Chimeras with multiple coherent regions

Emergence of chimeralike oscillation modes in excitable complex networks with preferentially cutting-rewiring operation

In this paper, the preferentially cutting-rewiring operation (PCRO) consisting of the cutting procedure and the rewiring procedure is proposed and is applied on an excitable Erdös-Rényi random network (EERRN), by which the structure of the initially homogeneous network changes dramatically, and lots of common leaves (CLs) are formed between the two hubs. Subsequently, besides the single-mode oscillations that can be usually observed in homogeneous excitable systems, a new kind of multi-mode oscillations composed of synchronous and asynchronous parts can self-organize to emerge, which are similar to the coherent and incoherent clusters in traditional chimera states and are consequently named as the chimeralike oscillation modes (CLOMs). Importantly, by utilizing the dominant phase-advanced driving method, both the mechanisms for the formation and the emergence of CLOMs in EERRNs with PCRO are well explained, among which the CL is exposed to play a key role in forming the CLOMs. Furthermore, the PCRO-induced CLOM phenomena can also be observed in other paradigmatic network models or with other paradigmatic excitable dynamics, which definitely confirms that the PCRO is an universal method in inducing the CLOMs in excitable complex networks. Our contributions may shed lights on a new perspective of the emergence of CLOMs in complex systems and would have great impacts in related fields.

Order parameter dynamics in complex systems: From models to data

Collective ordering behaviors are typical macroscopic manifestations embedded in complex systems and can be ubiquitously observed across various physical backgrounds. Elements in complex systems may self-organize via mutual or external couplings to achieve diverse spatiotemporal coordinations. The order parameter, as a powerful quantity in describing the transition to collective states, may emerge spontaneously from large numbers of degrees of freedom through competitions. In this minireview, we extensively discussed the collective dynamics of complex systems from the viewpoint of order-parameter dynamics. A synergetic theory is adopted as the foundation of order-parameter dynamics, and it focuses on the self-organization and collective behaviors of complex systems. At the onset of macroscopic transitions, slow modes are distinguished from fast modes and act as order parameters, whose evolution can be established in terms of the slaving principle. We explore order-parameter dynamics in both model-based and data-based scenarios. For situations where microscopic dynamics modeling is available, as prototype examples, synchronization of coupled phase oscillators, chimera states, and neuron network dynamics are analytically studied, and the order-parameter dynamics is constructed in terms of reduction procedures such as the Ott-Antonsen ansatz, the Lorentz ansatz, and so on. For complicated systems highly challenging to be well modeled, we proposed the eigen-microstate approach (EMP) to reconstruct the macroscopic order-parameter dynamics, where the spatiotemporal evolution brought by big data can be well decomposed into eigenmodes, and the macroscopic collective behavior can be traced by Bose-Einstein condensation-like transitions and the emergence of dominant eigenmodes. The EMP is successfully applied to some typical examples, such as phase transitions in the Ising model, climate dynamics in earth systems, fluctuation patterns in stock markets, and collective motion in living systems.

Short-lived chimera states

In the classic Kuramoto system of coupled two-dimensional rotators, chimera states characterized by the coexistence of synchronous and asynchronous groups of oscillators are long-lived because the average lifetime of these states increases exponentially with the system size. Recently, it was discovered that, when the rotators in the Kuramoto model are three-dimensional, the chimera states become short-lived in the sense that their lifetime scales with only the logarithm of the dimension-augmenting perturbation. We introduce transverse-stability analysis to understand the short-lived chimera states. In particular, on the unit sphere representing three-dimensional (3D) rotations, the long-lived chimera states in the classic Kuramoto system occur on the equator, to which latitudinal perturbations that make the rotations 3D are transverse. We demonstrate that the largest transverse Lyapunov exponent calculated with respect to these long-lived chimera states is typically positive, making them short-lived. The transverse-stability analysis turns the previous numerical scaling law of the transient lifetime into an exact formula: the "free" proportional constant in the original scaling law can now be precisely determined in terms of the largest transverse Lyapunov exponent. Our analysis reinforces the speculation that in physical systems, chimera states can be short-lived as they are vulnerable to any perturbations that have a component transverse to the invariant subspace in which they live.

Amplitude-mediated chimera states in nonlocally coupled Stuart-Landau oscillators

Chimera states achieve the coexistence of coherent and incoherent subgroups through symmetry breaking and emerge in physical, chemical, and biological systems. We show the presence of amplitude-mediated multicluster chimera states in nonlocally coupled Stuart-Landau oscillators. We clarify the prerequisites for having different types of chimera states by analytically and numerically studying how phase transitions occur between these states. Our results demonstrate how the oscillation amplitudes interact with the phase degrees of freedom in chimera states and significantly advance our understanding of the generation mechanisms of such states in coupled oscillator systems.

Frequency chimera state induced by differing dynamical timescales

We report the occurrence of a self-emerging frequency chimera state in spatially extended systems of coupled oscillators, where the coherence and incoherence are defined with respect to the emergent frequency of the oscillations. This is generated by the local coupling among nonlinear oscillators evolving under differing dynamical timescales starting from random initial conditions. We show how they self-organize to structured patterns with spatial domains of coherence that are in frequency synchronization, coexisting with domains that are incoherent in frequencies. Our study has relevance in understanding such patterns observed in real-world systems like neuronal systems, power grids, social and ecological networks, where differing dynamical timescales is natural and realistic among the interacting systems.

Identification of chimera using machine learning

Chimera state refers to the coexistence of coherent and non-coherent phases in identically coupled dynamical units found in various complex dynamical systems. Identification of chimera, on one hand, is essential due to its applicability in various areas including neuroscience and, on the other hand, is challenging due to its widely varied appearance in different systems and the peculiar nature of its profile. Therefore, a simple yet universal method for its identification remains an open problem. Here, we present a very distinctive approach using machine learning techniques to characterize different dynamical phases and identify the chimera state from given spatial profiles generated using various different models. The experimental results show that the performance of the classification algorithms varies for different dynamical models. The machine learning algorithms, namely, random forest, oblique random forest based on Tikhonov, axis-parallel split, and null space regularization achieved more than 96% accuracy for the Kuramoto model. For the logistic maps, random forest and Tikhonov regularization based oblique random forest showed more than 90% accuracy, and for the Hénon map model, random forest, null space, and axis-parallel split regularization based oblique random forest achieved more than 80% accuracy. The oblique random forest with null space regularization achieved consistent performance (more than 83% accuracy) across different dynamical models while the auto-encoder based random vector functional link neural network showed relatively lower performance. This work provides a direction for employing machine learning techniques to identify dynamical patterns arising in coupled non-linear units on large-scale and for characterizing complex spatiotemporal patterns in real-world systems for various applications.

Self-adaptation of chimera states

Chimera states in spatiotemporal dynamical systems have been investigated in physical, chemical, and biological systems, and have been shown to be robust against random perturbations. How do chimera states achieve their robustness? We uncover a self-adaptation behavior by which, upon a spatially localized perturbation, the coherent component of the chimera state spontaneously drifts to an optimal location as far away from the perturbation as possible, exposing only its incoherent component to the perturbation to minimize the disturbance. A systematic numerical analysis of the evolution of the spatiotemporal pattern of the chimera state towards the optimal stable state reveals an exponential relaxation process independent of the spatial location of the perturbation, implying that its effects can be modeled as restoring and damping forces in a mechanical system and enabling the articulation of a phenomenological model. Not only is the model able to reproduce the numerical results, it can also predict the trajectory of drifting. Our finding is striking as it reveals that, inherently, chimera states possess a kind of "intelligence" in achieving robustness through self-adaptation. The behavior can be exploited for the controlled generation of chimera states with their coherent component placed in any desired spatial region of the system.

Emergence of chimeras through induced multistability

Chimeras, namely coexisting desynchronous and synchronized dynamics, are formed in an ensemble of identically coupled identical chaotic oscillators when the coupling induces multiple stable attractors, and further when the basins of the different attractors are intertwined in a complex manner. When there is coupling-induced multistability, an ensemble of identical chaotic oscillators-with global coupling, or also under the influence of common noise or an external drive (chaotic, periodic, or quasiperiodic)-inevitably exhibits chimeric behavior. Induced multistability in the system leads to the formation of distinct subpopulations, one or more of which support synchronized dynamics, while in others the motion is asynchronous or incoherent. We study the mechanism for the emergence of such chimeric states, and we discuss the generality of our results.

Inducing isolated-desynchronization states in complex network of coupled chaotic oscillators

In a recent study about chaos synchronization in complex networks [Nat. Commun. 5, 4079 (2014)NCAOBW2041-172310.1038/ncomms5079], it is shown that a stable synchronous cluster may coexist with vast asynchronous nodes, resembling the phenomenon of a chimera state observed in a regular network of coupled periodic oscillators. Although of practical significance, this new type of state, namely, the isolated-desynchronization state, is hardly observed in practice due to its strict requirements on the network topology. Here, by the strategy of pinning coupling, we propose an effective method for inducing isolated-desynchronization states in symmetric networks of coupled chaotic oscillators. Theoretical analysis based on eigenvalue analysis shows that, by pinning a group of symmetric nodes in the network, there exists a critical pinning strength beyond which the group of pinned nodes can completely be synchronized while the unpinned nodes remain asynchronous. The feasibility and efficiency of the control method are verified by numerical simulations of both artificial and real-world complex networks with the numerical results in good agreement with the theoretical predictions.

Phase-flip chimera induced by environmental nonlocal coupling

We report the emergence of a collective dynamical state, namely, the phase-flip chimera, from an ensemble of identical nonlinear oscillators that are coupled indirectly via the dynamical variables from a common environment, which in turn are nonlocally coupled. The phase-flip chimera is characterized by the coexistence of two adjacent out-of-phase synchronized coherent domains interspersed by an incoherent domain, in which the nearby oscillators are in out-of-phase synchronized states. Attractors of the coherent domains are either from the same or from different basins of attractions, depending on whether they are periodic or chaotic. The conventional chimera precedes the phase-flip chimera in general. Further, the phase-flip chimera emerges after the completely synchronized evolution of the ensemble, in contrast to conventional chimeras, which emerge as an intermediate state between completely incoherent and coherent states. We have also characterized the observed dynamical transitions using the strength of incoherence, probability distribution of the correlation coefficient, and framework of the master stability function.

Phase oscillators in modular networks: The effect of nonlocal coupling

We study the dynamics of nonlocally coupled phase oscillators in a modular network. The interactions include a phase lag, α. Depending on the various parameters the system exhibits a number of different dynamical states. In addition to global synchrony there can also be modular synchrony when each module can synchronize separately to a different frequency. There can also be multicluster frequency chimeras, namely coherent domains consisting of modules that are separately synchronized to different frequencies, coexisting with modules within which the dynamics is desynchronized. We apply the Ott-Antonsen ansatz in order to reduce the effective dimensionality and thereby carry out a detailed analysis of the different dynamical states.

Emergence of multicluster chimera states

A remarkable phenomenon in spatiotemporal dynamical systems is chimera state, where the structurally and dynamically identical oscillators in a coupled networked system spontaneously break into two groups, one exhibiting coherent motion and another incoherent. This phenomenon was typically studied in the setting of non-local coupling configurations. We ask what can happen to chimera states under systematic changes to the network structure when links are removed from the network in an orderly fashion but the local coupling topology remains invariant with respect to an index shift. We find the emergence of multicluster chimera states. Remarkably, as a parameter characterizing the amount of link removal is increased, chimera states of distinct numbers of clusters emerge and persist in different parameter regions. We develop a phenomenological theory, based on enhanced or reduced interactions among oscillators in different spatial groups, to explain why chimera states of certain numbers of clusters occur in certain parameter regions. The theoretical prediction agrees well with numerics.

Effect of asymmetry parameter on the dynamical states of nonlocally coupled nonlinear oscillators

We show that coexisting domains of coherent and incoherent oscillations can be induced in an ensemble of any identical nonlinear dynamical systems using nonlocal rotational matrix coupling with an asymmetry parameter. Further, a chimera is shown to emerge in a wide range of the asymmetry parameter in contrast to near π/2 values of it employed in earlier works. We have also corroborated our results using the strength of incoherence in the frequency domain (S(ω)) and in the amplitude domain (S), thereby distinguishing the frequency and amplitude chimeras. The robust nature of the asymmetry parameter in inducing chimeras in any generic dynamical system is established using ensembles of identical Rössler oscillators, Lorenz systems, and Hindmarsh-Rose neurons in their chaotic regimes.

Chimera states in systems of nonlocal nonidentical phase-coupled oscillators

Chimera states consisting of domains of coherently and incoherently oscillating nonlocally coupled phase oscillators in systems with spatial inhomogeneity are studied. The inhomogeneity is introduced through the dependence of the oscillator frequency on its location. Two types of spatial inhomogeneity, localized and spatially periodic, are considered and their effects on the existence and properties of multicluster and traveling chimera states are explored. The inhomogeneity is found to break up splay states, to pin the chimera states to specific locations, and to trap traveling chimeras. Many of these states can be studied by constructing an evolution equation for a complex order parameter. Solutions of this equation are in good agreement with the results of numerical simulations.

Mechanism for intensity-induced chimera states in globally coupled oscillators

We identify the mechanism behind the existence of intensity-induced chimera states in globally coupled oscillators. We find that the effect of intensity in the system is to cause multistability by increasing the number of fixed points. This in turn increases the number of multistable attractors, and we find that their stability is determined by the strength of coupling. This causes the coexistence of different collective states in the system depending upon the initial state. We demonstrate that intensity-induced chimera is generic to both periodic and chaotic systems. We discuss possible applications of our results to real-world systems like the brain and spin torque nano-oscillators.

Multicluster and traveling chimera states in nonlocal phase-coupled oscillators

Chimera states consisting of domains of coherently and incoherently oscillating identical oscillators with nonlocal coupling are studied. These states usually coexist with the fully synchronized state and have a small basin of attraction. We propose a nonlocal phase-coupled model in which chimera states develop from random initial conditions. Several classes of chimera states have been found: (a) stationary multicluster states with evenly distributed coherent clusters, (b) stationary multicluster states with unevenly distributed clusters, and (c) a single cluster state traveling with a constant speed across the system. Traveling coherent states are also identified. A self-consistent continuum description of these states is provided and their stability properties analyzed through a combination of linear stability analysis and numerical simulation.

Coherence and incoherence in an optical comb

We demonstrate a coexistence of coherent and incoherent modes in the optical comb generated by a passively mode-locked quantum dot laser. This is experimentally achieved by means of optical linewidth, radio frequency spectrum, and optical spectrum measurements and confirmed numerically by a delay-differential equation model showing excellent agreement with the experiment. We interpret the state as a chimera state.

Observation and characterization of chimera states in coupled dynamical systems with nonlocal coupling

By developing the concepts of strength of incoherence and discontinuity measure, we show that a distinct quantitative characterization of chimera and multichimera states which occur in networks of coupled nonlinear dynamical systems admitting nonlocal interactions of finite radius can be made. These measures also clearly distinguish between chimera or multichimera states (both stable and breathing types) and coherent and incoherent as well as cluster states. The measures provide a straightforward and precise characterization of the various dynamical states in coupled chaotic dynamical systems irrespective of the complexity of the underlying attractors.

Chimera states: the existence criteria revisited

Chimera states, representing a spontaneous breakup of a population of identical oscillators that are identically coupled, into subpopulations displaying synchronized and desynchronized behavior, have traditionally been found to exist in weakly coupled systems and with some form of nonlocal coupling between the oscillators. Here we show that neither the weak-coupling approximation nor nonlocal coupling are essential conditions for their existence. We obtain, for the first time, amplitude-mediated chimera states in a system of globally coupled complex Ginzburg-Landau oscillators. We delineate the dynamical origins for the formation of such states from a bifurcation analysis of a reduced model equation and also discuss the practical implications of our discovery of this broader class of chimera states.