lattice (CML), a class of systems that has received sustained theoretical interest for the past three decades. Although the dynamics and statistical physics of CML systems have been theoretically explored, very few (if any) experimental realizations exist [17][18][19][20][21][22] . In our experiments, we create CML dynamics by using a liquid-crystal spatial light modulator (SLM) to control the polarization properties of an optical wavefront. We may electronically introduce any desired coupling topology including nearest neighbour, nonlocal, small world and scale free. In this work, we impose periodic boundary conditions for both onedimensional (1D) and 2D nonlocally coupled maps. Thus, we have developed a powerful experimental technique to observe the parallel evolution of the dynamics of arrays of coupled maps numbering up to thousands or more depending on the goals of the experiment. Figure 1 shows the experimental set-up of the optical CML. Polarization optics create a nonlinear relationship between the spatially dependent phase shift applied by the SLM and the intensity of the light falling on the camera: I (φ) = (1 − cos(φ))/2. The operation of the experimental apparatus is described in the Methods. Both the SLM and the camera frames are partitioned into an M ×M array of square regions. These regions correspond to nodes in the network of coupled maps. Time evolution of the network is achieved by iteratively updating the phase applied by each region of the SLM in a way that depends on the intensity measured by the camera.We present results for two different coupling schemes shown schematically in Fig. 1b,c. In the 1D configuration, the elements in the array are arranged as a ring with periodic boundary conditions. The site highlighted in white is updated based on the sites indicated in blue. As the elements are coupled diffusively to their neighbours within a range R in either one or two dimensions with periodic boundary conditions, the coupling...

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...

We refute an often invoked theorem which claims that a periodic orbit with an odd number of real Floquet multipliers greater than unity can never be stabilized by time-delayed feedback control in the form proposed by Pyragas. Using a generic normal form, we demonstrate that the unstable periodic orbit generated by a subcritical Hopf bifurcation, which has a single real unstable Floquet multiplier, can in fact be stabilized. We derive explicit analytical conditions for the control matrix in terms of the amplitude and the phase of the feedback control gain, and present a numerical example. Our results are of relevance for a wide range of systems in physics, chemistry, technology, and life sciences, where subcritical Hopf bifurcations occur.

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics. This counterintuitive phenomenon was first observed in systems of identical oscillators with symmetric coupling topology. Can one overcome these limitations? To address this question, we discuss the robustness of chimera states in networks of FitzHugh-Nagumo oscillators. Considering networks of inhomogeneous elements with regular coupling topology, and networks of identical elements with irregular coupling topologies, we demonstrate that chimera states are robust with respect to these perturbations and analyze their properties as the inhomogeneities increase. We find that modifications of coupling topologies cause qualitative changes of chimera states: additional random links induce a shift of the stability regions in the system parameter plane, gaps in the connectivity matrix result in a change of the multiplicity of incoherent regions of the chimera state, and hierarchical geometry in the connectivity matrix induces nested coherent and incoherent regions.

We investigate the spatio-temporal dynamics of coupled chaotic systems with nonlocal interactions, where each element is coupled to its nearest neighbors within a finite range. Depending upon the coupling strength and coupling radius, we find characteristic spatial patterns such as wave-like profiles and study the transition from coherence to incoherence leading to spatial chaos. We analyze the origin of this transition based on numerical simulations and support the results by theoretical derivations identifying a critical coupling strength and a scaling relation of the coherent profiles. To demonstrate the universality of our findings we consider time-discrete as well as time-continuous chaotic models realized as logistic map and Rössler or Lorenz system, respectively. Thereby we establish the coherence-incoherence transition in networks of coupled identical oscillators.

We study synchronization in delay-coupled oscillator networks using a master stability function approach. Within a generic model of Stuart-Landau oscillators (normal form of supercritical or subcritical Hopf bifurcation), we derive analytical stability conditions and demonstrate that by tuning the coupling phase one can easily control the stability of synchronous periodic states. We propose the coupling phase as a crucial control parameter to switch between in-phase synchronization or desynchronization for general network topologies or between in-phase, cluster, or splay states in unidirectional rings. Our results are robust even for slightly nonidentical elements of the network.

The influence of time delay in systems of two coupled excitable neurons is studied in the framework of the FitzHugh-Nagumo model. A time delay can occur in the coupling between neurons or in a self-feedback loop. The stochastic synchronization of instantaneously coupled neurons under the influence of white noise can be deliberately controlled by local time-delayed feedback. By appropriate choice of the delay time, synchronization can be either enhanced or suppressed. In delay-coupled neurons, antiphase oscillations can be induced for sufficiently large delay and coupling strength. The additional application of time-delayed self-feedback leads to complex scenarios of synchronized in-phase or antiphase oscillations, bursting patterns or amplitude death.