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Coupled oscillators exhibit intriguing dynamical states characterized by the coexistence of coherent and incoherent domains known as chimera states. Similar behaviors have been observed in coupled systems and continuous media. Here we investigate the transition from motionless to traveling chimera states in continuous media. Based on a prototype model for pattern formation, we observe coexistence between motionless and traveling chimera states. The spatial disparity of chimera states allows us to reveal the motion mechanism. The propagation of chimera states is described by their median and centroidal point. The mobility of these states depends on the size of the incoherent domain. The bifurcation diagram of traveling chimeras is elucidated.

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... Although the chimera state was originally observed in regular networks with various dimensions, the conception of chimera extended to the steady 9 or time-varying complex network, 10 multilayer network, 11 and even continuous media. 12 Various factors determining and adjusting chimera state like initial value selection, 13 time delay, 14 noise, 15 and network topology 16 were well studied. Some methods to construct and control chimera states were also developed. ...

Chimera, the coexistence state of synchronization and non-synchronization, widely exists in complex networks. It has a great potentially explanatory power for the unihemispheric sleep of birds and some mammals, in which the synchronizations of the hemispheres of the cerebral cortex are evolving alternately. In this study, a coupled nonlinear oscillator system with a topology of the modular complex network was constructed to simulate the left and right hemispheres of the brain. The results showed that a stable chimera, an alternating chimera, and a breathing chimera were produced when the coupling strength and connection probability of the left and right hemispheres were changed. Further, we studied the effect of noise on rich synchronous patterns and found that the alternating chimera was robust to Gaussian white noise when the strength was not very large. Finally, our study was extended to a complex network with three sub-networks, and an alternating chimera could exist in two or three sub-networks. Our research provides a deeper insight into the mechanism of brain function like unihemispheric sleep.

... The nonlocal coupling among oscillators is considered as a necessary condition for existence of chimera state, but the following researchers demonstrate that chimera state can also exist in network of oscillators with the nearest neighboring coupling (Bera et al. 2016) or global coupling (Banerjee 2015). The system which can exhibit chimera state is expanded from regular networks (Shepelev and Anishchenko 2020) with different dimensions to the complex networks or real cerebral networks (Zhu et al. 2014;Tang et al. 2019;Huo et al. 2019) and time-varying networks (Buscarino et al. 2015) or even continuous media (Alvarez-Socorroab et al. 2021). Chimera also plays an important role in transition of reaction-diffusion systems (Li et al. 2021). ...

Feed-forward effect gives rise to synchronization in neuron firing in deep layers of multiple neuronal network. But complete synchronization means the loss of encoding ability. In order to avoid the contradiction, we ask whether partial synchronization (coexistence of disordered and synchronized neuron firing emerges, also called chimera state) as a compromise strategy can achieve in the feed-forward multiple-layer network. The answer is YES. In order to manifest our argument, we design a multi-layer neuronal network in which neurons in every layer are arranged in a ring topology and neuron firing propagates within (intra-) and across (inter-) the multiply layers. Emergence of chimera state and other patterns highly depends on initial condition of neuronal network and strength of feed-forward effect. Chimera state, cluster and synchronization intra- and inter- layers are displayed by sequence through layers when initial values are elaborately chosen to guarantee emergence of chimera state in the first layer. All type of patterns except chimera state propagates down toward deeper layers in different speeds varying with strength of feed-forward effect. If chimera state already exists in every layer, feed-forward effect with strong and moderate strength spoils chimera states in deep layers and they can only survive in first few layers. When the effect is small enough, chimera states will propagate down toward deeper layers. Indeed, chimera states could exist and transit to deeper layers in a regular multiple network under very strict conditions. The results help understanding better the neuron firing propagating and encoding scheme in a feed-forward neuron network.

... Later, it has been established that chimera states are possible stable states (Pecora et al., 2014;Omel'chenko, 2018;Laing, 2019) in an ensemble of identical oscillators and with symmetry in the connectivity matrix or the topology of a network. By this time, this phenomenon has been widely explored in single-layer networks (Abrams and Strogatz, 2004;Sethia et al., 2008;Laing, 2009;Hagerstrom et al., 2012;Martens et al., 2013;Omelchenko et al., 2013;Gopal et al., 2014;Hart et al., 2019;Majhi et al., 2019;Parastesh et al., 2020;Wang and Liu, 2020), multilayer networks (Ghosh and Jalan, 2016;Maksimenko et al., 2016;Sawicki et al., 2018;Ruzzene et al., 2020), and 3D networks (Maistrenko et al., 2015;Kasimatis et al., 2018;Kundu et al., 2019) with different forms of chimeras such as traveling chimera (Bera et al., 2016a;Omel'chenko, 2019;Dudkowski et al., 2019;Alvarez-Socorro et al., 2021) and spiral chimera (Martens et al., 2010;Gu et al., 2013). Various dynamical models (Hizanidis et al., 2015;Banerjee et al., 2016;Bera et al., 2016b;Saha et al., 2019) with different coupling schemes (Meena et al., 2016;Bera et al., 2017) and global coupling (Sethia and Sen, 2014;Yeldesbay et al., 2014;Hens et al., 2015a;Mishra et al., 2015) have been used for observing chimera patterns. ...

We present an exemplary system of three identical oscillators in a ring interacting repulsively to show up chimera patterns. The dynamics of individual oscillators is governed by the superconducting Josephson junction. Surprisingly, the repulsive interactions can only establish a symmetry of complete synchrony in the ring, which is broken with increasing repulsive interactions when the junctions pass through serials of asynchronous states (periodic and chaotic) but finally emerge into chimera states. The chimera pattern first appears in chaotic rotational motion of the three junctions when two junctions evolve coherently, while the third junction is incoherent. For larger repulsive coupling, the junctions evolve into another chimera pattern in a periodic state when two junctions remain coherent in rotational motion and one junction transits to incoherent librational motion. This chimera pattern is sensitive to initial conditions in the sense that the chimera state flips to another pattern when two junctions switch to coherent librational motion and the third junction remains in rotational motion, but incoherent. The chimera patterns are detected by using partial and global error functions of the junctions, while the librational and rotational motions are identified by a libration index. All the collective states, complete synchrony, desynchronization, and two chimera patterns are delineated in a parameter plane of the ring of junctions, where the boundaries of complete synchrony are demarcated by using the master stability function.

... Later it has been established that chimera states are possible stable states [15][16][17] in an enemble of identical oscillators and with a symmetry in the connectivity matrix or the topology of a network. By this time, this phenomenon has been widely explored in single-layer [1][2][3][4][5][6][7][8][9][10][11], multilayer networks [18][19][20][21] and 3D networks [22][23][24] with different forms of chimeras such as traveling chimera [25][26][27], spiral chimera [28,29]. Various dynamical models [30][31][32][33] with different coupling schemes [34,35], and global coupling [36][37][38][39], have been used for observing chimera patterns. ...

We present an exemplary system of three identical oscillators in a ring interacting repulsively to show up chimera patterns. The dynamics of individual oscillators is governed by the superconducting Josephson junction. Surprisingly, the repulsive interactions establish a symmetry of compelete synchrony in the ring, which is broken with increasing interactions when the junctions pass through serials of asynchronous states (periodic and chaotic), but finally emerge into chimera states. The chimera pattern appears in chaotic rotational motion of the three junctions when two junctions evolve coherently while the third junction is incoherent. For larger repulsive coupling, the junctions evolves into another chimera pattern in a periodic state when two junctions remain coherent in rotational motion and one transits to incoherent librational motion. This chimera pattern is sensitive to initial conditions, in the sense, that the chimera state flips to another pattern when two junctions switch to coherent librational motion and the third junction remains in rotational motion, but incoherent. The chimera patterns are detected by using partial and global error functions of the junctions while the librational and rotational motions are identified by a libration index. All the collective states, complete synchrony, desynchronization and two chimera patterns, are delineated in a parameter plane of the ring of junctions, where the boundaries of complete synchrony are demarcated by using the master stability function.

... After the foundation of chimera state in 2002 [27], it became the focus of many researchers in a variety of dynamical systems such as the mechanical [28], optical [29], and chemical [30] oscillators and neuronal models [25,31,32]. Furthermore, these studies have represented the chimeras with different spatiotemporal patterns and properties, including the amplitude chimera [33] and traveling chimera [34]. ...

The fractional calculus in the neuronal models provides the memory properties. In the fractional-order neuronal model, the dynamics of the neuron depends on the derivative order, which can produce various types of memory-dependent dynamics. In this paper, the behaviors of the coupled fractional-order FitzHugh–Nagumo neurons are investigated. The effects of the coupling strength and the derivative order are under consideration. It is revealed that the level of the synchronization is decreased by decreasing the derivative order, and the chimera state emerges for stronger couplings. Furthermore, the patterns of the formed chimeras rely on the order of the derivatives.

... For instance, instead of nonlocal couplings (which has been regarded as a necessary condition for generating chimera states), recent studies show that chimera states can also be generated in systems with global [24,25] or local couplings [27,28,32,34,[37][38][39][40] . Meanwhile, the concept of chimera states has been largely generalized and a variety of chimera-like states have been reported, e.g., clustered chimeras [5,41] , amplitude and amplitude mediated chimeras [34,42] , alternating chimeras [43] , chimera death [31,44] , spiral wave chimeras [35,36] , switching chimeras [45] and traveling chimeras [37,46] . In addition, besides regular networks, chimera-like states have been also reported and studied in networks of complex structures [47][48][49][50][51][52][53][54] . ...

Spiral wave chimeras (SWCs), which combine the features of spiral waves and chimera states, are a new type of dynamical patterns emerged in spatiotemporal systems due to the spontaneous symmetry breaking of the system dynamics. In generating SWC, the conventional wisdom is that the dynamical elements should be coupled in a nonlocal fashion. For this reason, it is commonly believed that SWC is excluded from the general reaction-diffusion (RD) systems possessing only local couplings. Here, by an experimentally feasible model of three-component FitzHugh-Nagumo-type RD system, we demonstrate that, even though the system elements are locally coupled, stable SWCs can still be observed in a wide region in the parameter space. The properties of SWCs are explored, and the underlying mechanisms are analyzed from the point view of coupled oscillators. Transitions from SWC to incoherent states are also investigated, and it is found that SWCs are typically destabilized in two scenarios, namely core breakup and core expansion. The former is characterized by a continuous breakup of the single asynchronous core into a number of small asynchronous cores, whereas the latter is featured by the continuous expansion of the single asynchronous core to the whole space. Remarkably, in the scenario of core expansion, the system may develop into an intriguing state in which regular spiral waves are embedded in a completely disordered background. This state, which is named shadowed spirals, manifests from a new perspective the coexistence of incoherent and coherent states in spatiotemporal systems, generalizing therefore the traditional concept of chimera states. Our studies provide an affirmative answer to the observation of SWCs in typical RD systems, and pave a way to the realization of SWCs in experiments.

In this paper, analytical and numerical studies of the influence of the long-range interaction parameter on the excitability threshold in a ring of FitzHugh-Nagumo (FHN) system are investigated. The long-range interaction is introduced to the network to model regulation of the Gap junctions or hemichannels activity at the connexins level, which provides links between pre-synaptic and post-synaptic neurons. Results show that the long-range coupling enhances the range of the threshold parameter. We also investigate the long-range effects on the network dynamics, which induces enlargement of the oscillatory zone before the excitable regime. When considering bidirectional coupling, the long-range interaction induces traveling patterns such as traveling waves, while when considering unidirectional coupling, the long-range interaction induces multi-chimera states. We also studied the difference between the dynamics of coupled oscillators and coupled excitable neurons. We found that, for the coupled system, the oscillation period decreases with the increasing of the coupling parameter. For the same values of the coupling parameter, the oscillation period of the Oscillatory dynamics is greater than the oscillation period of the excitable dynamics. The analytical approximation shows good agreement with the numerical results.

Spiral wave chimeras (SWCs), which combine the features of spiral waves and chimera states, are a new type of dynamical patterns emerged in spatiotemporal systems due to the spontaneous symmetry breaking of the system dynamics. In generating SWC, the conventional wisdom is that the dynamical elements should be coupled in a nonlocal fashion. For this reason, it is commonly believed that SWC is excluded from the general reaction-diffusion (RD) systems possessing only local couplings. Here, by an experimentally feasible model of three-component FitzHugh-Nagumo-type RD system, we demonstrate that, even though the system elements are locally coupled, stable SWCs can still be observed in a wide region in the parameter space. The properties of SWCs are explored, and the underlying mechanisms are analyzed from the point view of coupled oscillators. Transitions from SWC to incoherent states are also investigated, and it is found that SWCs are typically destabilized in two scenarios, namely core breakup and core expansion. The former is characterized by a continuous breakup of the single asynchronous core into a number of small asynchronous cores, whereas the latter is featured by the continuous expansion of the single asynchronous core to the whole space. Remarkably, in the scenario of core expansion, the system may develop into an intriguing state in which regular spiral waves are embedded in a completely disordered background. This state, which is named shadowed spirals, manifests from a new perspective the coexistence of incoherent and coherent states in spatiotemporal systems, generalizing therefore the traditional concept of chimera states. Our studies provide an affirmative answer to the observation of SWCs in RD systems, and pave a way to the realization of SWCs in experiments.

Chimera states, or coherence–incoherence patterns in systems of symmetrically coupled identical oscillators, have been the subject of intensive study for the last two decades. In particular it is now known that the continuum limit of phase-coupled oscillators allows an elegant mathematical description of these states based on a nonlinear integro-differential equation known as the Ott–Antonsen equation. However, a systematic study of this equation usually requires a substantial computational effort. In this paper, we consider a special class of nonlocally coupled phase oscillator models where the above analytical approach simplifies significantly, leading to a semi-analytical description of both chimera states and of their linear stability properties. We apply this approach to phase oscillators on a one-dimensional lattice, on a two-dimensional square lattice and on a three-dimensional cubic lattice, all three with periodic boundary conditions. For each of these systems we identify multiple symmetric coherence–incoherence patterns and compute their linear stability properties. In addition, we describe how chimera states in higher-dimensional models are inherited from lower-dimensional models and explain how they can be grouped according to their symmetry properties and global order parameter.

Light polarization is an inherent property of the coherent laser output that finds
applications, for example, in vision, imaging, spectroscopy, cosmology, and
communications. We report here on light polarization dynamics that repeatedly switches
between a stationary state of polarization and an irregularly pulsating polarization. The
reported dynamics is found to result from the onset of chimeras. Chimeras in nonlinear
science refer to the counterintuitive coexistence of coherent and incoherent dynamics in
an initially homogeneous network of coupled nonlinear oscillators. The existence of
chimera states has been evidenced only recently in carefully designed experiments using
either mechanical, optomechanical, electrical, or optical oscillators. Interestingly, the
chimeras reported here originate from the inherent coherent properties of a commercial
laser diode. The spatial and temporal properties of the chimeras found in light
polarization are controlled by the laser diode and feedback parameters, leading, e.g., to
multistability between chimeras with multiple heads and to turbulent chimeras.

We investigate the phenomenon of traveling chimera states in the ring of self-excited coupled pendula suspended on the horizontally oscillating wheel. The bifurcation scenario of chimera creation and destruction is discussed, and the influence of the suspension’s parameters on possible behavior is studied. We describe the properties of the investigated states, analyzing the traveling time as well as the dynamics of the pendula, depending on their position within different pattern’s regions. The energy transfer method has been used to present how the units cooperate with each other, making the chimera arising possible. Unlike other studies on traveling chimera states, we examine the typical mechanical system including simple topology of coupling, which suggests that the described behavior can be observed naturally in the models of coupled dynamical oscillators.

Coupled nonlinear oscillators can present complex spatiotemporal behaviors. Here, we report the coexistence of coherent and incoherent domains, called chimera states, in an array of identical Duffing oscillators coupled to their nearest neighbors. The chimera states show a significant variation of amplitude in the desynchronized domain. These intriguing states are observed in the bistability region between a homogeneous state and a spatiotemporal chaotic one. These dynamical behaviors are characterized by their Lyapunov spectra and their global phase coherence order parameter. The local coupling between oscillators prevents one domain from invading the other one. Depending on initial conditions, a family of chimera states appear, organized in a snaking-like diagram.

We consider a paradigmatic nonvariational scalar Swift-Hohenberg equation that describes short wavenumber or large wavelength pattern forming systems. This work unveils evidence of the transition from stable stationary to moving localized structures in one spatial dimension as a result of a parity breaking instability. This behavior is attributed to the nonvariational character of the model. We show that the nature of this transition is supercritical. We characterize analytically and numerically this bifurcation scenario from which emerges asymmetric moving localized structures. A generalization for two-dimensional settings is discussed.

The defining property of chimera states is the coexistence of coherent and incoherent domains in systems that are structurally and spatially homogeneous. The recent realization that such states might be common in oscillator networks raises the question of whether an analogous phenomenon can occur in continuous media. Here, we show that chimera states can exist in continuous systems even when the coupling is strictly local, as in many fluid and pattern forming media. Using the complex Ginzburg-Landau equation as a model system, we characterize chimera states consisting of a coherent domain of a frozen spiral structure and an incoherent domain of amplitude turbulence. We show that in this case, in contrast with discrete network systems, fluctuations in the local coupling field play a crucial role in limiting the coherent regions. We suggest these findings shed light on new possible forms of coexisting order and disorder in fluid systems.

We consider coupled-waveguide resonators subject to optical injection. The dynamics of this simple device are described by the discrete Lugiato–Lefever equation. We show that chimera-like states can be stabilized, thanks to the discrete nature of the coupled-waveguide resonators. Such chaotic localized structures are unstable in the continuous Lugiato–Lefever model; this is because of dispersive radiation from the tails of localized structures in the form of two counter-propagating fronts between the homogeneous and the complex spatiotemporal state. We characterize the formation of chimera-like states by computing the Lyapunov spectra. We show that localized states have an intermittent spatiotemporal chaotic dynamical nature. These states are generated in a parameter regime characterized by a coexistence between a uniform steady state and a spatiotemporal intermittency state.

We consider three different two-dimensional networks of nonlocally coupled heterogeneous phase oscillators. These networks were previously studied with identical oscillators, and a number of spatiotemporal patterns were found, mostly as a result of direct numerical simulation. Here we take the continuum limit of an infinite number of oscillators and use the Ott--Antonsen ansatz to derive continuum level evolution equations for order parameter--like quantities. Most of the patterns previously found in these networks correspond to relative fixed points of these evolution equations, and we show the following results of extensive numerical investigations of these fixed points: their existence and stability, and the bifurcations involved in their loss of stability as parameters are varied. Our results answer a number of questions posed by previous authors who studied these networks, and provide a better understanding of these networks' dynamics.

Multistable systems exhibit a rich front dynamics between equilibria. In one-dimensional scalar gradient systems, the spread of the fronts is proportional to the energy difference between equilibria. Fronts spreading proportionally to the energetic difference between equilibria is a characteristic of one-dimensional scalar gradient systems. Based on a simple nonvariational bistable model, we show analytically and numerically that the direction and speed of front propagation is led by nonvariational dynamics. We provide experimental evidence of nonvariational front propagation between different molecular orientations in a quasi-one-dimensional liquid-crystal light valve subjected to optical feedback. Free diffraction length allows us to control the variational or nonvariational nature of this system. Numerical simulations of the phenomenological model have quite good agreement with experimental observations.

We report on the emergence of robust multi-clustered chimera states in a dissipative-driven system
of symmetrically and locally coupled identical SQUID oscillators. The "snake-like" resonance curve
of the single SQUID (Superconducting QUantum Interference Device) is the key to the formation
of the chimera states and is responsible for the extreme multistability exhibited by the coupled
system that leads to attractor crowding at the geometrical resonance frequency. Until now, chimera
states were mostly believed to exist for nonlocal coupling. Our findings provide theoretical evidence
that nearest neighbor interactions is indeed capable of supporting such states in a wide parameter
range. SQUID metamaterials are the subject of intense experimental investigations and we are
highly confident that the complex dynamics demonstrated in this manuscript can be confirmed in
the laboratory.

The chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. In this paper, the first evidence of three-dimensional chimera states is reported for the Kuramoto model of phase oscillators in 3D grid topology with periodic boundary conditions. Systematic analysis of the dependence of the spatiotemporal dynamics on the range and strength of coupling shows that there are two principal classes of the chimera patterns which exist in large domains of the parameter space: (I) oscillating and (II) spirally rotating. Characteristic examples from the first class include coherent as well as incoherent balls, tubes, crosses, and layers in incoherent or coherent surrounding; the second class includes scroll waves with incoherent, randomized rolls of different modality and dynamics. Numerical simulations started from various initial conditions indicate that the states are stable over the integration time. Videos of the dynamics of the chimera states are presented in supplementary material. It is concluded that three-dimensional chimera states, which are novel spatiotemporal patterns involving the coexistence of coherent and incoherent domains, can represent one of the inherent features of nature.

Chimera is a rich and fascinating class of self-organized solutions developed
in high dimensional networks having non-local and symmetry breaking coupling
features. Its accurate understanding is expected to bring important insight in
many phenomena observed in complex spatio-temporal dynamics, from living
systems, brain operation principles, and even turbulence in hydrodynamics. In
this article we report on a powerful and highly controllable experiment based
on optoelectronic delayed feedback applied to a wavelength tunable
semiconductor laser, with which a wide variety of Chimera patterns can be
accurately investigated and interpreted. We uncover a cascade of higher order
Chimeras as a pattern transition from N to N - 1 clusters of chaoticity.
Finally, we follow visually, as the gain increases, how Chimera is gradually
destroyed on the way to apparent turbulence-like system behaviour.

An analytical mechanism that support localized spatio-temporal chaos is provided. We consider a simple model-the Nagumo Kuramoto model-which contains the crucial ingredients for observing localized spatio-temporal chaos, namely, the spatio-temporal chaotic pattern and its coexistence with a uniform state. This model allows us to unveil the front dynamics and to show that it can be described by a chaotic motor corresponding to the deterministic counterpart of a Brownian motor. Front interaction is identified as the mechanism at the origin of the localized spatio-temporal chaotic structures.

The phenomenon of chimera states in the systems of coupled, identical oscillators has attracted a great deal of recent theoretical and experimental interest. In such a state, different groups of oscillators can exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Here, considering the coupled pendula, we find another pattern, the so-called imperfect chimera state, which is characterized by a certain number of oscillators which escape from the synchronized chimera's cluster or behave differently than most of uncorrelated pendula. The escaped elements oscillate with different average frequencies (Poincare rotation number). We show that imperfect chimera can be realized in simple experiments with mechanical oscillators, namely Huygens clock. The mathematical model of our experiment shows that the observed chimera states are controlled by elementary dynamical equations derived from Newton's laws that are ubiquitous in many physical and engineering systems.

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.

Detailed experimental and numerical results are presented about the pattern formation mechanism of spatially organized partially synchronized states in a networked chemical system with oscillatory metal dissolution. Numerical simulations of the reaction system are used to identify experimental conditions (heterogeneity, network topology, and coupling time-scale) under which the chemical reactions, which take place in a network, are split into coexisting coherent and incoherent domains through the chimera mechanism. Experiments are carried out with a network of twenty electrodes arranged in a ring with seven nearest neighbor couplings in both directions along the ring. The patterns are characterized by analyzing the oscillation frequencies and entrainments to the mean field of the phases of oscillations. The chimera state forms from two domains of elements: the chimera core in which the elements have identical frequencies and are entrained to their corresponding mean field and the chimera shell where the elements exhibit desynchrony with each other and the mean field. The experiments point out the importance of low level of heterogeneities (e.g., surface conditions) and optimal level of coupling strength and time-scale as necessary components for the realization of the chimera state. For systems with large heterogeneities, a ‘remnant’ chimera state is identified where the pattern is strongly affected by the presence of frequency clusters. The exploration of dynamical features with networked reactions could open up ways for identification of novel types of patterns that cannot be observed with reaction diffusion systems (with localized interactions) or with reactions under global constraints, coupling, or feedback.

We study experimentally and theoretically the dynamics of networks of
non-locally coupled electronic oscillators that are described by a
Kuramoto-like model. The experimental networks show long complex transients
from random initial conditions on the route to network synchronization. The
transients display complex behaviors, including resurgence of chimera states,
which are network dynamics where order and disorder coexists. The spatial
domain of the chimera state moves around the network and alternates with
desynchronized dynamics. The fast timescale of our oscillators (on the order of
100 ns) allows us to study the scaling of the transient time of large networks
of more than a hundred nodes. We find that the average transient time increases
exponentially with the network size and can be modeled as a Poisson process.
This exponential scaling is a result of a synchronization rate that follows a
power law of the phase-space volume.

We report a novel mechanism for the formation of chimera states, a peculiar
spatiotemporal pattern with coexisting synchronized and incoherent domains
found in ensembles of identical oscillators. Considering Stuart-Landau
oscillators we demonstrate that a nonlinear global coupling can induce this
symmetry breaking. We find chimera states also in a spatially extended system,
a modified complex Ginzburg-Landau equation. This theoretical prediction is
validated with an oscillatory electrochemical system, the electrooxidation of
silicon, where the spontaneous formation of chimeras is observed without any
external feedback control.

We propose a route to spatiotemporal chaos for one-dimensional stationary patterns, which is a natural extension of the quasiperiodicity route for low-dimensional chaos to extended systems. This route is studied through a universal model of pattern formation. The model exhibits a scenario where stationary patterns become spatiotemporally chaotic through two successive bifurcations. First, the pattern undergoes a subcritical Andronov-Hopf bifurcation leading to an oscillatory pattern. Subsequently, a secondary bifurcation gives rise to an oscillation with an incommensurable frequency with respect to the former one. This last bifurcation is responsible for the spatiotemporally chaotic behavior. The Lyapunov spectrum enables us to identify the complex behavior observed as spatiotemporal chaos, and also from the larger Lyapunov exponents characterize the above instabilities.

Dynamical processes in many engineered and living systems take place on complex networks of discrete dynamical units. We present laboratory experiments with a networked chemical system of nickel electrodissolution in which synchronization patterns are recorded in systems with smooth periodic, relaxation periodic, and chaotic oscillators organized in networks composed of up to twenty dynamical units and 140 connections. The reaction system formed domains of synchronization patterns that are strongly affected by the architecture of the network. Spatially organized partial synchronization could be observed either due to densely connected network nodes or through the 'chimera' symmetry breaking mechanism. Relaxation periodic and chaotic oscillators formed structures by dynamical differentiation. We have identified effects of network structure on pattern selection (through permutation symmetry and coupling directness) and on formation of hierarchical and 'fuzzy' clusters. With chaotic oscillators we provide experimental evidence that critical coupling strengths at which transition to identical synchronization occurs can be interpreted by experiments with a pair of oscillators and analysis of the eigenvalues of the Laplacian connectivity matrix. The experiments thus provide an insight into the extent of the impact of the architecture of a network on self-organized synchronization patterns.

Networks of nonlocally coupled phase oscillators can support chimera
states in which identical oscillators evolve into distinct groups that
exhibit coexisting synchronous and incoherent behaviours despite
homogeneous coupling. Similar nonlocal coupling topologies implemented
in networks of chaotic iterated maps also yield dynamical states
exhibiting coexisting spatial domains of coherence and incoherence. 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.
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
coherence-incoherence transition that gives rise to these chimera states
through experiment, theory and simulation.

We consider a paradigmatic spatially extended model of non-locally coupled phase oscillators which are uniformly distributed within a one-dimensional interval and interact depending on the distance between their sites' modulo periodic boundary conditions. This model can display peculiar spatio-temporal patterns consisting of alternating patches with synchronized (coherent) or irregular (incoherent) oscillator dynamics, hence the name coherence–incoherence pattern, or chimera state. For such patterns we formulate a general bifurcation analysis scheme based on a hierarchy of continuum limit equations. This provides the possibility of classifying known coherence–incoherence patterns and of suggesting directions for the search for new ones.

Time-delayed systems are found to display remarkable temporal patterns the dynamics of which split into regular and chaotic components repeating at the interval of a delay. This novel long-term behavior for delay dynamics results from strongly asymmetric nonlinear delayed feedback driving a highly damped harmonic oscillator dynamics. In the corresponding virtual space-time representation, the behavior is found to develop as a chimeralike state, a new paradigmatic object from the network theory characterized by the coexistence of synchronous and incoherent oscillations. Numerous virtual chimera states are obtained and analyzed, through experiment, theory, and simulations.

The synchronization of coupled oscillators is a fascinating manifestation of self-organization that nature uses to orchestrate essential processes of life, such as the beating of the heart. Although it was long thought that synchrony and disorder were mutually exclusive steady states for a network of identical oscillators, numerous theoretical studies in recent years have revealed the intriguing possibility of "chimera states," in which the symmetry of the oscillator population is broken into a synchronous part and an asynchronous part. However, a striking lack of empirical evidence raises the question of whether chimeras are indeed characteristic of natural systems. This calls for a palpable realization of chimera states without any fine-tuning, from which physical mechanisms underlying their emergence can be uncovered. Here, we devise a simple experiment with mechanical oscillators coupled in a hierarchical network to show that chimeras emerge naturally from a competition between two antagonistic synchronization patterns. We identify a wide spectrum of complex states, encompassing and extending the set of previously described chimeras. Our mathematical model shows that the self-organization observed in our experiments is controlled by elementary dynamical equations from mechanics that are ubiquitous in many natural and technological systems. The symmetry-breaking mechanism revealed by our experiments may thus be prevalent in systems exhibiting collective behavior, such as power grids, optomechanical crystals, or cells communicating via quorum sensing in microbial populations.

The existence, stability properties, and dynamical evolution of localized spatiotemporal chaos are studied. We provide evidence of spatiotemporal chaotic localized structures in a liquid crystal light valve experiment with optical feedback. The observations are supported by numerical simulations of the Lifshitz model describing the system. This model exhibits coexistence between a uniform state and a spatiotemporal chaotic pattern, which emerge as the necessary ingredients to obtain localized spatiotemporal chaos. In addition, we have derived a simplified model that allows us to unveil the front interaction mechanism at the origin of the localized spatiotemporal chaotic structures.

Single-neuron recordings from behaving primates have established a link between working memory processes and information-specific neuronal persistent activity in the prefrontal cortex. Using a network
model endowed with a columnar architecture and based on the physiological properties of cortical neurons and synapses, we have examined the synaptic mechanisms of selective persistent activity underlying spatial working memory in the prefrontal cortex. Our model reproduces the phenomenology of the oculomotor delayed-response experiment of Funahashi et al. (S. Funahashi, C.J. Bruce and P.S. Goldman-Rakic, Mnemonic coding of visual space in the monkey's dorsolateral prefrontal cortex. J Neurophysiol
61:331–349, 1989). To observe stable spontaneous and persistent activity, we find that recurrent synaptic excitation should be primarily mediated by NMDA receptors, and that overall recurrent synaptic interactions should be dominated by inhibition. Iso- directional tuning of adjacent pyramidal cells and interneurons can be accounted for by a structured pyramid-to-interneuron connectivity. Robust memory storage against random drift of the tuned persistent activity and against distractors (intervening stimuli during the delay period) may be enhanced by neuromodulation of recurrent synapses. Experimentally testable predictions concerning the neural basis of working memory are discussed.

We present a model and nonlinear analysis which account for the clustering behaviors of arid vegetation ecosystems, the formation
of localized bare soil spots (sometimes also called fairy circles) in these systems and the attractive or repulsive interactions governing their spatio-temporal evolution. Numerical solutions
of the model closely agree with analytical predictions.

Metastability occurs whenever, under given external conditions (as defined by the Reynolds number in hydrodynamics) more than one linearly stable solution may exist. In variational problems, the preferred state is usually claimed to be the one with the lowest energy. But in hydrodynamics, and except for very special conditions no energy functional exists. From an analogy between amplitude and reaction diffusion equations it is argued that the border separating in space two possible solutions of the flow equations (as turbulent and laminar in pipe flows or boundary layers) moves with a constant mean velocity, depending on the control parameter. The sign of this velocity allows to decide which state is stable and which is metastable. Possible physical implications of these ideas for various problems of hydrodynamics are discussed.

We show that the advection of optical localized structures is accompanied by the emission of vortices, with phase singularities appearing in the wake of the drifting structure. Localized structures are obtained in a light-valve experiment and made to drift by a mirror tilt in the feedback loop. Pairs of oppositely charged vortices are detected for small drifts, whereas for large drifts a vortex array develops. Observations are supported by numerical simulations and linear stability analysis of the system equations and are expected to be generic for a large class of translated optical patterns.

The coexistence of coherent and incoherent domains in discrete coupled oscillators, chimera state, has been attracted the attention of the scientific community. Here we investigate the macroscopic dynamics of the continuous counterpart of this phenomenon. Based on a prototype model of pattern formation, we study a family of localized states. These localized solutions can be characterized by their sizes, and positions, and Yorke-Kaplan dimension. Chimera states in continuous media correspond to chaotic localized states. As a function of parameters and their size, the position of these chimera states can be bounded or unbounded. This allows us to classify these solutions as wandering or confined walk. The wandering walk is characterized by a chaotic motion with a truncated Gaussian distribution in its displacement as well as memory effects.

Two-dimensional arrays of coupled waveguides or coupled microcavities allow us to confine and manipulate light. Based on a paradigmatic envelope equation, we show that these devices, subject to a coherent optical injection, support coexistence between a coherent and incoherent emission. In this regime, we show that two-dimensional chimera states can be generated. Depending on initial conditions, the system exhibits a family of two-dimensional chimera states and interaction between them. We characterize these two-dimensional structures by computing their Lyapunov spectrum and Yorke–Kaplan dimension. Finally, we show that two-dimensional chimera states are of spatiotemporal chaotic nature.

In this paper we consider the Ott–Antonsen equation describing the long time coarse-grained dynamics of a ring of nonlocally coupled phase oscillators. We study traveling wave solutions relevant to traveling chimera states in the coupled oscillator system. In particular, we derive and rigorously justify asymptotic formulas for traveling wave solutions in the case of small asymmetry of nonlocal coupling. We also show that the Ott–Antonsen equation provides a reliable description of traveling chimera states for heterogeneous oscillators (with Lorentzian distribution of natural frequencies), but fails to do this for identical oscillators.

Coupled oscillators can exhibit complex self-organization behavior such as phase turbulence, spatiotemporal intermittency, and chimera states. The latter corresponds to a coexistence of coherent and incoherent states apparently promoted by nonlocal or global coupling. Here we investigate the existence, stability properties, and bifurcation diagram of chimera-type states in a system with local coupling without different time scales. Based on a model of a chain of nonlinear oscillators coupled to adjacent neighbors, we identify the required attributes to observe these states: local coupling and bistability between a stationary and an oscillatory state close to a homoclinic bifurcation. The local coupling prevents the incoherent state from invading the coherent one, allowing concurrently the existence of a family of chimera states, which are organized by a homoclinic snaking bifurcation diagram.

The paper ‘The chemical basis of morphogenesis’ [Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952)] by Alan Turing remains hugely influential in the development of mathematical biology as a field of research and was his only published work in the area. In this paper I discuss the later development of his ideas as revealed by lesser-known archive material, in particular the draft notes for a paper with the title ‘Outline of development of the Daisy’.
These notes show that, in his mathematical work on pattern formation, Turing developed substantial insights that go far beyond Turing (1952). The model differential equations discussed in his notes are substantially different from those that are the subject of Turing (1952) and present a much more complex mathematical challenge. In taking on this challenge, Turing's work anticipates (i) the description of patterns in terms of modes in Fourier space and their nonlinear interactions, (ii) the construction of the well-known model equation usually ascribed to Swift and Hohenberg, published 23 years after Turing's death, and (iii) the use of symmetry to organise computations of the stability of symmetrical equilibria corresponding to spatial patterns.
This paper focuses on Turing's mathematics rather than his intended applications of his theories to phyllotaxis, gastrulation, or the unicellular marine organisms Radiolaria. The paper argues that this archive material shows that Turing encountered and wrestled with many issues that became key mathematical research questions in subsequent decades, showing a level of technical skill that was clearly both ahead of contemporary work, and also independent of it. His legacy in recognising that the formation of patterns can be understood through mathematical models, and that this mathematics could have wide application, could have been far greater than just the single paper of 1952.
A revised and substantially extended draft of ‘Outline of development of the Daisy’ is included in the Supplementary material.

We study an oscillatory medium with a nonlinear global coupling that gives rise to a harmonic mean-field oscillation with constant amplitude and frequency. Two types of cluster states are found, each undergoing a symmetry-breaking transition towards a related chimera state. We demonstrate that the diffusional coupling is non-essential for these complex dynamics. Furthermore, we investigate localized turbulence and discuss whether it can be categorized as a chimera state. (C) 2015 AIP Publishing LLC.

Chimera states, that is, dynamical regimes characterized by the existence of a symmetry-broken solution where a coherent domain and an incoherent one coexist, have been theoretically demonstrated and numerically found in networks of homogeneously coupled identical oscillators. In this work we experimentally investigate the behavior of a closed and an open chain of electronic circuits with neuron-like spiking dynamics and first neighbor connections. Experimental results show the onset of a regime that we call chimera states with quiescent and synchronous domains, where synchronization coexists with spatially patterned oscillation death. The whole experimental bifurcation scenario, showing how disordered states, synchronization, chimera states with quiescent and synchronous domains, and oscillatory death states emerge as coupling is varied, is presented.

Introduction; 1. Turbulence and dynamical systems; 2. Phenomenology of
turbulence; 3. Reduced models for hydrodynamic turbulence; 4. Turbulence
and coupled map lattices; 5. Turbulence in the complex Ginzburg-Landau
equation; 6. Predictability in high-dimensional systems; 7. Dynamics of
interfaces; 8. Lagrangian chaos; 9. Chaotic diffusion; Appendix A. Hopf
bifurcation; Appendix B. Hamiltonian systems; Appendix C. Characteristic
and generalised Lyapunov exponents; Appendix D. Convective
instabilities; Appendix E. Generalised fractal dimensions and
multifractals; Appendix F. Multiaffine fields; Appendix G. Reduction to
a finite-dimensional dynamical system; Appendix H. Directed percolation.

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.

We show spatial localized structures in degenerate optical parametric oscillators associated with bistability between two homogeneous solutions of the same amplitude and the opposite phase. These localized structures are principally different from the ones analyzed previously in nonlinear optics (including optical parametric oscillators), where bistability between different patterns (most often zero- and nonzero-field states) was at the root.

Populations of coupled oscillators may exhibit two coexisting
subpopulations, one with synchronized oscillations and the other with
unsynchronized oscillations, even though all of the oscillators are
coupled to each other in an equivalent manner. This phenomenon,
discovered about ten years ago in theoretical studies, was then further
characterized and named the chimera state after the Greek mythological
creature made up of different animals. The highly counterintuitive
coexistence of coherent and incoherent oscillations in populations of
identical oscillators, each with an equivalent coupling structure,
inspired great interest and a flurry of theoretical activity. Here we
report on experimental studies of chimera states and their relation to
other synchronization states in populations of coupled chemical
oscillators. Our experiments with coupled Belousov-Zhabotinsky
oscillators and corresponding simulations reveal chimera behaviour that
differs significantly from the behaviour found in theoretical studies of
phase-oscillator models.

We calculate the critical behavior of systems having a multicritical point of a new type, hereafter called a Lifshitz point, which separates ordered phases with k[over →]=0 and k[over →]≠0 along the λ line. For anisotropic systems, the correlation function is described in terms of four critical exponents, whereas for isotropic systems two exponents suffice. Critical exponents are calculated using an ε-type expansion.

We show that the interaction between a homogeneous bifurcation and diffusive instabilities can give rise to a rich variety of subharmonic and superharmonic 1D patterns. Stability domains deduced from pertinent amplitude equations are in good agreement with numerical integration of a symmetric reaction-diffusion system.

The effects of thermal fluctuations on the convective instability are considered. It is shown that the Langevin equations for hydrodynamic fluctuations are equivalent, near the instability, to a model for the crystallization of a fluid in equilibrium. Unlike the usual models, however, the free energy of the present system does not possess terms cubic in the order parameter, and therefore the system undergoes a second-order transition in mean-field theory. The effects of fluctuations on such a model were recently discussed by Brazovskii, who found a first-order transition in three dimensions. A similar argument also leads to a discontinuous transition for the convective model, which behaves two dimensionally for sufficiently large lateral dimensions. The magnitude of the jump is unobservably small, however, because of the weakness of the thermal fluctuations being considered. The relation of the present analysis to the work of Graham and Pleiner is discussed.

In this article we review the conditions for the appearance of localized states in a nonlinear optical system, with particular reference to the liquid crystal light valve (LCLV) experiment. The localized structures here described are of dissipative type; that is, they represent the localized solutions of a pattern-forming system. We discuss their features of stable addressable localized states, and we show that they dispose themselves on the nodes of highly symmetric lattices, as obtained by the introduction of an N-order rotation angle in the optical feedback loop. The stability is lost either on increase of the input light intensity or by the introduction of an extra small angle of rotation. The complex spatio-temporal dynamics that follows is characterized by oscillations in the position of the localized states. We discuss the origin of this permanent dynamics in relation to the non-variational character of the LCLV system, underlining the general character of such complex behaviours of localized states.

A Network of chaotic elements is investigated with the use of globally coupled maps. A simple coding of many attractors with clustering is shown. Through the coding, the attractors are organized so that their change exhibits bifurcation-like phenomena. A precision-dependent tree is constructed which leads to the similarity of our attractor with those of spin-glasses. Hierarchical dynamics is constructed on the tree, which leads to the dynamical change of trees and the temporal change of effective degrees of freedom. By a simple input on a site, we can switch among attractors and tune the strength of chaos. A threshold on a cluster size is found, beyond which a peculiar “posi-nega” switch occurs. Possible application to biological information processing is discussed with the emphasis on the fuzzy switch (chaotic search) and hierarchical code (categorization).

We show that a weak transverse spatial modulation in (2+1) nonlinear Schrödinger-type equation can result in nontrivial dynamics of a radially symmetric soliton. We provide examples of chaotic soliton motion in periodic media both for conservative and dissipative cases. We show that complex dynamics can persist even for soliton sizes greater than the modulation period.

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.

The dynamics of vegetation is formulated in terms of the allometric and structural properties of plants. Within the framework of a general and yet parsimonious approach, we focus on the relationship between the morphology of individual plants and the spatial organization of vegetation populations. So far, in theoretical as well as in field studies, this relationship has received only scant attention. The results reported remedy to this shortcoming. They highlight the importance of the crown/root ratio and demonstrate that the allometric relationship between this ratio and plant development plays an essential part in all matters regarding ecosystems stability under conditions of limited soil (water) resources. This allometry determines the coordinates in parameter space of a critical point that controls the conditions in which the emergence of self-organized biomass distributions is possible. We have quantified this relationship in terms of parameters that are accessible by measurement of individual plant characteristics. It is further demonstrated that, close to criticality, the dynamics of plant populations is given by a variational Swift-Hohenberg equation. The evolution of vegetation in response to increasing aridity, the conditions of gapped pattern formation and the conditions under which desertification takes place are investigated more specifically. It is shown that desertification may occur either as a local desertification process that does not affect pattern morphology in the course of its unfolding or as a gap coarsening process after the emergence of a transitory, deeply gapped pattern regime. Our results amend the commonly held interpretation associating vegetation patterns with a Turing instability. They provide a more unified understanding of vegetation self-organization within the broad context of matter order-disorder transitions.

We study numerically a Swift-Hohenberg equation describing, in the weak dispersion limit, nascent optical bistability with transverse effects. We predict that stable localized structures, and organized clusters of them, may form in the transverse plane. These structures consist of either kinks or dips. The number and spatial distribution of these localized structures are determined by the initial conditions while their peak (bottom) intensity remains essentially constant for fixed values of the system's parameters.