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The aim of the research is to study the influence of inhomogeneity in a control parameter of all partial elements in a ring of nonlocally coupled chaotic maps on the possibility of observing chimera states in the system and to compare the changes in regions of chimera realization using different methods of introducing the inhomogeneity. Methods. In this paper, snapshots of the system dynamics are constructed for various values of the parameters, as well as spatial distributions of cross-correlation coefficient values, which enable us to determine the regime observed in the system for these parameters. To improve the accuracy of the obtained results, the numerical studies are carried out for fifty different realizations of initial conditions of the ring elements. Results. It is shown that a fixed inhomogeneous distribution of the control parameters with increasing noise intensity leads to an increase in the range of the coupling strength where chimera states are observed. With this, the boundary lying in the region of strong coupling changes more significantly as compared with the case of weak coupling strength. The opposite effect is provided when the control parameters are permanently affected by noise. In this case increasing the noise intensity leads to a decrease in the interval of existence of chimera states. Additionally, the nature of the random variable distribution (normal or uniform one) does not strongly influence the observed changes in the ring dynamics. The regions of existence of chimera states are constructed in the plane of «coupling strength – noise intensity» parameters. Conclusion. We have studied how the region of existence of chimeras changes when the coupling strength between the ring elements is varied and when different characteristics of the inhomogeneous distribution of the control parameters are used. It has been shown that in order to increase the region of observing chimera states, the control parameters of the elements must be distributed inhomogeneously over the entire ensemble. To reduce this region, a constant noise effect on the control parameters should be used.

On November 30, 2020, at the age of 78, Vadim S. Anishchenko suddenly died. He was Head of the Department of Radiophysics and Nonlinear Dynamics, Saratov State University (SSU), Doctor of Physics and Mathematics, Professor, Honored Scientist of the Russian Federation, Laureate of the International Science Prize of the Alexander von Humboldt Foundation, Honorary Worker of Education of the Russian Federation, Honorary Professor of SSU, world-class specialist in the field of nonlinear dynamics, founder of one of the leading scientific schools in radiophysics and nonlinear dynamics. The memories of people who knew Vadim S. Anishchenko closely at different periods of his life are published, as a tribute to the memory of an outstanding personality.

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

Background and Objectives: As external and mutual synchronisation effects are conventional, the study of these phenomena in networks of nonlocally coupled chaotic maps is of much interest. In this paper we study the effects of synchronization in a two-layer network of nonlocally coupled discrete-time systems. Each layer represents a ring of nonlocally coupled logistic maps in the chaotic regime. Depending on the initial conditions and parameter values, they can demonstrate various spatiotemporal patterns, including amplitude and phase chimera structures. The network equations are being solved numerically for periodic boundary conditions and randomly distributed initial conditions. We consider interaction between identical and nonidentical ensembles for dissipative and inertial intercoupling. Materials and Methods: The analysis is carried out with the use of a set of programs in C++ which was developed for modelling dynamic systems with complex intercoupling definition. Synchronization effect is estimated by calculating root-mean-square deviations between the symmetric elements of the rings. The first characteristic to consider is the time deviation which is used to determine the clusters that respond to the impact and synchronize faster as the intercoupling strength grows. The second quantity is the time and ensemble deviation and it is used for quantifying the synchronization effect and estimating the synchronization region in dependence of the interlayer coupling. Results: The numerical research has shown that the effects of external and mutual synchronisation are clearly visible in the case of dissipative intercoupling. The best results were obtained for identical ensembles, however, nonidentical ensembles can be synchronized as well with a given accuracy. In the case of external synchronisation of nonidentical ensembles for dissipative intercoupling we have ascertained that amplitude chimera structures synchronize faster than the phase ones. Both systems are much harder or impossible to synchronize for the case of inertial intercoupling. Conclusion: The numerical results obtained in this paper allow to understand the course of synchronization effects of chimera states appearing in a two-layer network of nonlocally coupled chaotic oscillators for dissipative and inertial intercoupling.

This paper presents results of numerical statistical analysis of the effect of short-term localized noise of different intensity on the amplitude chimera lifetime in an ensemble of nonlocally coupled logistic maps in a chaotic regime. It is shown that a single and rather weak noise perturbation added only to the incoherence cluster of the amplitude chimera after its switching to the phase chimera mode is able to revive and stabilize the amplitude chimera, as well as to increase its lifetime to infinity. It is also analyzed how the amplitude chimera lifetime depends on the duration of noise influence of different intensity.

We investigate numerically the spatio-temporal dynamics of a 2D lattice of coupled discrete-time systems with nonlocal interaction. The individual map is given by a universal discrete system (the Nekorkin map) proposed for modeling the neural activity. The network behavior is studied for periodic and open boundary conditions. It is shown that for certain values of the nonlinear coupling parameters, rotating spiral waves and spiral wave chimeras can be observed in the considered lattice. We analyze and compare statistical and dynamical characteristics of the local oscillators from coherence and incoherence clusters of a spiral wave chimera.

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 30, 061104 (2020) and may be found at https://doi.org/10.1063/5.0008902. We study relay and complete synchronization in a heterogeneous triplex network of discrete-time chaotic oscillators. A relay layer and two outer layers, which are not directly coupled but interact via the relay layer, represent rings of nonlocally coupled two-dimensional maps. We consider for the first time the case when the spatiotemporal dynamics of the relay layer is completely different from that of the outer layers. Two different configurations of the triplex network are explored: when the relay layer consists of Lozi maps while the outer layers are given by Henon maps and vice versa. Phase and amplitude chimera states are observed in the uncoupled Henon map ring, while solitary state regimes are typical for the isolated Lozi map ring. We show for the first time relay synchronization of amplitude and phase chimeras, a solitary state chimera, and solitary state regimes in the outer layers. We reveal regimes of complete synchronization for the chimera structures and solitary state modes in all the three layers. We also analyze how the synchronization effects depend on the spatiotemporal dynamics of the relay layer and construct phase diagrams in the parameter plane of inter-layer vs intra-layer coupling strength of the relay layer.

We study the structure and properties of the chimera states in the ensembles of chaotic oscillators with nonlocal coupling. It is shown that the phase and amplitude chimera states in the ensembles of chaotic oscillators with nonhyperbolic and hyperbolic attractors can be obtained using the models in the forms of two-dimensional Henon and Lozi maps. The mechanisms of birth, the structure, and the lifetime of the phase and amplitude chimeras and the regime of solitary states are studied. The chimera states in two coupled ensembles of chaotic maps are considered. The possibility of realizing a new type of the chimera structure, namely, the chimera consisting of the solitary states is demonstrated. The effects of the external and mutual synchronizations of the chimera states are described by an example of two coupled ensembles of logistic maps with nonlocal coupling. A qualitative analogy of the obtained results with the classical effect of synchronization of periodic self-sustained oscillations is discussed.

We study numerically external synchronization of chimera states in a network of many unidirectionally coupled layers, each representing a ring of nonlocally coupled discretetime systems. The dynamics of each element in the network is described by either the logistic map or the bistable cubic map. We consider two cases: when all M unidirectionally coupled layers are identical and when (M − 1) identical layers differ from the first driving layer in their nonlocal coupling parameters. It is shown that the master chimera state in the first layer can be retranslating along the network with small distortions which are defined by a parameter mismatch. We also analyze the dependence of the mean-square deviation of the structure in the ith layer on the nonlocal coupling parameters.

We study the mechanism of appearance of a solitary state chimera in a ring of nonlocally coupled two-dimensional chaotic maps. We show that both solitary states and the solitary state chimera can be induced by noise which multiplicatively modulates the nonlocal coupling coefficient in the ensemble of nonlocally coupled Henon maps. It is demonstrated that these states can also arise due to multistability in the network. It has been firstly shown and explained the possibility of obtaining the solitary states and the solitary state chimera in the one-dimensional ring of nonlocally coupled Henon maps. Basins of attraction of coexisting attractors essentially depend on the spatiotemporal structure, which is realized in the ring, as well as on the position of a particular oscillator among the other oscillators. We have found out that the riddling effect of the basins of attraction can be observed when the network state shows an amplitude chimera.

We study the boundaries of existence of traveling waves and stationary spatial structures in an active medium model by varying the control parameters. The medium is represented by a ring of diffusively coupled FitzHugh–Nagumo neurons, which, when uncoupled, can demonstrate excitable, self-sustained oscillatory or bistable dynamics depending on control parameter values. The dynamical regimes realized in the medium are compared with those ones observed in an individual FitzHugh–Nagumo neuron. Possible bifurcations of traveling waves are analyzed when the dynamics of the medium elements changes. We also explore the influence of the relaxation level of FitzHugh–Nagumo neurons on the medium dynamics.

We consider ensembles of bistable elements with nonlocal interaction. It is shown that the bistability of units in the case of nonlocal interaction leads to the formation of chimera structures of a special type, which we have called double-well chimeras. Their distinctive feature consists in the formation of incoherence clusters with an irregular distribution of elements between two attractive sets existing in an individual element (two “potential wells”). Ensembles of different bistable units are considered, namely ensembles of cubic maps, FitzHugh–Nagumo oscillators in the regime of two stable equilibrium points and Chua’s circuits. The spatiotemporal behavior of the ensembles is studied in the cases of regular and chaotic dynamics in time, and different types of chimera structures are revealed.

We study the transition from coherence (complete synchronization) to incoherence (spatio-temporal chaos) in ensembles of nonlocally coupled chaotic maps with nonhyperbolic and hyperbolic attractors. As basic models of a partial element we use the Henon map and the Lozi map. We show that the transition to incoherence in a ring of coupled Henon maps occurs through the appearance of phase and amplitude chimera states. An ensemble of coupled Lozi maps demonstrates the coherence-incoherence transition via solitary states and no chimera states are observed in this case.

We consider in detail similarities and differences of the “coherence–incoherence” transition in ensembles of nonlocally coupled chaotic discrete-time systems with nonhyperbolic and hyperbolic attractors. As basic models we employ the Henon map and the Lozi map. We show that phase and amplitude chimera states appear in a ring of coupled Henon maps, while no chimeras are observed in an ensemble of coupled Lozi maps. In the latter, the transition to spatio-temporal chaos occurs via solitary states. We present numerical results for the coupling function which describes the impact of neighboring oscillators on each partial element of an ensemble with nonlocal coupling. Varying the coupling strength we analyze the evolution of the coupling function and discuss in detail its role in the “coherence–incoherence” transition in the ensembles of Henon and Lozi maps.

We study numerically the spatiotemporal dynamics and synchronization of a heterogeneous two-layer multiplex network where each layer is represented by a ring of nonlocally coupled FitzHugh–Nagumo neurons in the oscillatory regime. Being uncoupled, individual layers can show chimera states, solitary states and combined structures (the coexistence of chimera and solitary states) depending on the values of the intralayer coupling parameters and initial conditions. We choose different spatiotemporal patterns in the coupled layers and systematically study synchronization between them when the interlayer coupling is introduced through either the fast (activator) or the slow (inhibitor) variable of the FitzHugh–Nagumo oscillators. Our results enable to uncover the competitive behavior between the solitary states and the chimeras in the transition to synchronous regime in the considered network. We also analyze the synchronization peculiarities for two different types of the interlayer coupling by using the local and global synchronization measures.

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Shepelev, I. A., Muni, S. S., Schöll, E., & Strelkova, G. I. (2021). Repulsive inter-layer coupling induces anti-phase synchronization. In Chaos: An Interdisciplinary Journal of Nonlinear Science (Vol. 31, Issue 6, p. 063116). AIP Publishing. https://doi.org/10.1063/5.0054770 and may be found at https://doi.org/10.1063/5.0054770. We present numerical results for the synchronization phenomena in a bilayer network of repulsively coupled 2D lattices of van der Pol oscillators. We consider the cases when the network layers have either different or the same types of intra-layer coupling topology. When the layers are uncoupled, the lattice of van der Pol oscillators with a repulsive interaction typically demonstrates a labyrinth-like pattern, while the lattice with attractively coupled van der Pol oscillators shows a regular spiral wave structure. We reveal for the first time that repulsive inter-layer coupling leads to anti-phase synchronization of spatiotemporal structures for all considered combinations of intra-layer coupling. As a synchronization measure, we use the correlation coefficient between the symmetrical pairs of network nodes, which is always close to −1 in the case of anti-phase synchronization. We also study how the form of synchronous structures depends on the intra-layer coupling strengths when the repulsive inter-layer coupling is varied.

Correlation characteristics of chimera states have been calculated using the coefficient of mutual correlation of elements in a closed-ring ensemble of nonlocally coupled chaotic maps. Quantitative differences between the coefficients of mutual correlation for phase and amplitude chimeras are established for the first time.

The transition from coherence to incoherence in an ensemble of nonlocally coupled logistic maps is considered. Chimera states of two types (amplitude and phase) are found. The mechanism and conditions of their appearance are determined.

We explore numerically relay synchronization of wave structures in a heterogeneous three-layer network of coupled two-dimensional (2D) lattices of continuous-time systems. Remote layers, which are not directly connected but interact via a relay layer, consist of coupled van der Pol oscillators, while the middle layer is described by a lattice of interacting FitzHugh-Nagumo neurons. We show for the first time that already for weak inter-layer coupling, anti-phase relay synchronization of target wave patterns occurs in the considered network. This is a novel effect which can be observed in multiplex networks of interacting 2D lattices of oscillatory systems. Our numerical studies indicate that strong inter-layer coupling leads to in-phase synchronization of spatio-coherent structures in the network layers. We also analyze the impact of inter-layer coupling ranges in the relay and the remote layers on the relay synchronization effect.

We numerically study the spatiotemporal dynamics of a heterogeneous two-layer network of chaotic discrete-time systems. The multiplex network is described by uni- or bidirectionally coupled rings of Henon and Lozi maps with a nonlocal interaction. We examine in detail effects of weakly and strongly sparse inter-layer coupling upon effective synchronization and compare with the network dynamics in the case of non-sparse coupling between the rings. We show for the first time that effective synchronization is confidently observed in a finite range on the inter-layer coupling strength even when up to 50% links between the rings are missing. The impact of a different degree of sparseness of the inter-layer coupling is quantified by using the global synchronization measure. We also analyze the contribution of the intra- and inter-layer couplings in the synchronous structure formation.

In the present paper we study numerically the interaction of two 2-dimensional networks each representing a lattice of coupled van der Pol oscillators. The isolated lattices can demonstrate spiral and target wave structures for the cases of no-flux boundary conditions and random initial conditions. We explore the effect of mutual synchronization of target waves and target wave chimeras for dissipative and inertial inter-layer coupling between the lattices and for local and nonlocal intra-layer coupling between the lattice oscillators. We show for the first time the possibility of complete synchronization of target wave chimera states, including synchronization of incoherent cores. Our findings demonstrate that there is a difference with synchronization of spiral wave chimera states in which incoherent cores are not completely synchronized. The obtained results can have an important application and can be particularly used to study the dynamics of cardiac muscle in biophysics.

We study the spatiotemporal dynamics of coupled Lorenz systems with nonlocal interaction and for small values of the coupling strength. It is shown that due to the interaction the effective values of the control parameters can shift and the classical quasi-hyperbolic Lorenz attractor in an isolated element is transformed to a nonhyperbolic one. In this case, the network becomes multistable that is a typical property of nonhyperbolic chaotic systems. This fact gives rise to the appearance of chimera-like states, which have not been found in the studied network before. We also reveal and describe three different types of intermittency, both in time and in space, between various spatiotemporal structures in the network of nonlocally coupled Lorenz models.

The intrinsic features of systems with a line of equilibria are analyzed by studying of memristor-based Chua's oscillator. The analog modeling of the system is carried out together with its numerical simulation. The characteristics of stochastic oscillations in the system under study are explored in the presence of noise. The issues concerning the physical realization of a system with a line of equilibria are also considered.

We review rigorous and numerical results on the statistics of Poincare recurrences which are related to the modern development of the Poincare recurrence problem. We analyze and describe the rigorous results which are achieved both in the classical (local) approach and in the recently developed global approach. These results are illustrated by numerical simulation data for simple chaotic and ergodic systems. It is shown that the basic theoretical laws can be applied to noisy systems if the probability measure is ergodic and stationary. Poincare recurrences are studied numerically in nonautonomous systems. Statistical characteristics of recurrences are analyzed in the framework of the global approach for the cases of positive and zero topological entropy. We show that for the positive entropy, there is a relationship between the Afraimovich–Pesin dimension, Lyapunov exponents and the Kolmogorov–Sinai entropy either without and in the presence of external noise. The case of zero topological entropy is exemplified by numerical results for the Poincare recurrence statistics in the circle map. We show and prove that the dependence of minimal recurrence times on the return region size demonstrates universal properties for the golden and the silver ratio. The behavior of Poincare recurrences is analyzed at the critical point of Feigenbaum attractor birth. We explore Poincare recurrences for an ergodic set which is generated in the stroboscopic section of a nonautonomous oscillator and is similar to a circle shift. Based on the obtained results we show how the Poincare recurrence statistics can be applied for solving a number of nonlinear dynamics issues. We propose and illustrate alternative methods for diagnosing effects of external and mutual synchronization of chaotic systems in the context of the local and global approaches. The properties of the recurrence time probability density can be used to detect the stochastic resonance phenomenon. We also discuss how the fractal dimension of chaotic attractors can be estimated using the Poincare recurrence statistics.

In this paper we present numerical results for the spatio-temporal dynamics of a network of nonlocally coupled logistic maps. We show that temporal switchings between amplitude and phase chimera states can be observed in an autonomous ensemble. This alternation process, which we refer to as a temporally intermittent chimera, can be classified as a transient chimera. We have established that the intermittent regime has a finite lifetime and is finally transformed to the phase chimera, being a stationary and stable regime. We also demonstrate that such behavior is typical both for a separate element of an amplitude chimera cluster and for the whole amplitude chimera cluster.

We describe numerical results for the dynamics of networks of nonlocally coupled chaotic maps. Switchings in time between amplitude and phase chimera states have been first established and studied. It has been shown that in autonomous ensembles, a nonstationary regime of switchings has a finite lifetime and represents a transient process towards a stationary regime of phase chimera. The lifetime of the nonstationary switching regime can be increased to infinity by applying short-term noise perturbations.

We study numerically the statistics of recurrences in ergodic sets of the circle map type by using the multifractality analysis . We consider the standard circle map as well as the sets generated in stroboscopic sections of phase trajectories in a nonautonomous van der Pol oscillator and in a harmonically driven conservative nonlinear oscillator . The universal properties of the recurrence time dependence on the size of a return region are established. The numerical results demonstrate a full correspondence with the theoretical data obtained by Valentin Afraimovich.

We study the dynamics of a two-dimensional lattice of nonlocally coupled-map-based neuron models represented by Rulkov maps. It is firstly shown that this discrete-time neural network can exhibit spiral and target waves and corresponding chimera states when the control parameters (the coupling strength and the coupling radius) are varied. It is demonstrated that one-core, multicore, and ring-shaped core spiral chimeras can be realized in the network. We also reveal a novel type of chimera structure-a target wave chimera. We explore the transition from spiral wave chimeras to target wave structures when varying the coupling parameters. We report for the first time that the spiral wave regime can be suppressed by applying noise excitations, and the subsequent transition to the target wave mode occurs.

We investigate solitary states and solitary state chimeras in a ring of nonlocally coupled systems represented by FitzHugh-Nagumo neurons in the oscillatory regime. We perform a systematic study of solitary states in this network. In particular, we explore the phase space structure, calculate basins of attraction, analyze the region of existence of solitary states in the system's parameter space, and investigate how the number of solitary nodes in the network depends on the coupling parameters. We report for the first time the occurrence of solitary state chimera in networks of coupled time-continuous neural systems. Our results disclose distinctive features characteristic of solitary states in the FitzHugh-Nagumo model, such as the flat mean phase velocity profile. On the other hand, we show that the mechanism of solitary states' formation in the FitzHugh-Nagumo model similar to chaotic maps and the Kuramoto model with inertia is related to the appearance of bistability in the system for certain values of coupling parameters. This indicates a general, probably a universal desynchronization scenario via solitary states in networks of very different nature.

We study numerically forced synchronization of a heterogeneous multilayer network in the regime of a complex spatiotemporal pattern. Retranslating the master chimera structure in a driving layer along subsequent layers is considered, and the peculiarities of forced synchronization are studied depending on the nature and degree of heterogeneity of the network, as well as on the degree of asymmetry of the inter-layer coupling. We also analyze the possibility of synchronizing all the network layers with a given accuracy when the coupling parameters are varied.

We study numerically the dynamics of a network made of two coupled one-dimensional ensembles of discrete-time systems. The first ensemble is represented by a ring of nonlocally coupled Henon maps, and the second one - by a ring of nonlocally coupled Lozi maps. We find that the network of coupled ensembles can realize all the spatio-temporal structures which are observed both in the Henon map ensemble and in the Lozi map ensemble when uncoupled. Moreover, we reveal a new type of spatiotemporal structure, a solitary state chimera, in the considered network. We also establish and describe the effect of mutual synchronization of various complex spatiotemporal patterns in the system of two coupled ensembles of Henon and Lozi maps.

Characteristics of the Poincare return times are considered in a one-dimensional cubic map with a chaotic nonhyperbolic attractor. Two approaches, local one (Kac's theorem) and global one related with the AP-dimension estimation of return times, are used. The return times characteristics are studied in the presence of external noise. The characteristics of Poincare recurrences are compared with the form of probability measure and the complete correspondence of the obtained results with the mathematical theory is shown. The influence of the attractor crisis on the return time characteristics is also analyzed. The obtained results have a methodical and educational significance and can be used for solving a number of applied tasks.

In the present minireview, we analyze autonomous and non-autonomous oscillations of dynamical and stochastic systems in the framework of common concepts. We introduce the definition of an attractor for a non-autonomous system. We also propose the definition of self-sustained oscillations, which can be applied for both autonomous and non-autonomous systems. We consider noise-induced oscillations and formulate the definition of stochastic self-sustained oscillations for this case. All the statements made in this work are illustrated by particular examples.

We study the spatiotemporal dynamics of a ring of nonlocally coupled FitzHugh-Nagumo oscillators in the bistable regime. A new type of chimera patterns has been found in the noise-free network and when isolated elements do not oscillate. The region of existence of these structures has been explored when the coupling range and the coupling strength between the network elements are varied., Comment: 5 figures