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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.

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... Since the seminal work of Kuramoto and Battogtokh [2] , the universal phenomenon of chimeras states has been intensively studied. Most of these studies focus on discrete systems of coupled oscillators and only recently the dynamical richness of chimera solutions in continuous models has been explored [23,24,26,27] . In continuous media, the chimera states can be understood as localized spatiotemporal complex patterns resulting from a symmetry breaking. ...

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.

... Furthermore, from the coupling scheme's perspective, the chimera states have been investigated considering diverse connection arrangements, ranging from simple nearest-neighbor coupling to time-varying and hierarchical topologies [72][73][74][75][76][77]. The chimera surveys' growth led to the detection of various types of chimera, other than the initially observed chimera composed of phase-locked and phase distributed groups [78][79][80][81][82][83][84][85][86][87][88]. Different factors, such as the local dynamics of the elements, the coupling scheme and parameters, the time delay and noise, influence on the emergence of different chimera types. ...

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.

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.

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.

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

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

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

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

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

It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrodinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions).

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.

In this paper we report the occurrence of chimera patterns in a network of neuronal oscillators, which are coupled through {\it local}, synaptic {\it gradient} coupling. We discover a new chimera pattern, namely the {\it imperfect traveling chimera} where the incoherent traveling domain spreads into the coherent domain of the network. Remarkably, we also find that chimera states arise even for {\it one-way} local coupling, which is in contrast to the earlier belief that only nonlocal, global or nearest neighbor local coupling can give rise to chimera; this find further relaxes the essential connectivity requirement of getting a chimera state. We choose a network of identical bursting Hindmarsh-Rose neuronal oscillators and show that depending upon the relative strength of the synaptic and gradient coupling several chimera patterns emerge. We map all the spatiotemporal behaviors in parameter space and identify the transitions among several chimera patterns, in-phase synchronized state and global amplitude death state.

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.

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.

We study numerically the development of chimera states in networks of
nonlocally coupled oscillators whose limit cycles emerge from a Hopf
bifurcation. This dynamical system is inspired from population dynamics and
consists of three interacting species in cyclic reactions. The complexity of
the dynamics arises from the presence of a limit cycle and four fixed points.
When the bifurcation parameter increases away from the Hopf bifurcation the
trajectory approaches the heteroclinic invariant manifolds of the fixed points
producing spikes, followed by long resting periods. We observe chimera states
in this spiking regime as a coexistence of coherence (synchronization) and
incoherence (desynchronization) in a one-dimensional ring with nonlocal
coupling, and demonstrate that their multiplicity depends both on the system
and the coupling parameters. We also show that hierarchical (fractal) coupling
topologies induce traveling multichimera states. The speed of motion of the
coherent and incoherent parts along the ring is computed through the Fourier
spectra of the corresponding dynamics.

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

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.

Non-locally coupled, periodically arranged SQUIDs (Superconducting QUantum
Interference Devices) can form magnetic metamaterials exhibiting extraordinary
properties, including tuneability and dynamic multistability, which have been
experimentally observed. It is demonstrated numerically that they also exhibit
complex dynamic states in which clusters of SQUIDs with synchronous dynamics
coexist with clusters that exhibit asynchronous behavior. These {\em "chimera
states"} appear generically as a result of the non-local, dipole-dipole
magnetic coupling between SQUIDs, and they can be reached by randomly
initializing the system. They also affect measurable quantities and thus their
presence can in principle be detected with presently available experimental
set-ups.

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 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.

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 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 generalize the exact solution to the Bernoulli shift map. Under certain conditions, the gen- eralized functions can produce unpredictable dynamics. We use the properties of the generalized functions to show that certain dynamical systems can generate random dynamics. For instance, the chaotic Chua's circuit coupled to a circuit with a non-invertible I-V characteristic can generate unpredictable dynamics. In general, a nonperiodic time-series with truncated exponential behavior can be converted into unpredictable dynamics using non-invertible transformations. Using a new theoretical framework for chaos and randomness, we investigate some classes of coupled map lat- tices. We show that, in some cases, these systems can produce completely unpredictable dynamics. In a similar fashion, we explain why some wellknown spatiotemporal systems have been found to produce very complex dynamics in numerical simulations. We discuss real physical systems that can generate random dynamics.

Recent work on the behaviour of localised states in pattern forming partial
differential equations has focused on the traditional model Swift-Hohenberg
equation which, as a result of its simplicity, has additional structure --- it
is variational in time and conservative in space. In this paper we investigate
an extended Swift-Hohenberg equation in which non-variational and
non-conservative effects play a key role. Our work concentrates on aspects of
this much more complicated problem. Firstly we carry out the normal form
analysis of the initial pattern forming instability that leads to
small-amplitude localised states. Next we examine the bifurcation structure of
the large-amplitude localised states. Finally we investigate the temporal
stability of one-peak localised states. Throughout, we compare the localised
states in the extended Swift-Hohenberg equation with the analogous solutions to
the usual Swift-Hohenberg equation.

The cubic-quintic Swift-Hohenberg equation (SH35) provides a convenient order parameter description of several convective systems with reflection symmetry in the layer midplane, including binary fluid convection. We use SH35 with an additional quadratic term to determine the qualitative effects of breaking the midplane reflection symmetry on the properties of spatially localized structures in these systems. Our results describe how the snakes-and-ladders organization of localized structures in SH35 deforms with increasing symmetry breaking and show that the deformation ultimately generates the snakes-and-ladders structure familiar from the quadratic-cubic Swift-Hohenberg equation. Moreover, in nonvariational systems, such as convection, odd-parity convectons necessarily drift when the reflection symmetry is broken, permitting collisions among moving localized structures. Collisions between both identical and nonidentical traveling states are described.

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

We study the properties of 2D cavity solitons in a coherently driven optical resonator subjected to a delayed feedback. The delay is found to induce a spontaneous motion of a single cavity soliton that is stationary and stable otherwise. This behavior occurs when the product of the delay time and the feedback strength exceeds some critical value. We derive an analytical formula for the speed of a moving soliton. Numerical results are in good agreement with analytical predictions.

In this paper, we analyze a model of broad area vertical-cavity surface-emitting lasers subjected to frequency-selective optical feedback. In particular, we analyze the spatio-temporal regimes arising above threshold and the existence and dynamical properties of cavity solitons. We build the bifurcation diagram of stationary self-localized states, finding that branches of cavity solitons emerge from the degenerate Hopf bifurcations marking the homogeneous solutions with maximal and minimal gain. These branches collide in a saddle-node bifurcation, defining a maximum pump current for soliton existence that lies below the threshold of the laser without feedback. The properties of these cavity solitons are in good agreement with those observed in recent experiments.

Coherent behavior in ensembles of globally coupled maps is investigated in the limit of infinite number of elements. A self-consistent approach based on a nonlinear Frobenius-Perron equation is proposed for such systems, and a possibility of quasiperiodic and chaotic behavior of the mean field is demonstrated. For the study of finite ensembles a noisy nonlinear Frobenius-Perron equation is derived. Previous observations of violations of the law of large numbers are explained.

The interaction of two neighboring modulational instabilities in a coherently driven semiconductor cavity is investigated. First, an asymptotic reduction of the general equations is performed in the limit of a nearly vertical input-output characteristic. Next, a normal form is derived in the limit where the two instabilities are close to one other. An infinity of branches of periodic solutions are found to emerge from the unstable portion of the homogeneous branch. These branches have a nontrivial envelope in the bifurcation diagram that can either smoothly join the two instability points or form an isolated branch of solutions.

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 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 aim of this paper is to review some of the mechanisms which leads to stable localized patterns 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.

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

Preface; 1. Introduction to techniques; 2. Generating functions I; 3.
Generating functions II: recurrence, sites visited, and the role of
dimensionality; 4. Boundary conditions, steady state, and the
electrostatic analogy; 5. Variations on the random walk; 6. The shape of
a random walk; 7. Path integrals and self-avoidance; 8. Properties of
the random walk: introduction to scaling; 9. Scaling of walks and
critical phenomena; 10. Walks and the O(n) model: mean field theory and
spin waves; 11. Scaling, fractals, and renormalization; 12. More on the
renormalization group; References; Index.

We investigate the emergence of localized coherent behavior in systems consisting
of two populations of social agents possessing a condition for non-interacting states,
mutually coupled through global interaction fields. We employ two examples of such
dynamics: (i) Axelrod’s model for social influence, and (ii) a discrete version of a
bounded confidence model for opinion formation. In each case, the global interac-
tion fields correspond to the statistical mode of the states of the agents in each
population. In both systems we find localized coherent states for some values of
parameters, consisting of one population in a homogeneous state and the other in a
disordered state. This situation can be considered as a social analogue to a chimera
state arising in two interacting populations of oscillators. In addition, other asymp-
totic collective behaviors appear in both systems depending on parameter values: a
common homogeneous state, where both populations reach the same state; different
homogeneous states, where both population reach homogeneous states different from
each other; and a disordered state, where both populations reach inhomogeneous
states.

Systems driven far from thermodynamic equilibrium can create dissipative structures through the spontaneous breaking of symmetries. A particularly fascinating feature of these pattern-forming systems is their tendency to produce spatially confined states. These localized wave packets can exist as propagating entities through space and/or time. Various examples of such systems will be dealt with in this book, including localized states in fluids, chemical reactions on surfaces, neural networks, optical systems, granular systems, population models, and Bose-Einstein condensates.This book should appeal to all physicists, mathematicians and electrical engineers interested in localization in far-from-equilibrium systems. The authors - all recognized experts in their fields - strive to achieve a balance between theoretical and experimental considerations thereby giving an overview of fascinating physical principles, their manifestations in diverse systems, and the novel technical applications on the horizon.

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

We present a coupled-cavity model of a laser with frequency-selective
feedback, and use it to analyze and explain the existence of stationary
and dynamic spatial solitons in the device. Particular features of
soliton addressing in this system are discussed. We demonstrate the
advantages of our model with respect to the common Lang-Kobayashi
approximation.

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

A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency 0 of the instability.

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

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

The mutual information I is examined for a model dynamical system and for chaotic data from an experiment on the Belousov-Zhabotinskii reaction. An N logN algorithm for calculating I is presented. As proposed by Shaw, a minimum in I is found to be a good criterion for the choice of time delay in phase-portrait reconstruction from time-series data. This criterion is shown to be far superior to choosing a zero of the autocorrelation function.

The title statement is numerically shown for a globally coupled chaotic system. With an increasing number of elements, N, the distribution of the mean field approaches a Gaussian distribution, but the decrease of its mean-square deviation with N stops for large N. This violation of the law of large numbers is found to be caused by the emergence of a subtle coherence among elements, as is measured by the mutual information. With the inclusion of noise, the law of large numbers is restored. The mean-square deviation decreases in proportion to N-β with an exponent β<1 depending on the noise strength.