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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.

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... Since this first report of chimeras in systems with purely local coupling [15], many other authors have reported chimeras in systems with only local coupling [16,17,18,19]. Some have studied systems where each node in isolation has complex dynamics (e.g. ...

... and thus the system is described by (17) and ...

... One of the important parameters in a network of Winfree oscillators is the level of heterogeneity in natural frequencies, given here by δ. Varying δ and following both fixed points and periodic solutions of (17) and (18) we obtain Fig. 6. As in an all-to-all coupled network, for small κ and large δ the system has a stable spatially-uniform fixed point, indicated by the solid blue curve. ...

We study networks in the form of a lattice of nodes with a large number of phase oscillators and an auxiliary variable at each node. The only interactions between nodes are nearest-neighbour. The Ott/Antonsen ansatz is used to derive equations for the order parameters of the phase oscillators at each node, resulting in a set of coupled ordinary differential equations. Chimeras are steady states of these equations and we follow them as parameters are varied, determining their stability and bifurcations. In two-dimensional domains, we find that spiral wave chimeras and rotating waves have significantly different properties than those in networks with nonlocal coupling. Chimeras are unusual spatiotemporal patterns in networks of oscillators characterised by having some oscillators synchronised while the remainder are incoherent. Many studies of chimeras involve networks with nonlocal coupling, while a few consider only local coupling. Most of the studies of locally coupled networks show just the results of numerical simulations. We consider networks of phase oscillators with local interactions through an auxiliary field. Using the Ott/Antonsen ansatz we derive and study equations for the dynamics of such networks, determining the stability and bifurcations of chimera states. In several cases we find fundamental differences between solutions in locally coupled and nonlocally coupled networks.

... While initially considered peculiar to networks with weak nonlocal coupling, recent theoretical studies have predicted that chimera-like states can emerge even in systems with purely local coupling. 8,9 Here we report on experimental observations of chimera-like states in a system with local coupling -a coherently-driven Kerr nonlinear optical resonator. 18 We show that artificially engineered discreteness -realised by suitably modulating the coherent driving field -allows for the nonlinear localisation of spatiotemporal complexity, and we demonstrate unprecedented control over the existence, characteristics, and dynamics of the resulting chimera-like states. ...

... The resulting hybrid states, first identified in 2002 by Kuramoto and Battogtokh, 3 have come to be known as chimera states 1 and inspired a burgeoning field of research. [4][5][6][7][8][9][10][11][12][13][14][15][16][17] Chimeras were first identified in studies of coupled oscillators, but similar coexistence of coherent and incoherent domains has also been identified for a range of other systems; the term chimera-like state has been coined to generalize the concept to coupled (discrete) or extended (continuous) dynamical systems. 9 Studies have shown that chimeras and chimera-like states are ubiquitous, manifesting themselves in a variety of nonlinear systems, including human brain networks. ...

... [4][5][6][7][8][9][10][11][12][13][14][15][16][17] Chimeras were first identified in studies of coupled oscillators, but similar coexistence of coherent and incoherent domains has also been identified for a range of other systems; the term chimera-like state has been coined to generalize the concept to coupled (discrete) or extended (continuous) dynamical systems. 9 Studies have shown that chimeras and chimera-like states are ubiquitous, manifesting themselves in a variety of nonlinear systems, including human brain networks. 25 Experimental observations have been reported, e.g., in chemical, 11,16 optical, 10,14,15,17 electronic, 12 and mechanical 13 systems. ...

Chimera states -- named after the mythical beast with a lion's head, a goat's body, and a dragon's tail -- correspond to spatiotemporal patterns characterised by the coexistence of coherent and incoherent domains in coupled systems. They were first identified in 2002 in theoretical studies of spatially extended networks of Stuart-Landau oscillators, and have been subject to extensive theoretical and experimental research ever since. While initially considered peculiar to networks with weak nonlocal coupling, recent theoretical studies have predicted that chimera-like states can emerge even in systems with purely local coupling. Here we report on the first experimental observations of chimera-like states in a system with local coupling -- a coherently-driven Kerr nonlinear optical resonator. We show that artificially engineered discreteness -- realised by suitably modulating the coherent driving field -- allows for the nonlinear localisation of spatiotemporal complexity, and we demonstrate unprecedented control over the existence, characteristics, and dynamics of the resulting chimera-like states. Moreover, using ultrafast time lens imaging, we resolve the chimeras' picosecond-scale internal structure in real time.

... 28 Experimentally, chimera states have been reported in a chemical system, 29 optoelectronic oscillators, 30,31 and a mechanical oscillator network. 32 Recently, it has been established that chimera-like and chimera states can occur in systems with local (nearest neighbors) [33][34][35][36] or global coupling. [37][38][39] The aim of this paper is to investigate the formation of chimera states in an array of Duffing oscillators chain coupled to their nearest neighbors. ...

... Hence, in this region, one expects to observe chimera states, since the coupling prevents the most favorable state and invades the less favorable one. 34 The coupling induces a periodic poten- tial over the dynamics of the interface between domains. This periodic potential is well known as the Peierls-Nabarro potential. ...

... A rigorous way of characterizing the complexity of chimera states is through the use of the Lyapunov spectrum. 34,35 This spectrum provides information about permanent dynamic with exponential sensitivity to initial conditions. 47 When the largest Lyapunov exponent is nega- tive, the system has a stationary equilibrium, such as uniform or pattern states. ...

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.

... Driven oscillators exhibit a plethora of behaviors, including chaos [1][2][3], synchronization [4], and chimeras [5][6][7]. In the case of magnetic media, the magnetization exhibits damped precessions around a net magnetic field. ...

... A positive exponent demonstrates the presence of chaos in the system. Since this type of VCMA-driven oscillator is proposed as a unit of magnetic networks, we expect that the found multistability may play a role in forming collective behaviors, such as front propagation and chimeras [5][6][7]. ...

The control of magnetization dynamics has allowed numerous technological applications. Magnetization dynamics can be excited by, e.g., alternating magnetic fields, charge and spin currents, and a voltage-induced control of interfacial properties. An example of the last mechanism is the voltage-controlled magnetic anisotropy effect, which can induce magnetization precessions and switchings with low-power consumption. Time-dependent voltage-controlled magnetic anisotropy can induce complex dynamic behaviors for magnetization. This work studies the magnetization dynamics of a single magnetic nano-oscillator forced with a time-dependent voltage-controlled magnetic anisotropy. Unexpectedly, the oscillator displays multistable regimes, i.e., distinct initial conditions evolve towards different oscillatory states. When voltage is changed the oscillatory state exhibits period-doubling route to chaos. The chaotic behavior is numerically demonstrated by the determination of the largest Lyapunov exponent.

... In experimental studies, chimera states have been generated successfully in coupled systems such as optical oscillators [17] , chemical oscillators [18,19] , mechanical oscillators [20] , electronic oscillators [21] , electrochemical systems [22] and lasers [23] . With these studies, the conditions for generating chimera states as adopted in the seminal works have been largely relaxed [24][25][26][27][28][29] , and the concept of chimera states has been largely broadened and generalized [30][31][32][33][34][35][36] . 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] . ...

... With these studies, the conditions for generating chimera states as adopted in the seminal works have been largely relaxed [24][25][26][27][28][29] , and the concept of chimera states has been largely broadened and generalized [30][31][32][33][34][35][36] . 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] . ...

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.

... Since its discovery, chimera state has inspired extensive theoretical and experimental studies during the past two decades, with the systems investigated ranging from physical to chemical and to biological systems . With these studies, the strict conditions for generating chimera states as adopted in the seminal works have been largely relaxed [28][29][30][31][32][33], and the concept of chimera state has been largely broadened and generalized [34][35][36][37][38][39][40]. 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 [4,28,30] or local couplings [31,32,36,38,[41][42][43][44]; and, besides regular networks, a variety of chimera-like states have been reported and studied in networks of complex structures [45][46][47][48][49][50][51]. ...

... With these studies, the strict conditions for generating chimera states as adopted in the seminal works have been largely relaxed [28][29][30][31][32][33], and the concept of chimera state has been largely broadened and generalized [34][35][36][37][38][39][40]. 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 [4,28,30] or local couplings [31,32,36,38,[41][42][43][44]; and, besides regular networks, a variety of chimera-like states have been reported and studied in networks of complex structures [45][46][47][48][49][50][51]. In particular, chimera-like states have been observed in complex network of coupled neurons [49,50], and are regarded as having important implications to the neuronal * Corresponding author. ...

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.

... These intrigued states correspond to breaking symmetry solution without bistability. Extension of chimera states in bistable systems was proposed in several coupled systems [26][27][28][29] , which was usually denominated chimera-like states. Depending on the initial condition, these states have different size and exhibit a family of solutions with the coexistence of coherent and incoherent domains. ...

... Initially, even though the existence of chimera states was attributed to the nonlocal nature of the coupling [24] . Subsequently, chimera states have been observed in systems that are coupled globally [36][37][38] , and locally [26][27][28]39,40,21] . In all these studies, domains remain motionless. ...

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.

... Coupled oscillators are primordial to understanding the propagation of waves in continuous and discrete media [1]. An out-of-equilibrium coupled oscillators chain, that is coupled oscillators under the influence of injection and dissipation of energy, shows a variety of different phenomena [2][3][4] such as phase turbulence [2], synchronization [3], defects turbulence [5], random occurrence of coherence events [6], defect-mediated turbulence [7], spatiotemporal intermittency [8], quasiperiodicity in extended systems [9], chimera states [10,11], and particle-type solutions [12]. Depending on the nonlinearity, the oscillators may have more than one equilibrium, a phenomenon known as multistability. ...

... Introducing the above ansatz (10) in Eq. (9), considering the linear leading terms in U , and averaging in a period of the front propagation, after straightforward calculations, we get ...

Coupled oscillators can exhibit complex spatiotemporal dynamics. Here, we study the propagation of nonlinear waves into an unstable state in dissipative coupled oscillators. To this, we consider the dissipative Frenkel–Kontorova model, which accounts for a chain of coupled pendulums or Josephson junctions and coupling superconducting quantum interference devices. As a function of the dissipation parameter, the front that links the stable and unstable state is characterized by having a transition from monotonous to non-monotonous profile. In the conservative limit, these traveling nonlinear waves are unstable as a consequence of the energy released in the propagation. Traveling waves into unstable states are peculiar of dissipative coupling systems. When the coupling and the dissipation parameter are increased, the average front speed decreases. Based on an averaging method, we analytically determine the front speed. Numerical simulations show a quite fair agreement with the theoretical predictions. To show that our results are generic, we analyze a chain of coupled logistic equations. This model presents the predicted dynamics, opening the door to investigate a wider class of systems.

... (18) pointing out the similarity between the classical Kuramoto model and the spatially extended systems (1) and (5). On the other hand, there are several differences making the Ott-Antonsen equation (26) much simpler than its spatially extended counterpart (18). First, the integral operator H is a rank-1 operator whereas the operator G has, in general, an infinite-dimensional range. ...

... Another dynamical mechanism leading to the emergence of chimera states in locally coupled oscillatory systems was described in [26]. It relies on the inherent bistability of the oscillators, when each of them can settle onto either a fixed point or a stable periodic orbit. ...

Chimera states are self-organized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, Ott-Antonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed.

... Another avenue of research on chimera states has focused on understanding the network topologies and coupling mechanisms that can facilitate their emergence. Although chimera states have been proved to arise in 1D rings where every node is connected to its two neighbors (a condition named local coupling [27,28]) as well in the "opposite" case of a complete network where each node is connected to all the other ones (global coupling [29,30]), the nonlocal coupling configuration proved to be particularly important for the onset of chimera states. Indeed, a large amount of literature has demonstrated that chimera states are commonly observed within such setting [16,19]. ...

Chimera states are dynamical states where regions of synchronous trajectories coexist with incoherent ones. A significant amount of research has been devoted to studying chimera states in systems of identical oscillators, nonlocally coupled through pairwise interactions. Nevertheless, there is increasing evidence, also supported by available data, that complex systems are composed of multiple units experiencing many-body interactions that can be modeled by using higher-order structures beyond the paradigm of classic pairwise networks. In this work we investigate whether phase chimera states appear in this framework, by focusing on a topology solely involving many-body, nonlocal, and nonregular interactions, hereby named nonlocal d-hyperring, (d+1) being the order of the interactions. We present the theory by using the paradigmatic Stuart-Landau oscillators as node dynamics, and we show that phase chimera states emerge in a variety of structures and with different coupling functions. For comparison, we show that, when higher-order interactions are “flattened” to pairwise ones, the chimera behavior is weaker and more elusive.

... According to the literature, the first observation of chimera state was made by Kuramoto and Battogtokh in the complex Ginzburg-Landau equation with a weak nonlocal coupling (Kuramoto and Battogtokh, 2002). Subsequently, it has been investigated in a variety of models like phase oscillators, chemical oscillators, planar oscillators and many other different types of networks (Kuramoto and Battogtokh, 2002;Majhi et al., 2016;Nkomo et al., 2016;Clerc et al., 2016). Chimera states are well known for nonlocally coupled systems, but recently they also have been found in feedback-delayed oscillators and in globally coupled networks (Schmidt et al., 2014;Dudkowski et al., 2014;Gopal et al., 2014;Schmidt and Krischer, 2015). ...

The main objective of this thesis is the investigation and the control of the dynamics of networks. The resolution of this problem required three key concepts that we have adopted: the dead zone, ampliﬁcation and optimal control. Due to its numerous applications in telecommunication as well as in human relations, the concept of dead zone could be a better way to model the coupling between two or more systems. For example, in systems using the magnetic ﬁeld for the coupling, the magnetic ﬁeld may vary with the distance between systems, thus making the coupling between these systems intermittent. The studies carried out in this thesis show that this dead zone not only improves the transition to synchronization in the case of two coupled systems, but can also allow the transition from one behaviour to another (chimera states, multi-chimera states, clusters, synchronization). An implementation in Pspice using electronic components allowed us to design a circuit modeling this dead zone and thus showed the feasibility of the synchronization obtained by the numerical resolution in MATLAB. The existence of synchronization in certain devices such as gear systems, pulley-belt systems improves eﬃciency. The improvement of the speed in these devices is done through the transmission ratio considered as the ampliﬁcation coeﬃcient in this work. Thus, the concept of ampliﬁcation is a key concept that is highly sought-after in many electronic, mechanical and even electrical systems. Obtaining this ampliﬁcation in linear systems is more obvious. However, most real systems are non-linear and therefore, to highlight this ampliﬁcation becomes a titanic task. In this work, the study of the inﬂuence of ampliﬁcation in the case of two similar coupled systems shows that it is possible to switch from phase-ﬂip to phase synchronisation or phase-lock. In the case of networks, this ampliﬁcation also makes it possible to switch from desynchronization to synchronization via clusters, splays etc. The stability of synchronization can be accessed via the master stability function. This convenient tool allows for treating the node dynamics and the network topology in two separated steps, and, thus, allows for a quite general treatment of diﬀerent network topologies. In this thesis, the network formed by the Rössler systems used gave us a type III MSF allowing us to highlight a synchronization zone and two desynchronization zones. Later, the application of the Hamilton-Bellman-Jacobi method enabled us to
manufacture an optimal controller that minimizes the transition time to a precise state based on the cost administered. Thus, the resulting controller allowed us to achieve synchronization with minimal time.

... In the synchronized group, all individual units stay near their common attractor, whereas the desynchronized group units approach the attractor only occasionally, taking large detours around it for the rest of the time. Another mechanism supporting the formation of chimera states is found in the case of bistable dynamical units with two coexisting qualitatively different attractors (e.g., a fixed point and a chaotic attractor) [25][26][27][28] . Synchronized and desynchronized groups are then identified by the fact near which attractor the unit moves. ...

Nonlinear systems possessing nonattracting chaotic sets, such as chaotic saddles, embedded in their state space may oscillate chaotically for a transient time before eventually transitioning into some stable attractor. We show that these systems, when networked with nonlocal coupling in a ring, are capable of forming chimera states, in which one subset of the units oscillates periodically in a synchronized state forming the coherent domain, while the complementary subset oscillates chaotically in the neighborhood of the chaotic saddle constituting the incoherent domain. We find two distinct transient chimera states distinguished by their abrupt or gradual termination. We analyze the lifetime of both chimera states, unraveling their dependence on coupling range and size. We find an optimal value for the coupling range yielding the longest lifetime for the chimera states. Moreover, we implement transversal stability analysis to demonstrate that the synchronized state is asymptotically stable for network configurations studied here.

... With these studies, the conditions of generating chimera states have been broaden and generalized. Particularly, it is believed for a long time that chimera states occur only in the nonlocal coupled systems [1, 2], but recent works show that the nonlocal coupling is not necessary at all and chimera states can be observed in systems with different coupling topologies [28], including global coupling [29,30] and local couping [31][32][33][34][35][36]. Further more, the concept of chimera states has been generalized and extended largely. ...

Chimera states are usually observed in oscillator systems with nonlocal couplings. Such a nonlocal coupling arises typically as oscillators are coupled via an external environment which changes so fast that it could be eliminated adiabatically. Here we for the first time report the existence of spiral wave chimera states in large populations of Stuart-Landau oscillators coupled via a slowly changing diffusive environment under which the adiabatic approximation breaks down. The transition from spiral wave chimeras to spiral waves with the smooth core and to unstable spiral wave chimeras as functions of the system parameters are exploited. The phenomenological mechanism for explaining the formation of spiral wave chimeras is also proposed. The existence of spiral wave chimeras and the underlying mechanism are further confirmed in a three-component FitzHugh-Nagumo system with the similar environmental coupling scheme. Our results provide important hints to seek chimera patterns in both laboratory and realistic chemical or biological systems.

... for different global (Kaneko, 1990), weak non-local (Kuramoto and Battogtokh, 2002) and local couplings (Clerc et al., 2016). The pioneering study by Hart et al. (2016) experimentally witnessed the existence of chimera in system of four globally coupled chaotic optoelectronic oscillators which is the minimal network of oscillators required to support the state. ...

Nonlinear phenomena emerging from the coupled behaviour of a network of oscillators have attracted considerable research attention over the years, of which, symmetry-breaking phenomena such as chimera, clustering, and weak chimera along with amplitude death and phase-flip bifurcation are some noteworthy examples. Understanding these global dynamical behaviour exhibited by a network of coupled oscillators has been a topic of extensive research in many fields of science and engineering. Various factors govern the resulting dynamical behaviour of such networks, including the number of oscillators and the coupling schemes between them. Although these factors are seldom significant in a large population of oscillators, a small change in these factors can drastically affect the global behaviour in small populations. Therefore, recent studies focus on the discovery of such phenomena in a system with minimum number of oscillators, as their insights can equally be applied to the dynamics of large, spatially extended systems. Moreover, the individual occurrence of the aforementioned phenomenon has been a focus of most studies in the past. Therefore, a collective existence of all these phenomena in single system is rarely observed.
The primary aim of the present project report is to provide the experimental existence of several phenomena, including symmetry-breaking phenomena such as clustering, weak chimera, and chimera along with theoretically reported states of in-phase and anti-phase chimera and other states of synchronization and oscillation quenching, in a network of two or more coupled candle-flame oscillators, exhibiting limit cycle oscillations in the uncoupled state. In the case of two candle-flame oscillators, we observe the transition from in-phase synchronization (IP) to the state of anti-phase synchronization (AP) via the intermediate state of amplitude death (AD) as the distance between the oscillators is increased. As the strength of coupling between the oscillators is increased by increasing the number of candles in an oscillator, we report a decrease in the span of AD region between the states of in-phase and anti-phase oscillations, leading up to the point of phase-flip bifurcation (PFB). Although theoretical research has postulated the coexistence of AD and PFB upon variation of different control parameters, such an occurrence has not been reported in practical systems. We provide the first experimental evidence of the coexistence of AD and PFB in a physical system, comprising of a coupled pair of candle-flame oscillators.
The investigation is further extended to present the first experimental study highlighting the existence of several states of symmetry-breaking coupled dynamics of an array of four non-locally coupled candle-flame oscillators, as the distance between them is varied. We also provide an experimental confirmation of different dynamical states, including in-phase, anti-phase, and multi-phase weak chimera. Hence by varying the coupling structure in a network of four oscillators, we observe the emergence of phase locking, amplitude quenching, and chimera states. In many practical systems, one (or more) among these states is considered undesirable. Therefore, we bring out the possibility of evading the undesirable occurrence of phase-locking, amplitude quenching, and chimera states by smart control of the coupling parameters.
In the subsequent analysis of the project report, we provide an experimental investigation on the effect of factors, such as number, coupling strength and topology of the oscillators, on the coupled behaviour of a minimal network of candle-flame oscillators. We found that when the oscillators are strongly coupled, the global behaviour of the network exhibits in-phase synchrony and amplitude death, which remains independent of the number and the topology of oscillators. However, when they are weakly coupled, the global behaviour of the network exhibits the occurrence of multiple stable states which alternately switch in time. In addition to such states of clustering, chimera, and weak chimera, we report the first experimental evidence of partial amplitude death in a network of candle-flame oscillators. We also show that the networks with closed-loop topology tend to hold global synchronization for longer duration as compared to those with open-loop topology.
Finally, the experimental results obtained from coupled pair and quadruplet of candle-flame oscillators are compared with a generic mathematical model of time-delay coupled Stuart-Landau oscillators. We observe that the dynamics observed in coupled Stuart-Landau oscillators show high similarity to those observed in candle-flame oscillators. Thus, we understood that coupled candle-flame oscillators exhibit a variety of dynamical behaviour depending on the number of oscillators, the coupling strength, and the topological arrangement of the oscillators. The results obtained can be used as a benchmark for several theoretical studies on coupled oscillators using generic mathematical oscillators. Further, due to the simple and generic nature of these oscillators, we can conclude that the results obtained from candle-flame oscillators can be extended to various other fields of engineering including biological, ecological, mechanical and electrochemical systems.

... Chimera states refer to a type of symmetry-breaking dynamical states in which oscillators spontaneously organize into coexisting domains with dramatically different behaviors, i.e., coherent and incoherent oscillations [1]. Chimera states were first numerically found in a nonlocally coupled complex Ginzburg-Landau equation [2] and, since then, they have been studied intensively in the past decade [3][4][5][6][7][8][9][10][11][12]. It has been found that chimera states can be built in neural oscillators [13,14], time-discrete maps [15], and systems allowing for only equilibrium [16]. ...

In a chimera state, domains composed of synchronized oscillators coexist with ones composed of desynchronized oscillators. It is a common view that, in a chimera state, oscillators in coherent domains always share the same mean phase velocity. However, recent studies have suggested that oscillators in different coherent domains may have different mean phase velocities. In this work, we study a ring of nonlocally coupled Brusselators. We find a two-frequency chimera state with mixed phase regularities in which Brusselators in adjacent coherent domains oscillate at different velocities. Moreover, Brusselators in coherent domains with higher mean phase velocity are nearly in phase. In contrast, Brusselators in coherent domains with lower mean phase velocity are randomly partitioned into two groups in antiphase. We find that the local mean fields in these two types of coherent domains perform different dynamics. Based on local mean fields, we provide an explanation for the formation of this type of chimera state. Furthermore, the stability diagrams of the two-frequency chimera state with mixed phase regularities are investigated in different parameter planes

... In addition, the study of chimera states currently attracts much interest (for reviews see [21,22]). Chimera states have been found in networks of nonlocally coupled phase oscillators [23,24], in systems with local [25][26][27][28] and global [29][30][31][32][33][34] interactions, and in networks of time-discrete maps [35][36][37][38]. ese states have been investigated in a diversity of contexts [39][40][41][42][43][44][45][46][47]. ...

We show that dynamical clustering, where a system segregates into distinguishable subsets of synchronized elements, and chimera states, where differentiated subsets of synchronized and desynchronized elements coexist, can emerge in networks of globally coupled robust-chaos oscillators. We describe the collective behavior of a model of globally coupled robust-chaos maps in terms of statistical quantities and characterize clusters, chimera states, synchronization, and incoherence on the space of parameters of the system. We employ the analogy between the local dynamics of a system of globally coupled maps with the response dynamics of a single driven map. We interpret the occurrence of clusters and chimeras in a globally coupled system of robust-chaos maps in terms of windows of periodicity and multistability induced by a drive on the local robust-chaos map. Our results show that robust-chaos dynamics does not limit the formation of cluster and chimera states in networks of coupled systems, as it had been previously conjectured.

... As a result of this mechanism, one expects to find a family of localized solutions organized through a snaking type bifurcation diagram [30] . The discrete counterpart of the snaking type bifurcation diagram for chimera states has also been reported [15][16][17][18] . ...

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.

... It was shown that the chimera state is formed in all three networks for a specific range of parameters. The requisites for the occurrence of chimera state in local coupling was investigated by Clerc et al. [296] in the network of nonlinear oscillators without time scales. This model shows the coexistence of a uniform steady-state and an incoherent state, which is the result of the homoclinic bifurcation. ...

Chimeras are this year coming of age since they were first observed by Kuramoto and Battogtokh in 2002 in a one-dimensional network of complex Ginzburg–Landau equations. What started as an observation of a peculiar coexistence of synchronized and desynchronized states, almost two decades latter turned out to be an important new paradigm of nonlinear dynamics at the interface of physical and life sciences. Chimeras have been observed in uni-hemispheric sleep of aquatic mammals and migratory birds, in electrocorticographic recordings of epileptic seizures, and in neural bump states that are central to the coding of working memory and visual orientation. Chimera states have also been observed experimentally in physical systems, for example in liquid crystal light modulators, and they have been linked to power grids outages and optomechanics. Here we present a major review of chimeras, dedicated to all aspects of their theoretical and practical existence. We cover different dynamical systems in which chimera states have been observed, different types of chimeras, and different mathematical methods used for their analysis. We also review the importance of network structure for the emergence of chimeras, as well as different schemes aimed at controlling the symmetry breaking spatiotemporal pattern. We conclude by outlining open challenges and opportunities for future research entailing chimeras.

... Later these states have been found in diverse oscillatory models with nonlocal interaction such as the FitzHugh-Nagumo, the van der Pol, the Stuart-Landau, the Hindmarsh-Rose and others [48,33,28,38,41,16,31,26,57,42]. However, the chimera structures have been discovered in systems with purely local coupling [23,10,47,25,44] and with global one [56,41]. The chimeras have been found in real experiments [14,48,51]. ...

The present work is devoted to the detailed quantification of the transition from spiral waves to spiral wave chimeras in a network of self-sustained oscillators with two-dimensional geometry. The basic elements of the networks are the van der Pol oscillator and the FitzHugh-Nagumo neuron. Both models are in the regime of relaxation oscillations. We analyze the regime by using the indices of local sensitivity which enables us to evaluate the sensitivity of each individual oscillator at finite time. Spi-ral waves are observed in both lattices when the interaction between elements have the local character. The dynamics of all the elements is regular. There are no high-sensitive regions. We have discovered that when the coupling becomes nonlocal, the features of the systems significantly changes. The oscillation regime of the spiral wave center element switches to chaotic one. Besides this, a region with high sensitivity occurs around this oscillator. Moreover, we show that the latter expands in space with elongation of the coupling range. As a result, an incoherence cluster of the spiral wave chimera is formed exactly within this high-sensitive area. Formation of this cluster is accompanied by the sharp increase in values of the maximal Lyapunov exponent to the positive region. Furthermore, we explore that the system can even switch to hyperchaotic regime, when several Lyapunov exponents becomes positive.

... Recently, chimera states have been numerically observed in a Duffing oscillators chain coupled to nearest neighbors [36] . The local coupling prevents the incoherent domain from invading the coherent one, allowing concurrently the existence of a family of chimera states, which are organized by a homoclinic snaking-like bifurcation diagram [37,38] . The effect of the local coupling on domain dynamics can be modeled considering the Peierls-Nabarro potential [39][40][41] . ...

Arrays of oscillators driven out-of-equilibrium can support the coexistence between coherent and incoherent domains that have become known as chimera states. Recently, we have reported such an intriguing self-organization phenomenon in a chain of locally coupled Duffing oscillators. Based on this prototype model, we reveal a generalization of chimera states corresponding to the coexistence of incoherent domains. These freak states emerge through a bifurcation in which the coherent domain of an existing chimera state experiences an instability giving rise to another incoherent state. Using Lyapunov exponents and Fourier analysis allows us to characterize the dynamical nature of these extended solutions. Taking the Kuramoto order parameter, we were able to compute the bifurcation diagram of freak chimera states.

... This phenomenon is intensively studied in recent years (see [13][14][15][16][17][18] ). However, the chimera cluster structures were also found in some models of ensembles with local coupling [19][20][21] . The effects of different topologies of coupling on the chimera state formation is observed in the review [22] . ...

We study impacts of external harmonic forces on chimera states in an ensemble of chaotic Rössler oscillators with nonlocal interaction. The main attention is paid to control the spatial structure by applying a targeted localized excitation on an incoherence cluster. This influence on a phase chimera enables us to eliminate the incoherence cluster and to realize the regime with a piecewise smooth spatial profile. The mechanism of elimination of the incoherence cluster of the phase chimera consists in-phase synchronization of all oscillators within the region of influence of the external force. This phenomenon is observed for a sufficiently wide range of the external force frequency, especially when its value is less than the natural frequency. Increasing the external force amplitude can lead to two scenarios depending on the dynamics of individual oscillators. In the case of regular dynamics, a strong force induces another type of the incoherence cluster within the region of the external force influence. The oscillator dynamics within this region becomes chaotic. Thus, the features of this cluster are similar to those for the incoherence cluster of an amplitude chimera. When the dynamics is chaotic, the force can cause the system to switch to the regime of a metastable spatial distribution with a qualitatively different character at different time intervals. It is impossible to eliminate the incoherence cluster of the amplitude chimera by means of the localized harmonic influence for any values of its parameters. The destruction of the amplitude chimera structure under the influence of the external force leads either to the intermittent regime or to inducing the stable incoherence cluster.

... Further experimental studies observed the existence of chimera in mechanical oscillators, such as coupled pendula [22], metronomes [13], electrochemical oscillators [5,23], electronic oscillators [24], optical oscillators [8] and in thermoacoustic oscillators [25]. Chimera states have also been predicted for different global [28], weak nonlocal [3], and local couplings [29]. The pioneering study by Hart et al. [8] experimentally witnessed the existence of chimera in four globally coupled chaotic optoelectronic oscillators. ...

Synchronization and chimera are examples of collective behavior observed in an ensemble of coupled nonlinear oscillators. Recent studies have focused on their discovery in systems with least possible number of oscillators. Here we present an experimental study revealing the synchronization route to weak chimera via quenching, clustering, and chimera states in a single system of four coupled candle-flame oscillators. We further report the discovery of multiphase weak chimera along with experimental evidence of the theoretically predicted states of in-phase chimera and antiphase chimera.

... To determine the type of chimera state and for estimating the synchronization of quasi-periodic oscillations, the degree of synchronization is evaluated by Lyapunov spectra [12,22] or by calculating the complex order parameter [23]. However, these indicators do not contain information about the complex structure of synchronization, which, in the case of chimeric synchronization, consists of a set of different individual synchronous patterns. ...

This paper presents a new method for evaluating the synchronization of quasi-periodic oscillations of two oscillators, termed “chimeric synchronization”. The family of metrics is proposed to create a neural network information converter based on a network of pulsed oscillators. In addition to transforming input information from digital to analogue, the converter can perform information processing after training the network by selecting control parameters. In the proposed neural network scheme, the data arrives at the input layer in the form of current levels of the oscillators and is converted into a set of non-repeating states of the chimeric synchronization of the output oscillator. By modelling a thermally coupled VO2-oscillator circuit, the network setup is demonstrated through the selection of coupling strength, power supply levels, and the synchronization efficiency parameter. The distribution of solutions depending on the operating mode of the oscillators, sub-threshold mode, or generation mode are revealed. Technological approaches for the implementation of a neural network information converter are proposed, and examples of its application for image filtering are demonstrated. The proposed method helps to significantly expand the capabilities of neuromorphic and logical devices based on synchronization effects.

... However, it also describes the motion of charged particles in a periodic crystal in the presence of an electric field, and arises in other spatially forced physical systems as well. In particular, the potential has been used to explain the existence of localized spatiotemporally chaotic solutions in both continuous [31] and discrete media [27,51]. The second term on the right-hand side of Eq. (8) corresponds to additive time-dependent forcing with amplitude ...

Driven dissipative many-body systems are described by differential equations for macroscopic variables which include fluctuations that account for ignored microscopic variables. Here, we investigate the effect of deterministic fluctuations, drawn from a system in a state of phase turbulence, on front dynamics. We show that despite these fluctuations a front may remain pinned, in contrast to fronts in systems with Gaussian white noise fluctuations, and explore the pinning-depinning transition. In the deterministic case, this transition is found to be robust but its location in parameter space is complex, generating a fractal-like structure. We describe this transition by deriving an equation for the front position, which takes the form of an overdamped system with a ratchet potential and chaotic forcing; this equation can, in turn, be transformed into a linear parametrically driven oscillator with a chaotically oscillating frequency. The resulting description provides an unambiguous characterization of the pinning-depinning transition in parameter space. A similar calculation for noise-driven front propagation shows that the pinning-depinning transition is washed out.

... Several authors have dealt with the classification of chimera states [18], including a direction with evaluating Lyapunov spectra [19,20], analyzing the instantaneous distribution of the amplitudes of the ensemble elements [21,22] and complex order parameters [23]. The terms of the amplitude and phase chimeras in an ensemble of chaotic oscillators are proposed in [19,21]. ...

The paper presents a new method for the classification of chimera states, which characterizes the synchronization of two coupled oscillators more accurately. As an example of method application, a neural network information converter based on a network of pulsed oscillators is used, which can convert input information from digital to analogue type and perform information processing after training the network by selecting control parameters. In the proposed neural network scheme, the data arrives at the input layer in the form of current levels of the oscillators and is converted into a set of non-repeating states of the chimera synchronization of the output oscillator. By modelling a thermally coupled VO2-oscillator circuit, the network setup is demonstrated through the selection of coupling strength, power supply levels and the synchronization efficiency parameter. The distribution of solutions depending on the operating mode of the oscillators, prethreshold mode or generation mode are revealed. Technological approaches for the implementation of a neural network information converter are proposed, and examples of its application for image filtering are demonstrated. The proposed method for the classification of chimera states helps significantly expand the capabilities of neuromorphic and logical devices based on synchronization effects.

... Initially it was thought that chimeras can be observed only in networks of non-locally coupled oscillators [1]. Later studies revealed that besides non-locally connected networks [3,4,[13][14][15][16], these states can be found in local [17][18][19] as well as in global [6,20] coupling topologies. Chimera patterns are analyzed in networks of Logistic maps with hierarchical connectivities [21]. ...

Complex networks of coupled maps of matrices (NCMM) are investigated in this paper. It is shown that a NCMM can evolve into two different steady states—the quiet state or the state of divergence. It appears that chimera states of spatiotemporal divergence do exist in the regions around the boundary lines separating these two steady states. It is demonstrated that digital image entropy can be used as an effective measure for the visualization of these regions of chimera states in different networks (regular, feed-forward, random, and small-world NCMM).

... Another example is the chimera identified in a globally coupled population of oscillators with delay feedback coupling [23]. Moreover, while initially found in nonlocally coupled oscillators, chimeras have also been noted in strictly locally coupled models of phase-amplitude oscillators in both networks [24][25][26][27][28] and continuous systems [29]. Our phase-phase oscillator model constitutes a particularly simple totally symmetric system (i.e., it is both vertex transitive and edge transitive in the language of graph theory) which exhibits chimeras. ...

Recent research has led to the discovery of fundamental new phenomena in network synchronization, including chimera states, explosive synchronization, and asymmetry-induced synchronization. Each of these phenomena has thus far been observed only in systems designed to exhibit that one phenomenon, which raises the questions of whether they are mutually compatible and, if so, under what conditions they co-occur. Here, we introduce a class of remarkably simple oscillator networks that concurrently exhibit all of these phenomena. The dynamical units consist of pairs of nonidentical phase oscillators, which we refer to as Janus oscillators by analogy with Janus particles and the mythological figure from which their name is derived. In contrast to previous studies, these networks exhibit (i) explosive synchronization with identical oscillators; (ii) extreme multistability of chimera states, including traveling, intermittent, and bouncing chimeras; and (iii) asymmetry-induced synchronization in which synchronization is promoted by random oscillator heterogeneity. These networks also exhibit the previously unobserved possibility of inverted synchronization transitions, in which a transition to a more synchronous state is induced by a reduction rather than an increase in the coupling strength. These various phenomena are shown to emerge under rather parsimonious conditions and even in locally connected ring topologies, which has the potential to facilitate their use to control and manipulate synchronization in experiments.

... Another example is the chimera identified in a globally coupled population of oscillators with delay feedback coupling [23]. Moreover, while initially found in nonlocally-coupled oscillators, chimeras have also been noted in strictly locally coupled-models of phase-amplitude oscillators in both networks [27][28][29][30][31] and continuous systems [32]. Our phase-phase oscillator model constitutes a particularly simple totally symmetric system (i.e., both vertex and edge transitive in the language of graph theory) which exhibits chimeras. ...

Recent research has led to the discovery of fundamental new phenomena in network synchronization, including chimera states, explosive synchronization, and asymmetry-induced synchronization. Each of these phenomena has thus far been observed only in systems designed to exhibit that one phenomenon, which raises the questions of whether they are mutually compatible and of what the required conditions really are. Here, we introduce a class of remarkably simple oscillator networks that concurrently exhibit all of these phenomena, thus ruling out previously assumed conditions. The dynamical units consist of pairs of non-identical phase oscillators, which we refer to as Janus oscillators by analogy with Janus particles and the mythological figure from which their name is derived. In contrast to previous studies, these networks exhibit: i) explosive synchronization in the absence of any correlation between the network structure and the oscillator's frequencies; ii) extreme multi-stability of chimera states, including traveling, intermittent, and bouncing chimeras; and iii) asymmetry-induced synchronization in which synchronization is promoted by random oscillator heterogeneity. These networks also exhibit the previously unobserved possibility of inverted synchronization transitions, in which transition to a more synchronous state is induced by a reduction rather than increase in coupling strength. These various phenomena are shown to emerge under rather parsimonious conditions, and even in locally connected ring topologies, which has the potential to facilitate their use to control and manipulate synchronization in experiments.

... Such states are typical of ensembles of nonlinear oscillators with nonlocal coupling [1,2,10,15,17,29]. In certain cases, they can be observed in ensembles with global [5,23,28] and local [6,11,24] interactions between elements. The main feature of chimera states is the simultaneous presence of clusters with different behavior (coherent and incoherent) in an ensemble of coupled identical elements. ...

The influence of noise on chimera states arising in ensembles of nonlocally coupled chaotic maps is studied. There are two types of chimera structures that can be obtained in such ensembles: phase and amplitude chimera states. In this work, a series of numerical experiments is carried out to uncover the impact of noise on both types of chimeras. The noise influence on a chimera state in the regime of periodic dynamics results in the transition to chaotic dynamics. At the same time, the transformation of incoherence clusters of the phase chimera to incoherence clusters of the amplitude chimera occurs. Moreover, it is established that the noise impact may result in the appearance of a cluster with incoherent behavior in the middle of a coherence cluster.

... The bifurcation diagram of the dissipative DB amplitudes as a function of Ω, shown as the branches formed by the red circles in Fig. 2, resembles the snaking bifurcation curves for spatially localized states in the Swift-Hohenberg equation [43,44]; however, snaking bifurcation curves also occur in discrete problems [45][46][47]. Interestingly, snaking bifurcation diagrams for chimera states have been obtained in the 1D extended Bogdanov-Takens lattice [48]. ...

A SQUID (Superconducting QUantum Interference Device) metamaterial on a Lieb lattice with nearest-neighbor coupling supports simultaneously stable dissipative breather families which are generated through a delicate balance of input power and intrinsic losses. Breather multistability is possible due to the peculiar snaking flux ampitude - frequency curve of single dissipative-driven SQUIDs, which for relatively high sinusoidal flux field amplitudes exhibits several stable and unstable solutions in a narrow frequency band around resonance. These breathers are very weakly interacting with each other, while multistability regimes with different number of simultaneously stable breathers persist for substantial intervals of frequency, flux field amplitude, and coupling coefficients. Moreover, the emergence of chimera states as well as novel temporally chaotic states exhibiting spatial homogeneity within each sublattice of the Lieb lattice is demonstrated.

Nonlinear systems possessing nonattracting chaotic sets, such as chaotic saddles, embedded in their state space may oscillate chaotically for a transient time before eventually transitioning into some stable attractor. We show that these systems, when networked with nonlocal coupling in a ring, are capable of forming chimera states, in which one subset of the units oscillates periodically in a synchronized state forming the coherent domain, while the complementary subset oscillates chaotically in the neighborhood of the chaotic saddle constituting the incoherent domain. We find two distinct transient chimera states distinguished by their abrupt or gradual termination. We analyze the lifetime of both chimera states, unraveling their dependence on coupling range and size. We find an optimal value for the coupling range yielding the longest lifetime for the chimera states. Moreover, we implement transversal stability analysis to demonstrate that the synchronized state is asymptotically stable for network configurations studied here.

We uncover the emergence of distinct sets of multistable chimera states in addition to chimera death and synchronized states in a smallest population of three globally coupled oscillators with mean-field diffusive coupling. Sequence of torus bifurcations result in the manifestation of distinct periodic orbits as a function of the coupling strength, which in turn result in the genesis of distinct chimera states constituted by two synchronized oscillators coexisting with an asynchronous oscillator. Two subsequent Hopf bifurcations result in homogeneous and inhomogeneous steady states resulting in desynchronized steady states and chimera death state among the coupled oscillators. The periodic orbits and the steady states lose their stability via a sequence of saddle-loop and saddle-node bifurcations finally resulting in a stable synchronized state. We have generalized these results to N coupled oscillators and also deduced the variational equations corresponding to the perturbation transverse to the synchronization manifold and corroborated the synchronized state in the two-parameter phase diagrams using its largest eigenvalue. Chimera states in three coupled oscillators emerge as a solitary state in N coupled oscillator ensemble.

We consider two populations of the globally coupled Sakaguchi-Kuramoto model with the same intra- and interpopulations coupling strengths. The oscillators constituting the intrapopulation are identical whereas the interpopulations are nonidentical with a frequency mismatch. The asymmetry parameters ensure the permutation symmetry among the oscillators constituting the intrapopulation and a reflection symmetry among the oscillators constituting the interpopulation. We show that the chimera state manifests by spontaneously breaking the reflection symmetry and also exists in almost in the entire explored range of the asymmetry parameter without restricting to the near π/2 values of it. The saddle-node bifurcation mediates the abrupt transition from the symmetry breaking chimera state to the symmetry-preserving synchronized oscillatory state in the reverse trace, whereas the homoclinic bifurcation mediates the transition from the synchronized oscillatory state to synchronized steady state in the forward trace. We deduce the governing equations of motion for the macroscopic order parameters employing the finite-dimensional reduction by Watanabe and Strogatz. The analytical saddle-node and homoclinic bifurcation conditions agree well with the simulations results and the bifurcation curves.

Chimera states are firstly discovered in nonlocally coupled oscillator systems. Such a nonlocal coupling arises typically as oscillators are coupled via an external environment whose characteristic time scale τ is so small (i.e., τ → o) that it could be eliminated adiabatically. Nevertheless, whether the chimera states still exist in the opposite situation (i.e., τ ≫ 1) is unknown. Here, by coupling large populations of Stuart—Landau oscillators to a diffusive environment, we demonstrate that spiral wave chimeras do exist in this oscillator-environment coupling system even when τ is very large. Various transitions such as from spiral wave chimeras to spiral waves or unstable spiral wave chimeras as functions of the system parameters are explored. A physical picture for explaining the formation of spiral wave chimeras is also provided. The existence of spiral wave chimeras is further confirmed in ensembles of FitzHugh—Nagumo oscillators with the similar oscillator-environment coupling mechanism. Our results provide an affirmative answer to the observation of spiral wave chimeras in populations of oscillators mediated via a slowly changing environment and give important hints to generate chimera patterns in both laboratory and realistic chemical or biological systems.

Chimera states have drawn great attention during the last several years. Multi-clustered chimera states with several coherent domains are one important type of chimera dynamics due to their relations with pattern formation. In this work, we study a ring of nonlocally coupled Brusselators. We find that the multi-stability of multi-clustered chimera states prevails in the model and that the phenomenon is insensitive to the coupling radius. The mechanisms behind the multi-stability of multi-clustered chimera states are explored. We find that there are two types of multi-clustered chimera states, one occurring at the coupling radius close to 0.5 and the other occurring at the coupling radius away from 0.5. The multi-stability of the former one originates from the periodic two-cluster dynamics in globally coupled Brusselators. The multi-stability of the latter one may be explained by linear growth rates of the perturbation, which are related but insensitive to different wave numbers to homogeneous states.

Chimeras are surprising yet important states in which domains of decoherent (asynchronous) and coherent (synchronous) oscillations co-exist. In this article, we report on the discovery of a new class of chimeras, called mixed-amplitude chimera states, in which the structures, amplitudes, and frequencies of the oscillations differ substantially in the decoherent and coherent regions. These mixed-amplitude chimeras exhibit domains of decoherent small-amplitude oscillations (phase waves) coexisting with domains of stable and coherent large-amplitude or mixed-mode oscillations (MMOs). They are observed in a prototypical bistable partial differential equation with oscillatory dynamics, spatially homogeneous kinetics, and purely local, isotropic diffusion. They are observed in parameter regimes immediately adjacent to regimes in which common large-amplitude solutions exist, such as trigger waves, spatially homogeneous MMOs, and sharp-interface solutions. Also, key singularities, folded nodes, and folded saddles arising commonly in multi-scale, bistable systems play important roles, and these have not previously been studied in systems with chimeras. The discovery of these mixed-amplitude chimeras is an important advance for understanding some processes in neuroscience, pattern formation, and physics, which involve both small-amplitude and large-amplitude oscillations. It may also be of use for understanding some aspects of electroencephalogram recordings from animals that exhibit unihemispheric slow-wave sleep.

Cellular automata are conceptual discrete dynamical systems useful in the theory of information. The spatiotemporal patterns that they produce are intimately related to computational mechanics in distributed complex systems. Here, we investigate their physical implementation in the framework of chimera states in which coherent and incoherent behavior coexist. Hence, chimera states were subject to quantitative and qualitative analyzes borrowing the same tools used to characterize cellular automata. Our results reveal the existence of cellular automata-type dynamics submerged in the dynamics exhibited by our optical chimera states. Thus, they share a panoply of attributes in terms of computational abilities.

Background and Objectives: One of the actual problems in nonlinear dynamics is the formation and interaction of complex spatial structures such as chimeras and solitary states arising in multicomponent systems. Chimera states are typical for ensembles of identical oscillators with regular, chaotic, and even stochastic behavior in a case of nonlocal interaction of the elements. They represent cluster structures, including groups of elements with synchronous and non-synchronous oscillations. Chimeras were discovered and investigated in real experiments, that indicates the possibility of observing such regimes in complex systems in living nature and in technology. Solitary states are less studied today. The regime of solitary states is characterized by the synchronous behavior of most elements of the ensemble, while individual oscillators behave in a “special state”. In the present work, an ensemble of phase oscillators with inertia (rotators) is chosen as the basic model for investigation. Such ensembles with a specific coupling topology are widely used in modeling the operation of energy networks. Ensembles of rotators with nonlocal coupling are characterized by both chimera states and solitary state regimes. The problem of interaction of ensembles of rotators with nonlocal coupling and synchronization of complex spatial structures (chimeras and solitary states) formed in them has not been studied yet. Materials and Methods: A two-layer multiplex network of rotators with a nonlocal character of intralayer interactions is considered. Each layer consists of 100 elements with the same value of the coupling coefficient and coupling phase shift for each element within one layer. The interlayer coupling is symmetric. At the initial stage, with a random choice of initial conditions, steady regimes (chimeras or solitary states) in non-interacting layers were found. Next, the interlayer coupling was introduced and the evolution of the layer dynamics in the selected initial regimes was studied. Four cases of interaction with various initial states of the layers were considered. In the first case, the two layers are completely identical and demonstrate slightly different chimera structures without interlayer coupling. Their evolution with the introduction and growth of the interlayer coupling is considered for two values of the coupling phase shift. It is shown that, starting from a certain threshold value of the interlayer coupling coefficient, the complete synchronization regime is established in the layers, and the coupling phase shift significantly affects the value of the synchronization threshold. In the second case, the previous experiment is reproduced for the two layers with a frequency mismatch. Chimera states established without interlayer interaction are characterized by significantly different average frequencies of the elements in the two layers. In the presence of non-identity of the layers (in this case, frequency mismatch), the regime of complete synchronization of spatial structures is impossible. However, with an increase in the interlayer coupling coefficient, effective synchronization can be obtained which corresponds to a slight difference in the phases of rotators in the interacting layers with full frequency synchronization. In the third case, we consider the interaction between the layers in the solitary state regimes with different spatial structures. In this case, a frequency mismatch is also introduced for the elements of the two layers. For solitary states, the effective synchronization regime with an increase in the interlayer coupling is also established. In both layers the same configurations of solitary states are realized and frequency synchronization is observed. In the fourth case, a heterogeneous multiplex network is considered, in which one layer is in the chimera state, the second layer shows the solitary state mode. With a certain strength of the interlayer coupling the complex structures are destroyed in both layers of the network and a spatially uniform regimes are established. In this case, all the rotators of the two layers rotate at the same frequency, and the difference in the regimes in the layers reduces to a small phase shift, the same for all pairs of coupled rotators of the two layers. Conclusion: The effects of synchronization in the multiplex network were established for two layers in the regimes of complex spatio-temporal dynamics, such as chimera states and solitary states. The influence of the frequency mismatch of the network elements and the phase shift in the interlayer coupling on the synchronization phenomena was studied.

The present work is devoted to the detailed quantification of the transition from spiral waves to spiral wave chimeras in a network of self-sustained oscillators with two-dimensional geometry. The basic elements of the network under consideration are the van der Pol oscillator or the FitzHugh – Nagumo neuron. Both of the models are in the regime of relaxation oscillations. We analyze the regime by using the indices of local sensitivity, which
enables us to evaluate the sensitivity of each oscillator at a finite time. Spiral waves are observed in both lattices when the interaction between elements has a local character. The dynamics of all the elements is regular. There are no pronounced high-sensitive regions. We have discovered that, when the coupling becomes nonlocal, the features of the system change significantly. The oscillation regime of the spiral wave center element switches to a chaotic one. Besides, a region with high sensitivity occurs around the wave center oscillator. Moreover, we show that the latter expands in space with elongation of the coupling range. As a result, an incoherence cluster of the spiral wave chimera is formed exactly within this high-sensitive area. A sharp increase in the values of the maximal Lyapunov exponent in the positive region leads to the formation of the incoherence cluster. Furthermore, we find that the system can even switch to a hyperchaotic regime when several Lyapunov exponents become positive.

We consider two diffusively coupled populations of identical oscillators, where the oscillators in each population are coupled with a common dynamic environment. Existence and stability of a variety of stationary states are analyzed on the basis of the Ott-Antonsen reduction method, which reveals that the chimera state occurs under the diffusive coupling scheme. Furthermore, we find an exotic symmetry-breaking behavior, the so-called the heterosynchronous state, in which each population exhibits in-phase coherence, while the order parameters of two populations rotate at different phase velocities. The chimera and heterosynchronous states emerge from bistabilities of distinct states for decoupled population and occur as a unique continuation for weak diffusive couplings. The heterosynchronous state is caused by an indirect coupling scheme via dynamic environments and could occur for a finite-size system as well, even for the system that consists of one oscillator per population.

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.

We review numerical results of studies of the complex dynamics of one- and double-dimensional networks (ensembles) of nonlocally coupled identical chaotic oscillators in the form of discrete- and continuous-time systems, as well as lattices of coupled ensembles. We show that these complex networks can demonstrate specific types of spatio-temporal patterns in the form of chimera states, known as the coexistence of spatially localized domains of coherent (synchronized) and incoherent (asynchronous) dynamics in a network of nonlocally coupled identical oscillators. We describe phase, amplitude, and double-well chimeras and solitary states; their basic characteristics are analyzed and compared. We focus on two basic discrete-time models, Hénon and Lozi maps, which can be used to describe typical chimera structures and solitary states in networks of a wide range of chaotic oscillators. We discuss the bifurcation mechanisms of their appearance and evolution. In conclusion, we describe effects of synchronization of chimera states in coupled ensembles of chaotic maps.

While chimera states themselves are usually believed to be chaotic transients, the involvement of chaos behind their self-organization is not properly distinguished or studied. In this work, we demonstrate that small chimeras in the local flux dynamics of an array of magnetically coupled superconducting quantum interference devices (SQUIDs) driven by an external field are born through transiently chaotic dynamics. We deduce analytic expressions for small chimeras and synchronous states which correspond to nonchaotic attractors in the model. We also numerically study the bifurcations underlying the multistability responsible for their generation. Transient chaos manifests itself in the short-term flux oscillations with erratically fluctuating amplitudes, exponential escape time distribution, and irregular dependence of the escape time to initial conditions. We classify the small chimera states in terms of the position of the nonsynchronized member and numerically construct their basin of attraction. The basin is shown to possess an interesting structure consisting of both ordered and fractal parts, which again can be attributed to transient chaos.

We investigate the occurrence of collective dynamical states such as transient amplitude chimera, stable amplitude chimera and imperfect breathing chimera states in a \textit{locally coupled} network of Stuart-Landau oscillators. In an imperfect breathing chimera state, the synchronized group of oscillators exhibits oscillations with large amplitudes while the desynchronized group of oscillators oscillates with small amplitudes and this behavior of coexistence of synchronized and desynchronized oscillations fluctuates with time. Then we analyze the stability of the amplitude chimera states under various circumstances, including variations in system parameters and coupling strength, and perturbations in the initial states of the oscillators. For an increase in the value of the system parameter, namely the nonisochronicity parameter, the transient chimera state becomes a stable chimera state for a sufficiently large value of coupling strength. In addition, we also analyze the stability of these states by perturbing the initial states of the oscillators. We find that while a small perturbation allows one to perturb a large number of oscillators resulting in a stable amplitude chimera state, a large perturbation allows one to perturb a small number of oscillators to get a stable amplitude chimera state. We also find the stability of the transient and stable amplitude chimera states as well as traveling wave states for appropriate number of oscillators using Floquet theory. In addition, we also find the stability of the incoherent oscillation death states.

We study the dynamics of mobile, locally coupled identical oscillators in the presence of coupling delays. We find different kinds of chimera states in which coherent in-phase and antiphase domains coexist with incoherent domains. These chimera states are dynamic and can persist for long times for intermediate mobility values. We discuss the mechanisms leading to the formation of these chimera states in different mobility regimes. This finding could be relevant for natural and technological systems composed of mobile communicating agents.

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.

Coupled dissipative nonlinear oscillators exhibit complex spatiotemporal dynamics. Frenkel-Kontorova is a prototype model of coupled nonlinear oscillators, which exhibits coexistence between stable and unstable state. This model accounts for several physical systems such as the movement of atoms in condensed matter and magnetic chains, dynamics of coupled pendulums, and phase dynamics between superconductors. Here, we investigate kinks propagation into an unstable state in the Frenkel-Kontorova model with dissipation. We show that unlike point-like particles ?-kinks spread in a pulsating manner. Using numerical simulations, we have characterized the shape of the ?-kink oscillation. Different parts of the front propagate with the same mean speed, oscillating with the same frequency but different amplitude. The asymptotic behavior of this propagation allows us to determine the minimum mean speed of fronts analytically as a function of the coupling constant. A generalization of the Peierls-Nabarro potential is introduced to obtain an effective continuous description of the system. Numerical simulations show quite fair agreement between the Frenkel-Kontorova model and the proposed continuous description.

This paper is concerned with the spatiotemporal dynamics of the 2D lattice of cubic maps with nonlocal coupling. Different types of chimera structures have been found. Also, the underexplored regime of solitary states has been found. It is shown that the solitary states are typical of a large coupling radius. The possibility of detecting such a regime increases with the transition to global interaction, while chimera states disappear.

The recently discovered chimera state involves the coexistence of
synchronized and desynchronized states for a group of identical oscillators.
This fascinating chimera state has until now been found only in non-local or
globally coupled oscillator systems. In this work, we for the first time show
numerical evidence of the existence of spiral wave chimeras in
reaction-diffusion systems where each element is locally coupled by diffusion.
This spiral wave chimera rotates inwardly, i.e., coherent waves propagate
toward the phase randomized core. A continuous transition from spiral waves
with smooth core to spiral wave chimeras is found as we change the local
dynamics of the system. Our findings on the spiral wave chimera in locally
coupled oscillator systems largely improve our understanding of the chimera
state and suggest that spiral chimera states may be found in natural systems
which can be modeled by a set of oscillators indirectly coupled by a diffusive
environment.

Two symmetrically coupled populations of $N$ oscillators with inertia $m$
display chaotic solutions with broken symmetry similar to experimental
observations with mechanical pendula. In particular, we report the first
evidence of intermittent chaotic chimeras, where one population is synchronized
and the other jumps erratically between laminar and turbulent phases. These
states have finite life-times diverging as a power-law with $N$ and $m$.
Lyapunov analyses reveal chaotic properties in quantitative agreement with
theoretical predictions for globally coupled dissipative systems.

Chimera states in spatially extended networks of oscillators have some
oscillators synchronised while the remainder are asynchronous. These states
have primarily been studied in networks with nonlocal coupling, and more
recently in networks with global coupling. Here we present three networks with
only local coupling (diffusive, to nearest neighbours) which are numerically
found to support chimera states. One of the networks is analysed using a
self-consistency argument in the continuum limit, and this is used to find the
boundaries of existence of a chimera state in parameter space.

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.

Chimera states, representing a spontaneous break-up of a population of
identical oscillators that are identically coupled, into sub-populations
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.

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 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.

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. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states with increasing number of intervals of irregularity, so-called chimera's heads. We report three scenarios for the chimera birth: (1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER, (2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and (3) via homoclinic transition with complex multistable dynamics including an "eight-like" limit cycle resulting eventually in a chimera state.

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.

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 investigate the possibility of obtaining chimera state solutions of the nonlocal complex Ginzburg-Landau equation (NLCGLE) in the strong coupling limit when it is important to retain amplitude variations. Our numerical studies reveal the existence of a variety of amplitude-mediated chimera states (including stationary and nonstationary two-cluster chimera states) that display intermittent emergence and decay of amplitude dips in their phase incoherent regions. The existence regions of the single-cluster chimera state and both types of two-cluster chimera states are mapped numerically in the parameter space of C_{1} and C_{2}, the linear and nonlinear dispersion coefficients, respectively, of the NLCGLE. They represent a new domain of dynamical behavior in the well-explored rich phase diagram of this system. The amplitude-mediated chimera states may find useful applications in understanding spatiotemporal patterns found in fluid flow experiments and other strongly coupled systems.

More than a decade ago, a surprising coexistence of synchronized and
asynchronous behavior called the chimera state was discovered in networks of
nonlocally coupled identical phase oscillators. In later years, chimeras were
found to occur in a variety of theoretical and experimental studies of chemical
and optical systems, as well as models of neuron dynamics. In this Letter, we
study two coupled populations of pendula represented by phase oscillators with
a second derivative term multiplied by a mass parameter m and treat the first
order derivative terms as dissipation with parameter {\epsilon}. We first
present numerical evidence showing that chimeras do exist in this system for
small mass values 0<m<<1 and {\epsilon} greater than a threshold value that
grows linearly with m. We then proceed to explain these states by reducing the
coherent population to a single damped pendulum equation driven parametrically
by oscillating averaged quantities related to the incoherent population.

We have identified the occurrence of chimera states for various coupling schemes in networks of two-dimensional and three-dimensional Hindmarsh–Rose oscillators, which represent realistic models of neuronal ensembles. This result, together with recent studies on multiple chimera states in nonlocally coupled FitzHugh–Nagumo oscillators, provide strong evidence that the phenomenon of chimeras may indeed be relevant in neuroscience applications. Moreover, our work verifies the existence of chimera states in coupled bistable elements, whereas to date chimeras were known to arise in models possessing a single stable limit cycle. Finally, we have identified an interesting class of mixed oscillatory states, in which desynchronized neurons are uniformly interspersed among the remaining ones that are either stationary or oscillate in synchronized motion.

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.

We study the dynamics of two symmetrically coupled populations of identical leaky integrate-and-fire neurons characterized by an excitatory coupling. Upon varying the coupling strength, we find symmetry-breaking transitions that lead to the onset of various chimera states as well as to a new regime, where the two populations are characterized by a different degree of synchronization. Symmetric collective states of increasing dynamical complexity are also observed. The computation of the the finite-amplitude Lyapunov exponent allows us to establish the chaoticity of the (collective) dynamics in a finite region of the phase plane. The further numerical study of the standard Lyapunov spectrum reveals the presence of several positive exponents, indicating that the microscopic dynamics is high-dimensional.

We present a straightforward and reliable continuous method for computing the full or partial Lyapunov spectrum associated with a dynamical system specified by a set of differential equations. We do this by introducing a stability parameter and augmenting the dynamical system with an orthonormal k-dimensional frame and a Lyapunov vector such that the frame is continuously Gram - Schmidt orthonormalized and at most linear growth of the dynamical variables is involved. We prove that the method is strongly stable when where is the kth Lyapunov exponent in descending order and we show through examples how the method is implemented. It extends many previous results.

Under drift forces, a monostable pattern propagates. However, examples of nonpropagative dynamics have been observed. We show that the origin of this pinning effect comes from the coupling between the slow scale of the envelope to the fast scale of the modulation of the underlying pattern. We evidence that this effect stems from spatial inhomogeneities in the system. Experiments and numerics on drifting pattern-forming systems subjected to inhomogeneous spatial pumping or boundary conditions confirm this origin of pinning dynamics.

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.

Dissipative localized structures exhibit intricate bifurcation diagrams. An adequate theory has been developed in one space dimension; however, discrepancies arise with the experiments. Based on an optical feedback with spatially modulated input beam, we set up a 1D forced configuration in a nematic liquid crystal layer. We characterize experimentally and theoretically the homoclinic snaking diagram of localized patterns, providing a reconciliation between theory and experiments.

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.

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 R\"ossler
systems reveal that intermediate, partially coherent states represent
characteristic spatio-temporal patterns at the transition from coherence to
incoherence.

We investigate chimera states in a ring of identical phase oscillators coupled in a time-delayed and spatially nonlocal fashion. We find novel clustered chimera states that have spatially distributed phase coherence separated by incoherence with adjacent coherent regions in antiphase. The existence of such time-delay induced phase clustering is further supported through solutions of a generalized functional self-consistency equation of the mean field. Our results highlight an additional mechanism for cluster formation that may find wider practical applications.

Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a spatiotemporal 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.

Models describing microscopic or mesoscopic phenomena in physics are inherently discrete, where the lattice spacing between fundamental components, such as in the case of atomic sites, is a fundamental physical parameter. The effect of spatial discreteness over front propagation phenomenon in an overdamped one-dimensional periodic lattice is studied. We show here that the study of front propagation leads in a discrete description to different conclusions that in the case of its, respectively, continuous description, and also that the results of the discrete model, can be inferred by effective continuous equations with a supplementary spatially periodic term that we have denominated Peierls-Nabarro drift, which describes the bifurcation diagram of the front speed, the appearance of particle-type solutions and their snaking bifurcation diagram. Numerical simulations of the discrete equation show quite good agreement with the phenomenological description.

The aim of this book is to develop a unified approach to nonlinear science, which does justice to its multiple facets and to the diversity and richness of the concepts and tools developed in this field over the years. Nonlinear science emerged in its present form following a series of closely related and decisive analytic, numerical and experimental developments that took place over the past three decades. It appeals to an extremely large variety of subject areas, but, at the same time, introduces into science a new way of thinking based on a subtle interplay between qualitative and quantitative techniques, topological and metric considerations and deterministic and statistical views. Special effort has been made throughout the book to illustrate both the development of the subject and the mathematical techniques, by reference to simple models. Each chapter concludes with a set of problems. This book will be of great value to graduate students in physics, applied mathematics, chemistry, engineering and biology taking courses in nonlinear science and its applications.

The dynamics of two symmetrically coupled populations of rotators is studied
for different values of the inertia. The system is characterized by different
types of solutions, which all coexist with the fully synchronized state. At
small inertia the system is no more chaotic and one observes mainly quasi-
periodic chimeras, while the usual (stationary) chimera state is not anymore
observable. At large inertia one observes two different kind of chaotic
solutions with broken symmetry: the intermittent chaotic chimera, characterized
by a synchronized population and a population displaying a turbulent behaviour,
and a second state where the two populations are both chaotic but whose
dynamics adhere to two different macroscopic attractors. The intermittent
chaotic chimeras are characterized by a finite life-time, whose duration
increases as a power-law with the system size and the inertia value. Moreover,
the chaotic population exhibits clear intermittent behavior, displaying a
laminar phase where the two populations tend to synchronize, and a turbulent
phase where the macroscopic motion of one population is definitely erratic. In
the thermodynamic limit these states survive for infinite time and the laminar
regimes tends to disappear, thus giving rise to stationary chaotic solutions
with broken symmetry contrary to what observed for chaotic chimeras on a ring
geometry.

We describe a turbulent state characterized by the presence of topological defects. This ``topological turbulence'' is likely to be experimentally observed in nonequilibrium systems.

We demonstrate emergence of a complex state in a homogeneous ensemble of
globally coupled identical oscillators, reminiscent of chimera states in
locally coupled oscillator lattices. In this regime some part of the ensemble
forms a regularly evolving cluster, while all other units irregularly oscillate
and remain asynchronous. We argue that chimera emerges because of effective
bistability which dynamically appears in the originally monostable system due
to internal delayed feedback in individual units. Additionally, we present two
examples of chimeras in bistable systems with frequency-dependent phase shift
in the global coupling.

Over the past two decades scientists, mathematicians, and engineers have come to understand that a large variety of systems exhibit complicated evolution with time. This complicated behavior is known as chaos. In the new edition of this classic textbook Edward Ott has added much new material and has significantly increased the number of homework problems. The most important change is the addition of a completely new chapter on control and synchronization of chaos. Other changes include new material on riddled basins of attraction, phase locking of globally coupled oscillators, fractal aspects of fluid advection by Lagrangian chaotic flows, magnetic dynamos, and strange nonchaotic attractors. This new edition will be of interest to advanced undergraduates and graduate students in science, engineering, and mathematics taking courses in chaotic dynamics, as well as to researchers in the subject.

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.

Chimera states occur spontaneously in populations of coupled photosensitive chemical oscillators. Experiments and simulations are carried out on nonlocally coupled oscillators, with the coupling strength decreasing exponentially with distance. Chimera states with synchronized oscillators, phase waves, and phase clusters coexisting with unsynchronized oscillators are analyzed. Irregular motion of the cores of asynchronous oscillators is found in spiral-wave chimeras.

Propagation failure (pinning) of traveling waves is studied in a discrete scalar reaction-diffusion equation with a piecewise linear, bistable reaction function. The critical points of the pinning transition, and the wavefront profile at the onset of propagation are calculated exactly. The scaling of the wave speed near the transition, and the leading corrections to the front shape are also determined. We find that the speed vanishes logarithmically close to the critical point, thus the model belongs to a different universality class than the standard Nagumo model, defined with a smooth, polynomial reaction function.

Calculations are made of the size of a dislocation and of the critical shear stress for its motion.