Spatiotemporal chaos and turbulence are universal concepts for the explanation of irregular behavior in various physical systems. Recently, a remarkable new phenomenon, called "chimera states," has been described, where in a spatially homogeneous system, regions of irregular incoherent motion coexist with regular synchronized motion, forming a self-organized pattern in a population of nonlocally coupled oscillators. Whereas most previous studies of chimera states focused their attention on the case of large numbers of oscillators employing the thermodynamic limit of infinitely many oscillators, here we investigate the properties of chimera states in populations of finite size using concepts from deterministic chaos. Our calculations of the Lyapunov spectrum show that the incoherent motion, which is described in the thermodynamic limit as a stationary behavior, in finite size systems appears as weak spatially extensive chaos. Moreover, for sufficiently small populations the chimera states reveal their transient nature: after a certain time span we observe a sudden collapse of the chimera pattern and a transition to the completely coherent state. Our results indicate that chimera states can be considered as chaotic transients, showing the same properties as type-II supertransients in coupled map lattices.

Chimera states are a recently new discovered dynamical phenomenon that appears in arrays of nonlocally coupled oscillators and displays a spatial pattern of coherent and incoherent regions. We report here an additional feature of this dynamical regime: an irregular motion of the position of the coherent and incoherent regions, i.e., we reveal the nature of the chimera as a spatiotemporal pattern with a regular macroscopic pattern in space, and an irregular motion in time. This motion is a finite-size effect that is not observed in the thermodynamic limit. We show that on a large time scale, it can be described as a Brownian motion. We provide a detailed study of its dependence on the number of oscillators N and the parameters of the system.

Chimera states are particular trajectories in systems of phase oscillators with non-local coupling that display a spatio-temporal pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hyperchaotic nature of the chimera states and show a correspondence of the Lyapunov dimension with the number of incoherent oscillators. Then, we pass to the thermodynamic limit equation and present an analytic approach to the spectrum of a corresponding linearized evolution operator. We show that in this setting, the chimera state is neutrally stable and that the continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems. Chimera states (see Figure 1) are remarkable spatio-temporal patterns where regions of synchrony coexist with regions of incoherent motion in a spatially homogeneous system of coupled oscillators. They constitute a new paradigm of dynamical behavior that can serve as a prototype for various physical phenomena, e.g. coexistence of synchronous and asynchronous neural activity (so called 'bump' states) [1, 2, 3, 4] or turbulent-laminar flow patterns [5]. For their mathematical description one has to employ concepts from the fields of pattern formation, deterministic chaos, and statistical physics. Indeed, starting with the pioneering work of Kuramoto [6], the thermodynamic limit of a large number of oscillators has been developed to a powerful tool for the investigation of chimera states. In this paper, we put our focus to the relation of chimera states in finite size systems to their thermodynamic limits. After a careful numerical study of the Lyapunov spectra for chimera trajectories in finite size systems, we compare our results with the spectral properties of the linearized evolution operator in the thermodynamic limit. We show that there the chimera states are neutrally stable and that their continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.

Recently, it has been shown that large arrays of identical oscillators with nonlocal coupling can have a remarkable type of solutions that display a stationary macroscopic pattern of coexisting regions with coherent and incoherent motions, often called chimera states. Here, we present a detailed numerical study of the appearance of such solutions in two-dimensional arrays of coupled phase oscillators. We discover a variety of stationary patterns, including circular spots, stripe patterns, and patterns of multiple spirals. Here, stationarity means that, for increasing system size, the locally averaged phase distributions tend to the stationary profile given by the corresponding thermodynamic limit equation.

We present a control scheme that is able to find and stabilize an unstable chaotic regime in a system with a large number of interacting particles. This allows us to track a high dimensional chaotic attractor through a bifurcation where it loses its attractivity. Similar to classical delayed feedback control, the scheme is noninvasive, however only in an appropriately relaxed sense considering the chaotic regime as a statistical equilibrium displaying random fluctuations as a finite size effect. We demonstrate the control scheme for so-called chimera states, which are coherence-incoherence patterns in coupled oscillator systems. The control makes chimera states observable close to coherence, for small numbers of oscillators, and for random initial conditions.