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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.

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... including chaos, multi-stability, and so on [9][10][11][12][13]16]. For instance, dynamic characteristics of two heterogeneous neurons coupled with multi-stable memory synapses were studied to understand the brain's information processing mechanism [16]. ...

... On the other hand, as a fascinating collective dynamical behaviors, synchronization has tackled extensive efforts recently. Various types of chaos synchronization have been investigated, such as complete synchronization [27][28][29], phase synchronization [30], projective synchronization [31,32], generalized synchronization [10,33,34], lag synchronization [15], and chimera [13,35]. For example, complete synchronization of a complex-valued chaotic network was achieved by adaptive feedback control [29]. ...

... When parameters are d = 3, c = 74 + 0.01 j, a = 1 − 0.06 j, b = 0.8, e = 1 and the initial value is Y 1 = (1 + 2 j, 1 + 1 j, 1, 1 + 2 j, 1 + 1 j, 2, 1 + 2 j, 1 + 1 j, 3), the amplitude control of network (8) with respect to α is shown in Fig. 8. Figure 8a shows the average absolute values of state variables x 11 , x 12 and x 13 with respect to parameter α from 0 to 5. When parameter α increases in the interval (0, 1.5), the average absolute values of variable x 13 remain unchanged, while those of the other four variables are decreasing. It can be concluded that the change of parameter α can only control the amplitude of state variables x 11 and x 12 except for x 13 . Figure 8b depicts the Lyapunov exponent spectrum with respect to α ∈ (0, 5). ...

Control and synchronization of a complex-valued laser network is a common question in mimicking the working mechanism of the biological brain. However, there is seldom study on geometric control of a complex-valued laser network. In this paper, a chain network of complex-valued Lorenz laser systems is investigated. Complex dynamics such as hyper-chaos, quasi-periodic orbits, and parametric attractors are revealed by the Lyapunov exponent, Kaplan–Yorke dimension, bifurcation diagram, phase portrait, and power spectrum. To control the hyper-chaotic signal, two kinds of new non-bifurcation control methods collectively known as the geometric control can be designed. One is amplitude control, which is to rescale the size of the hyper-chaotic attractor regularly by introducing a parameter; the other is rotation control, which is to achieve amplitude rescaling and offset boosting of the hyper-chaotic attractor simultaneously according to the designed rotation angle. The results show that these two geometric control methods have revised the amplitude and offset of the hyper-chaotic signal without switching the Lyapunov exponents. Moreover, the derived criterion of chaotic complete synchronization shows that strong coupling strength can lead to chaotic complete synchronization in the chain network. For relatively weak coupling, chaotic phase synchronization is also found.

... The chimera state was initially discovered in coupled phase oscillators [32]. After that, it became the center of attention of many researchers in different fields [33][34][35]. Several attempts have also been made to study the chimera state in neural networks with different structures and coupling schemes [31,[36][37][38][39][40][41]. ...

... Notwithstanding, the most common field of application of the Duffing equation seems to be mechanics, where its basic form is considered a mathematical model of motion of a single degree-of-freedom system, with linear damping and nonlinear stiffness. Obviously, it can also be used to describe one or more degrees of freedom in MDOF systems [17][18][19][20] . Generally, the Duffing equation is used to model nonlinear dynamic systems where the state changes exist as oscillations, so it is often called the Duffing oscillator. ...

Each Duffing equation has an unstable solution area with a boundary, which is also a line of bifurcation. Generally, in a system that can be modeled by the Duffing equation, bifurcations can occur at frequencies lower than the origin point frequency of the unstable solution area for a softening system and at higher frequencies for a hardening system. The main goal of this research is to determine the analytical formulas for the origin point of the unstable solution area of a system described by a forced Duffing oscillator with softening stiffness, taking damping into account. To achieve this goal, two systems of softening Duffing oscillators that differ strongly in their nonlinearity factor value have been selected and tested. For each system, for three combinations of linear and nonlinear stiffness coefficients with the same nonlinearity factor, bistability areas and unstable solution areas were determined for a series of damping coefficient values. For each case, curves determined for different damping values were grouped to obtain the origin point curve of the unstable solution, ultimately developing the target formulas.

... 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.

... Recently, Sharma [22] observed the states of partial amplitude death and phase-flip bifurcation in a system of three theoretical time-delay coupled relay oscillators. The study by Clerc et al. [29] observed the existence of freak chimera states characterized as the coexistence of incoherent domains in a system of Duffing oscillators with the nearest neighbor coupling scheme. Although all aforementioned studies separately provide insights on the role of coupling structure and number of oscillators on the dynamical behavior of a network, none of the experimental studies so far has comprehensively delineated the explicit dependence of the global behavior of the same network on the change in the number of oscillators, the coupling topology, and the strength of coupling between the oscillators. ...

Understanding the global dynamical behavior of a network of coupled oscillators has been a topic of immense research in many fields of science and engineering. Various factors govern the resulting dynamical behavior of such networks, including the number of oscillators and their coupling schemes. Although these factors are seldom significant in large populations, a small change in them can drastically affect the global behavior in small populations. In this paper, we perform an experimental investigation on the effect of these factors on the coupled behavior of a minimal network of candle-flame oscillators. We observe that strongly coupled oscillators exhibit the global behavior of in-phase synchrony and amplitude death, irrespective of the number and the topology of oscillators. However, when they are weakly coupled, their global behavior exhibits the intermittent occurrence of multiple stable states in time. We report the experimental discovery of partial amplitude death in a network of candle-flame oscillators, in addition to the observation of other dynamical states including clustering, chimera, and weak chimera. We also show that closed-loop networks tend to hold global synchronization for longer duration as compared to open-loop networks.

... 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.

A generalized Duffing oscillator is considered, which takes into account high-order derivatives and power nonlinearities. The Painlevé test is applied to study the integrability of the mathematical model. It is shown that the generalized Duffing oscillator passes the Painlevé test only in the case of the classic Duffing oscillator which is described by the second-order differential equation. However, in the general case there are the expansion of the general solution in the Laurent series with two arbitrary constants. This allows us to search for exact solutions of generalized Duffing oscillators with two arbitrary constants using the classical Duffing oscillator as the simplest equation. The algorithm of finding exact solutions is presented. Exact solutions for the generalized Duffing oscillator are found for equations of fourth, sixth, eighth and tenth order in the form of periodic oscillations and solitary pulse.

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

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

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

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

We studied the phenomenon of chimera states in networks of non–locally coupled externally excited oscillators. Units of the considered networks are bi–stable, having two co–existing attractors of different types (chaotic and periodic). The occurrence of chimeras is discussed, and the influence of coupling radius and coupling strength on their co–existence is analyzed (including typical bifurcation scenarios). We present a statistical analysis and investigate sensitivity of the probability of observing chimeras to the initial conditions and parameter values. Due to the fact that each unit of the considered networks is individually excited, we study the influence of the excitation failure on stability of observed states. Typical transitions are shown, and changes in network's dynamics are discussed. We analyze systems of coupled van der Pol–Duffing oscillators and the Duffing ones. Described chimera states are robust as they are observed in the wide regions of parameter values, as well as in other networks of coupled forced oscillators.

We show that amplitude chimeras in ring networks of Stuart-Landau oscillators with symmetry-breaking nonlocal coupling represent saddle-states in the underlying phase space of the network. Chimera states are composed of coexisting spatial domains of coherent and of incoherent oscillations. We calculate the Floquet exponents and the corresponding eigenvectors in dependence upon the coupling strength and range, and discuss the implications for the phase space structure. The existence of at least one positive real part of the Floquet exponents indicates an unstable manifold in phase space, which explains the nature of these states as long-living transients. Additionally, we find a Stuart-Landau network of minimum size $N=12$ exhibiting amplitude chimeras

Neuronal systems have been modeled by complex networks in different description levels. Recently, it has been verified that networks can simultaneously exhibit one coherent and other incoherent domain, known as chimera states. In this work, we study the existence of chimera states in a network considering the connectivity matrix based on the cat cerebral cortex. The cerebral cortex of the cat can be separated in 65 cortical areas organised into the four cognitive regions: visual, auditory, somatosensory-motor and frontolimbic. We consider a network where the local dynamics is given by the Hindmarsh-Rose model. The Hindmarsh-Rose equations are a well known model of neuronal activity that has been considered to simulate membrane potential in neuron. Here, we analyse under which conditions chimera states are present, as well as the affects induced by intensity of coupling on them. We observe the existence of chimera states in that incoherent structure can be composed of desynchronised spikes or desynchronised bursts. Moreover, we find that chimera states with desynchronised bursts are more robust to neuronal noise than with desynchronised spikes.

Kuramoto and Battogtokh [Nonlinear Phenom. Complex Syst. 5, 380 (2002)] discovered chimera states represented by stable coexisting synchrony and asynchrony domains in a lattice of coupled oscillators. After reformulation in terms of local order parameter, the problem can be reduced to partial differential equations. We find uniformly rotating periodic in space chimera patterns as solutions of a reversible ordinary differential equation, and demonstrate a plethora of such states. In the limit of neutral coupling they reduce to analytical solutions in form of one- and two-point chimera patterns as well as localized chimera solitons. Patterns at weakly attracting coupling are characterized by virtue of a perturbative approach. Stability analysis reveals that only simplest chimeras with one synchronous region are stable.

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

We propose a control scheme which can stabilize and fix the position of
chimera states in small networks. Chimeras consist of coexisting domains of
spatially coherent and incoherent dynamics in systems of nonlocally coupled
identical oscillators. Chimera states are generally difficult to observe in
small networks due to their short lifetime and erratic drifting of the spatial
position of the incoherent domain. The control scheme, like a tweezer, might be
useful in experiments, where usually only small networks can be realized.

In a recent work, we have studied networks of two-dimensional and three-dimensional Hindmarsh-Rose oscillators and have discovered some very interesting oscillatory phenomena, called chimera states, in which synchronized neuronal ensembles coexist with completely asynchronous ones. In this paper, we summarize our work in connection with other studies on nonlocally coupled FitzHugh-Nagumo oscillators, examine the occurrence of chimera states in coupled bistable elements and point out that mixed oscillatory states also exist, in which desynchronized neurons are interspersed among neurons that oscillate in synchronous fashion. We also demonstrate, by a preliminary study, that it is possible to control these states by varying an external current parameter applied to the main potential variable in order to observe new phenomena that may be relevant in neuroscience applications.

Chimera states are complex spatiotemporal patterns in networks of identical
oscillators, characterized by the coexistence of synchronized and
desynchronized dynamics. Here we propose to extend the phenomenon of chimera
states to the quantum regime, and uncover intriguing quantum signatures of
these states. We calculate the quantum fluctuations about semiclassical
trajectories and demonstrate that chimera states in the quantum regime can be
characterized by bosonic squeezing, weighted quantum correlations, and measures
of mutual information. Our findings reveal the relation of chimera states to
quantum information theory, and give promising directions for experimental
realization of chimera states in quantum systems.

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

We discuss the occurrence of chimera states in networks of nonlocally coupled bistable oscillators, in which individual subsystems are characterized by the coexistence of regular (a fixed point or a limit cycle) and chaotic attractors. By analyzing the dependence of the network dynamics on the range and strength of coupling, we identify parameter regions for various chimera states, which are characterized by different types of chaotic behavior at the incoherent interval. Besides previously observed chimeras with space-temporal and spatial chaos in the incoherent intervals we observe another type of chimera state in which the incoherent interval is characterized by a central interval with standard space-temporal chaos and two narrow side intervals with spatial chaos. Our findings for the maps as well as for time-continuous van der Pol-Duffing's oscillators reveal that this type of chimera states represents characteristic spatiotemporal patterns at the transition from coherence to incoherence.

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

We report the existence of a chimera state in an assembly of identical
nonlinear oscillators that are globally linked to each other in a simple planar
cross-coupled form. The rotational symmetry breaking of the coupling term
appears to be responsible for the emergence of these collective states that
display a characteristic coexistence of coherent and incoherent behaviour. Our
finding, seen in both a collection of van der Pol oscillators and chaotic
Rossler oscillators, further simplifies the existence criterion for chimeras
and thereby broadens the range of their applicability to real world situations.

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

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

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

Networks of nonlocally coupled phase oscillators can support chimera
states in which identical oscillators evolve into distinct groups that
exhibit coexisting synchronous and incoherent behaviours despite
homogeneous coupling. Similar nonlocal coupling topologies implemented
in networks of chaotic iterated maps also yield dynamical states
exhibiting coexisting spatial domains of coherence and incoherence. In
these discrete-time systems, the phase is not a continuous variable, so
these states are generalized chimeras with respect to a broader notion
of incoherence. Chimeras continue to be the subject of intense
theoretical investigation, but have yet to be realized experimentally.
Here we show that these chimeras can be realized in experiments using a
liquid-crystal spatial light modulator to achieve optical nonlinearity
in a spatially extended iterated map system. We study the
coherence-incoherence transition that gives rise to these chimera states
through experiment, theory and simulation.

We 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.

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

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

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

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.

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.

We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec \cite{O_68}, which provides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been extensively used as an indicator of chaos, and the algorithm of the so--called `standard method', developed by Benettin et al. \cite{BGGS_80b}, for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although, we are mainly interested in finite--dimensional conservative systems, i. e. autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed. Comment: 74 pages, 8 figures, accepted for publication in Lecture Notes in Physics

Chimera states are remarkable spatiotemporal patterns in which coherence coexists with incoherence. As yet, chimera states have been considered as nongeneric, since they emerge only for particular initial conditions. In contrast, we show here that in a network of globally coupled oscillators delayed feedback stimulation with realistic (i.e., spatially decaying) stimulation profile generically induces chimera states. Intriguingly, a bifurcation analysis reveals that these chimera states are the natural link between the coherent and the incoherent states.

Neuronal systems have been modelled by complex networks in different description levels. Recently, it has been verified that the networks can simultaneously exhibit one coherent and other incoherent domain, known as chimera states. In this work, we study the existence of chimera-like states in a network considering the connectivity matrix based on the cat cerebral cortex. The cerebral cortex of the cat can be separated in 65 cortical areas organised into the four cognitive regions: visual, auditory, somatosensory-motor and frontolimbic. We consider a network where the local dynamics is given by the Hindmarsh–Rose model. The Hindmarsh–Rose equations are a well known model of the neuronal activity that has been considered to simulate the membrane potential in neuron. Here, we analyse under which conditions chimera-like states are present, as well as the effects induced by intensity of coupling on them. We identify two different kinds of chimera-like states: spiking chimera-like state with desynchronised spikes, and bursting chimera-like state with desynchronised bursts. Moreover, we find that chimera-like states with desynchronised bursts are more robust to neuronal noise than with desynchronised spikes.

We review chimera patterns, which consist of coexisting spatial domains of coherent (synchronized) and incoherent (desynchronized) dynamics in networks of identical oscillators. We focus on chimera states involving amplitude as well as phase dynamics, complex topologies like small-world or hierarchical (fractal), noise, and delay. We show that a plethora of novel chimera patterns arise if one goes beyond the Kuramoto phase oscillator model. For the FitzHugh-Nagumo system, the Van der Pol oscillator, and the Stuart-Landau oscillator with symmetry-breaking coupling various multi-chimera patterns including amplitude chimeras and chimera death occur. To test the robustness of chimera patterns with respect to changes in the structure of the network, regular rings with coupling range R, small-world, and fractal topologies are studied. We also address the robustness of amplitude chimera states in the presence of noise. If delay is added, the lifetime of transient chimeras can be drastically increased.

We explore the bifurcation transition from coherence to incoherence in ensembles of nonlocally coupled chaotic systems. It is firstly shown that two types of chimera states, namely, amplitude and phase, can be found in a network of coupled logistic maps, while only amplitude chimera states can be observed in a ring of continuous-time chaotic systems. We reveal a bifurcation mechanism by analyzing the evolution of space-time profiles and the coupling function with varying coupling coefficient and formulate the necessary and sufficient conditions for realizing the chimera states in the ensembles.

A “chimera state” is a dynamical pattern that occurs in a network of coupled identical oscillators when the symmetry of the oscillator population is broken into synchronous and asynchronous parts. We report the experimental observation of chimera and cluster states in a network of four globally coupled chaotic opto-electronic oscillators. This is the minimal network that can support chimera states, and our study provides new insight into the fundamental mechanisms underlying their formation. We use a unified approach to determine the stability of all the observed partially synchronous patterns, highlighting the close relationship between chimera and cluster states as belonging to the broader phenomenon of partial synchronization. Our approach is general in terms of network size and connectivity. We also find that chimera states often appear in regions of multistability between global, cluster, and desynchronized states.

We study the existence of chimera states in a network of locally coupled chaotic and limit-cycle oscillators. The necessary condition for chimera state in purely local coupled oscillators is discussed. At first, we numerically observe the existence of chimera or multichimera states in the locally coupled Hindmarsh–Rose neuron model. We find that delay time in the nonlinear local coupling reduces the domain of the coherent island in the parameter space of the synaptic coupling strength and time delay, and thus the coherent region can be completely eliminated once the time delay exceeds a certain threshold. We then consider another form of nonlinearity in the local coupling, and the existence of chimera states is observed in the time-delayed Mackey–Glass system and in a Van der Pol oscillator. We also discuss the effect of time delay in local coupling for the existence of chimera states in Mackey–Glass systems. The nonlinearity present in the coupling function plays a key role in the emergence of chimera or multichimera states. A phase diagram for the chimera state is identified over a wide parameter space.

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

The Duffing Equation: Nonlinear Oscillators and their Behaviour brings together the results of a wealth of disseminated research literature on the Duffing equation, a key engineering model with a vast number of applications in science and engineering, summarizing the findings of this research. Each chapter is written by an expert contributor in the field of nonlinear dynamics and addresses a different form of the equation, relating it to various oscillatory problems and clearly linking the problem with the mathematics that describe it. The editors and the contributors explain the mathematical techniques required to study nonlinear dynamics, helping the reader with little mathematical background to understand the text. The Duffing Equation provides a reference text for postgraduate and students and researchers of mechanical engineering and vibration / nonlinear dynamics as well as a useful tool for practising mechanical engineers. Includes a chapter devoted to historical background on Georg Duffing and the equation that was named after him. Includes a chapter solely devoted to practical examples of systems whose dynamic behaviour is described by the Duffing equation. Contains a comprehensive treatment of the various forms of the Duffing equation. Uses experimental, analytical and numerical methods as well as concepts of nonlinear dynamics to treat the physical systems in a unified way.

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

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.

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

Preface 1. Introduction Part I. Synchronization Without Formulae: 2. Basic notions: the self-sustained oscillator and its phase 3. Synchronization of a periodic oscillator by external force 4. Synchronization of two and many oscillators 5. Synchronization of chaotic systems 6. Detecting synchronization in experiments Part II. Phase Locking and Frequency Entrainment: 7. Synchronization of periodic oscillators by periodic external action 8. Mutual synchronization of two interacting periodic oscillators 9. Synchronization in the presence of noise 10. Phase synchronization of chaotic systems 11. Synchronization in oscillatory media 12. Populations of globally coupled oscillators Part III. Synchronization of Chaotic Systems: 13. Complete synchronization I: basic concepts 14. Complete synchronization II: generalizations and complex systems 15. Synchronization of complex dynamics by external forces Appendix 1. Discovery of synchronization by Christiaan Huygens Appendix 2. Instantaneous phase and frequency of a signal References Index.

The properties of dislocations are calculated by an approximate method due to Peierls. The width of a dislocation is small, displacements comparable with the interatomic distance being confined to a few atoms. The shear stress required to move a dislocation in an otherwise perfect lattice is of the order of a thousandth of the "theoretical" shear strength. The energy and effective mass of a single dislocation increase logarithmically with the size of the specimen. A pair of dislocations of opposite sign in the same glide plane cannot be in stable equilibrium unless they are separated by a distance of the order of 10 000 lattice spacings. If an external shear stress is applied there is a critical separation of the pair of dislocations at which they are in unstable equilibrium. The energy of this unstable state is the activation energy for the formation of a pair of dislocations. It depends on the external shear, and for practical stresses is of the order of 7 electron volts per atomic plane.
The size and energy of dislocations in real crystals are unlikely to differ greatly from those calculated: the stress required to move a dislocation and the critical separation of two dislocations may be seriously in error.

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

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

Arrays of identical oscillators can display a remarkable spatiotemporal pattern in which phase-locked oscillators coexist with drifting ones. Discovered two years ago, such "chimera states" are believed to be impossible for locally or globally coupled systems; they are peculiar to the intermediate case of nonlocal coupling. Here we present an exact solution for this state, for a ring of phase oscillators coupled by a cosine kernel. We show that the stable chimera state bifurcates from a spatially modulated drift state, and dies in a saddle-node bifurcation with an unstable chimera state.

Tweezers for chimeras in small networks

- I Omelchenko
- Oe
- A Zakharova
- M Wolfrum
- E Schöll

Omelchenko I, Omel'chenko OE, Zakharova A, Wolfrum M, Schöll E. Tweezers for chimeras in small networks. Phys Rev Lett 2016;116::114101. doi: 10.
1103/PhysRevLett.116.114101.

Lyapunov exponents: a tool to explore complex dynamics

- A Pikovsky
- A Politi

Pikovsky A, Politi A. Lyapunov exponents: a tool to explore complex dynamics. Cambridge University Press; 2016.