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In this paper, the encryption improvement via modulation of the fractional order chaotic oscillators state variables, is presented. A network of N-coupled fractional order Lü chaotic oscillators, is synchronized. A voice message is encrypted via additive encryption, by using a state variable of the synchronized network. The selected state variable is modulated and used to encrypt the message again. The results are compared.

We study the dynamics of a multilayer network of chaotic oscillators subject to an amplification. Previous studies have proven that multilayer networks present phenomena such as synchronization, cluster and chimera states. Here we consider a network with two layers of Roessler chaotic oscillators as well as applications to multilayer networks of chaotic jerk and Lienard oscillators. Intralayer coupling is considered to be all to all in the case of Roessler oscillators, a ring for jerk oscillators and global mean field coupling in the case of Lienard, the interlayer coupling is unidirectional in all these three cases. The second layer has an amplification coefficient. An in depth study on the case of a network of Roessler oscillators using master stability function and order parameter leads to several phenomena such as complete synchronization, generalized, cluster and phase synchronization with amplification. For the case of Roessler oscillators, we note that there are also certain values of coupling parameters and amplification where the synchronization does not exist or the synchronization can exist but without amplification. Using other systems with different topologies, we obtain some interesting results such as chimera state with amplification, cluster state with amplification and complete synchronization with amplification., 13 pages, 8 figures. To appear in Chaos: An Interdisciplinary Journal of Nonlinear Science

The type of transition from asynchronous behavior to the generalized synchronization regime in mutually coupled chaotic oscillators has been studied. To separate the epochs of the synchronous and asynchronous motion in time series of mutually coupled chaotic oscillators, a method based on the local Lyapunov exponent calculation has been proposed. The efficiency of the method has been testified using the examples of unidirectionally coupled dynamical systems for which the type of transition is well known. The transition to generalized synchronization regime in mutually coupled systems has been shown to be an on-off intermittency as well as in the case of the unidirectional coupling.

Spiral wave chimeras with spatially extended unsynchronized cores have been reported in nonlocally coupled phase oscillators and chaotic oscillators. Here, we investigate the two-dimensional nonlocally coupled FitzHugh–Nagumo oscillators with open boundary conditions. Rich dynamics of spiral wave chimeras is discovered numerically. Besides outwardly propagating and inwardly propagating single-spiral wave chimeras, we find the double-spiral wave chimera with two spiral chimeras rotating with each other and the breakup of spiral wave chimeras. We also find a transition between a normal spiral wave with a phase singularity at its center and spiral wave chimera.

We report enhancing of complete synchronization in identical chaotic oscillators when their interaction is mediated by a mismatched oscillator. The identical oscillators now interact indirectly through the intermediate relay oscillator. The induced heterogeneity in the intermediate oscillator plays a constructive role in reducing the critical coupling for a transition to complete synchronization. A common lag synchronization emerges between the mismatched relay oscillator and its neighboring identical oscillators that leads to this enhancing effect. We present examples of one-dimensional open array, a ring, a star network and a two-dimensional lattice of dynamical systems to demonstrate how this enhancing effect occurs. The paradigmatic R\"ossler oscillator is used as a dynamical unit, in our numerical experiment, for different networks to reveal the enhancing phenomenon., Comment: 10 pages, 7 figures

In this paper, inspired by a newly proposed two-dimensional nonlinear oscillator with an infinite number of coexisting attractors, a modified nonlinear oscillator is proposed. The original system has an exciting feature of having layer&ndash, layer coexisting attractors. One of these attractors is self-excited while the rest are hidden. By forcing this system with its twin, a new four-dimensional nonlinear system is obtained which has an infinite number of coexisting torus attractors, strange attractors, and limit cycle attractors. The entropy, energy, and homogeneity of attractors&rsquo, images and their basin of attractions are calculated and reported, which showed an increase in the complexity of attractors when changing the bifurcation parameters.

We explore numerically inter-layer synchronization of spatiotemporal patterns in a heterogeneous bilayer network of nonlocally coupled Rossler oscillators. Both layers exhibit phase chimera regimes with similar cluster structures, but there is a mean frequency mismatch between the layers. We show that an entrainment of the mean frequency in the interacting layers is observed already for sufficiently weak inter-layer coupling even when the frequency mismatch is large. At the same time, frequency synchronization does not mean structure synchronization. Our numerical study indicates that inter-layer synchronization takes place within noticeably shorter ranges of the frequency mismatch and the inter-layer coupling strength than the frequency synchronization regime. Besides, we show that the weakest chaotic behavior is observed when both frequency and structure synchronizations occur, while the most developed chaotic oscillations are typical when the spatiotemporal structures in the layers are desynchronized.

We study synchronization of two chaotic oscillators coupled with time delay in a master-slave configuration and with delayed positive feedback in the slave oscillator which acts as memory. The dynamics of the slave oscillator is analyzed with bifurcation diagrams of the peak value of the system variable with respect to the coupling and feedback strengths and two delay times. For small coupling, when the oscillators’ phases synchronize, memory can induce bistability and stabilize periodic orbits, whereas for stronger coupling it is not possible. The delayed feedback signal impairs synchronization, simultaneously enhancing coherence of the slave oscillator.

We show that a reservoir computer is an effective tool for model-free prediction of extreme events in deterministic chaotic systems. This prediction allows us to suppress unwanted extreme events, by applying weak control perturbations to the system at times preceding expected extreme events. The effectiveness of such a prediction and prevention strategy is demonstrated for a system of globally coupled FitzHugh-Nagumo neurons and for a system of two almost identical unidirectionally coupled chaotic oscillators.

Many systems exhibit both attractive and repulsive types of interactions, which may be dynamic or static. A detailed understanding of the dynamical properties of a system under the influence of dynamically switching attractive or repulsive interactions is of practical significance. However, it can also be effectively modeled with two coexisting competing interactions. In this work, we investigate the effect of time-varying attractive-repulsive interactions as well as the hybrid model of coexisting attractive-repulsive interactions in two coupled nonlinear oscillators. The dynamics of two coupled nonlinear oscillators, specifically limit cycles as well as chaotic oscillators, are studied in detail for various dynamical transitions for both cases. Here, we show that dynamic or static attractive-repulsive interactions can induce an important transition from the oscillatory to steady state in identical nonlinear oscillators due to competitive effects. The analytical condition for the stable steady state in dynamic interactions at the low switching time period and static coexisting interactions are calculated using linear stability analysis, which is found to be in good agreement with the numerical results. In the case of a high switching time period, oscillations are revived for higher interaction strength.

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.

In exploring oscillator synchronization, a general observation is that as the oscillators become nonidentical, e.g., introducing parameter mismatch among the oscillators, the propensity for synchronization will be deteriorated. Yet in realistic systems, parameter mismatch is unavoidable and even worse in some circumstances, the oscillators might follow different types of dynamics. Considering the significance of synchronization to the functioning of many realistic systems, it is natural to ask the following question: Can synchronization be achieved in networked oscillators of clearly different parameters or dynamics? Here, by the model of networked chaotic oscillators, we are able to demonstrate and argue that, despite the presence of parameter mismatch (or different dynamics), stable synchronization can still be achieved on symmetric complex networks. Specifically, we find that when the oscillators are configured on the network in such a way that the symmetric nodes have similar parameters (or follow the same type of dynamics), cluster synchronization can be generated. The stabilities of the cluster synchronization states are analyzed by the method of symmetry-based stability analysis, with the theoretical predictions in good agreement with the numerical results. Our study sheds light on the interplay between symmetry and cluster synchronization in complex networks and give insights into the functionalities of realistic systems where nonidentical nonlinear oscillators are presented and cluster synchronization is crucial.

We introduce a type of cyclic coupling to investigate synchronization of chaotic oscillators. We derive analytical solutions of the critical coupling for stable synchronization under the cyclic coupling for the Rossler system and the Lorenz oscillator as paradigmatic illustration. Based on the master stability function (MSF) approach, the analytical results on critical coupling are verified numerically. An enhancing effect in terms of lowering the critical coupling or enlarging the synchronization window in a critical coupling space is noticed. The cyclic coupling is also applied in other models, Hindmarsh-Rose model, Sprott system, Chen system and forced Duffing system to confirm the enhancing effect. The cyclic coupling allows tuning of two coupling constants in reverse directions when an optimal control of synchronization is feasible.

The dynamics of indirectly coupled Lorenz circuits is investigated experimentally. The in-phase and anti-phase synchronization of indirectly coupled chaotic oscillators reported in Phys. Rev. E 81, 046216 (2010) is verified by physical experiments with electronic circuits. Two chaotic systems coupled through a common dynamic environment shows the verity of synchronization behaviours such as anti-phase synchronization, in-phase synchronization, identical synchronization, anti-synchronization, etc.

An artificial neuronal network composed by 2D interconnected chaotic oscillators is explored for brain waves (EEG) simulation. For the inverse problem solution a parallel real-coded genetic algorithm (PRCGA) is proposed. In order to conduct thorough comparison between the simulated and target signal characteristics, a spectrum analysis of the signals is undertaken. A good matching between the theoretical and experimental EEG signals has been achieved. Numerical results of calculations are presented and discussed.

A thermoacoustic chaotic oscillator is a fluid system that presents thermally induced chaotic oscillations of a gas column. This study experimentally reports a bifurcation diagram when two thermoacoustic chaotic oscillators are dissipatively coupled to each other. The two-parameter bifurcation diagram is constructed by varying the frequency mismatch and the coupling strength. Complete chaos synchronization is observed in the region with a frequency mismatch of less than 1% of the uncoupled oscillator. In other regions, synchronization between quasiperiodic oscillations and that between limit-cycle oscillations and amplitude death are observed as well as asynchronous states.

Experiments involving the Mercury Beating Heart (MBH) oscillator, exhibiting irregular (aperiodic) dynamics, are performed. In the first set of experiments, control over irregular dynamics of the MBH oscillator was obtained via a superimposed periodic voltage signal. These irregular (aperiodic) dynamics were recovered once the control was switched off. Subsequently, two MBH oscillators were coupled to attain synchronization of their aperiodic oscillations. Finally, two uncoupled MBH oscillators were subjected, repeatedly, to a common stochastic forcing, resulting in an enhancement of their mutual phase correlation. Published by AIP Publishing.

This paper presents a comparative study oil design and implementation of Wien type chaotic oscillators. By making a collection of almost all Wien bridge-based chaotic circuits, we have investigated these oscillators in terms of chaotic dynamics, circuit structures, active building blocks, nonlinear element structures and operating frequency by using PSpice simulations and laboratory experiments. In addition to this comparative investigation, we present our two basic experimental contributions to referred implementations. While the first of our experimental contributions consists of the experimentally implementation of CFOA-based Chua's circuit modified for very high chaotic oscillations, the scope of the second is to experimentally implement a Wien type high frequency chaos generator, which has the diode-inductor composite, in the inductorless form by using CFOA-based synthetic inductor. (c) 2008 Elsevier Ltd. All rights reserved.

Synchronization of chaotic oscillators in a generalized sense leads to richer behavior than identical chaotic oscillations in coupled systems. In this paper, we study the generalized synchronization of uni-directionally coupled nuclear spin generator system.

This work deals with the partial synchronization problem of two different chaotic oscillators considering model uncertainties in the slave system via control approach. The slave system is forced to follow the master signal via a linearizing controller based on model uncertainty reconstructor which leads to proportional–integral–derivative (PID) control structure. This reconstructor is related with a proportional–derivative (PD) reduced-order observer, it would be considered as a sub-slave system for the original slave of the synchronization procedure. The asymptotic performance of the synchronization methodology is proven via the dynamic of the synchronization error. Numerical experiment illustrates the closed-loop behavior of the proposed methodology.

We present an approach to generate (multivariate) twin surrogates (TS) based on recurrence properties. This technique generates surrogates which correspond to an independent copy of the underlying system, i.e. they induce a trajectory of the underlying system starting at different initial conditions. We show that these surrogates are well suited to test for complex synchronisation and exemplify this for the paradigmatic system of Rossler oscillators. The proposed test enables to assess the statistical relevance of a synchronisation analysis from passive experiments which are typical in natural systems.

The dynamics behaviors of coupled Saito's oscillators are investigated intensively. Various phenomena are explored such as antisynchronization (AS), hysteresis, bistability, riddled basin and coexistence of chaotic AS and lag AS. With increasing coupling intensity, the coupled chaotic oscillators undergo a transition from phase synchronization to AS. With decreasing coupling intensity, they will transit from AS to antiphase synchronization. The necessary condition for AS is explored and the stability of AS is studied.

We study the anti-phase synchronization (APS) in a system of two coupled chaotic oscillators. The necessary condition and the stability analysis for the APS are given theoretically. The APS state in specific systems such as Chua circuits and Lorenz oscillators are numerically studied. The different types of transitions to APS in both the systems are found.

The dynamics of fractional-order systems have attracted increasing attentions in recent years. In this paper, the synchronization of two coupled nonlinear fractional order chaotic oscillators is numerically demonstrated using the master–slave synchronization scheme. It is shown that fractional-order chaotic oscillators can be synchronized with appropriate coupling strength.

The dynamics of fractional-order systems have attracted increasing attentions in recent years. In this paper, we study the synchronization technique based on the master?slave synchronization scheme and apply it to the synchronization of two coupled nonlinear fractional-order electronic chaotic oscillators. Simulations show that two coupled fractional-order chaotic oscillators can be brought to an exact synchronization with appropriate coupling strength. It is interesting that the synchronization rate of the fractional-order chaotic oscillators is slower than its integer-order counterpart; however, with the increase of system order, the curves of synchronization error can be smoothened, which indicates that the master?slave synchronization of two coupled fractional order oscillators can be smoothened and stabilized.

The effect of internal noise on phase synchronization of two coupled chemical chaotic oscillators is investigated numerically using chemical Langevin equations. It is found that internal noise can enhance the phase synchronization, and there exists an optimal internal noise level such that the best phase synchronization is achieved. Since the magnitude of the internal noise is changed via the variation of the system size, these phenomena also indicate the existence of an optimal system size.

We developed the control technique for nonlinear oscillators when simple feedback alters the oscillation energy. The control does not require any computation nor knowledge of the system equations. Depending on the control perturbation parameters, two scenarios take place: (i) changing (linearly or nonlinearly) the oscillator damping, and (ii) suppressing (enhancing) the driving force. In the case of large deviations between phases of the chaotic oscillator and the driving force, only first scenario holds. Generalization of the technique to the case of oscillator networks is considered.

We have observed anti-synchronization phenomena in coupled identical chaotic oscillators. Anti-synchronization can be characterized by the vanishing of the sum of relevant variables. We have qualitatively analyzed its base mechanism by using the dynamics of the difference and the sum of the relevant variables in coupled chaotic oscillators. Near the threshold of the synchronization and anti-synchronization transition, we have obtained the novel characteristic relation.

This work provides a methodology for the control of complex non-linear systems by compensating for the non-linear part using support vector machines (SVM) and subsequently developing simple linear feedback law for control. The method tested for the benchmark Rossler and Lorenz chaotic oscillators shows excellent performance.

Two-dimensional (2D) lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rossler systems the cluster synchronization regimes are stable and robust against up to 10%-15% parameter mismatch and against small noise.

We numerically study the dynamic behaviour of coupled chaotic oscillators in a so-called semiquantum chaotic system in which one is classical and the other is quantum mechanical. Fourier spectra of the classical oscillator and the ground-state wavefunction of the quantum part have been investigated, when small pulse perturbations are applied. It is found that semiquantum chaos can also be successfully controlled by the nonfeedback method.

An open plus nonlinear closed loop control law is presented for chaotic oscillations described by a set of non-autonomous second-order ordinary differential equations. It is proven that the basins of entrainment are global when the right-hand sides of the equations are given by arbitrary polynomial functions. The forced Duffing oscillator and the forced van der Pol oscillator are treated as numerical examples to demonstrate the applications of the method.

The improvement of phase synchronization between two coupled chaotic oscillators achieved using a pulse as the external force in a glow discharge plasma is experimentally demonstrated. The effect of coupling and pulse application on the dynamic behaviors of systems is investigated. From the viewpoint of the phase of oscillations, experimental results show that phase synchronization between two coupled chaotic oscillators is improved by applying a pulse., Journal of the Physical Society of Japan, 86(9), 095003; 2017

The phenomenon of amplitude death has been explored using a variety of different coupling strategies in the last two decades. In most of the work, the basic coupling arrangement is considered to be static over time, although many realistic systems exhibit significant changes in the interaction pattern as time varies. In this article, we study the emergence of amplitude death in a dynamical network composed of time-varying interaction amidst a collection of random walkers in a finite region of three dimensional space. We consider an oscillator for each walker and demonstrate that depending upon the network parameters and hence the interaction between them, global oscillation in the network gets suppressed. In this framework, vision range of each oscillator decides the number of oscillators with which it interacts. In addition, with the use of an appropriate feedback parameter in the coupling strategy, we articulate how the suppressed oscillation can be resurrected in the systems' parameter space. The phenomenon of amplitude death and the resurgence of oscillation is investigated taking limit cycle and chaotic oscillators for broad ranges of parameters, like interaction strength k between the entities, vision range r and the speed of movement v., Comment: 7 pages, 7 figures (Accepted for publication in Europhysics Letters, July 2017)

We show that two identical chaotic oscillators can evolve in antiphase synchronization regime when noncontinuous coupling between them is introduced. As an example, we consider dynamics of two mechanical oscillators coupled by impacts.

Existing works on coherence resonance, i.e., the phenomenon of noise-enhanced temporal regularity, focus on excitable dynamical systems such as those described by the FitzHugh-Nagumo equations. We extend the scope of coherence resonance to an important class of dynamical systems: coupled chaotic oscillators. In particular, we show that, when a system of coupled chaotic oscillators is under the influence of noise, the degree of temporal regularity of dynamical variables characterizing the difference among the oscillators can increase and reach a maximum value at some optimal noise level. We present numerical results illustrating the phenomenon and give a physical theory to explain it.

We report experimental evidence of phase synchronization of high dimensional chaotic oscillators in a laboratory experiment. The experiment consists of a thermocapillary driven convective cell in a time dependent chaotic regime. The synchronized states emerge as a consequence of a localized temperature perturbation to the heater. The transition to phase synchronization is studied as a function of the external perturbations. The existence and stability conditions for this phenomenon are discussed.

The synchronization transition is described for a system of two asymmetrically coupled chaotic oscillators. Such a system can represent the two-cluster state in a large ensemble of globally coupled oscillators. It is shown that the transition can be typically mediated by a transcritical transversal bifurcation. The latter has a hard brunch that dominates the global dynamics, so that the synchronization transition is normally hard. For a particular example of coupled logistic maps a diversity of transition scenaria includes both local and global riddling. In the case of small non-identity of the interacting systems the riddling is shown to turn into an exterior or interior crisis.

The synchronization of two different chaotic oscillators is studied, based on an open-loop control – the entrainment control. We consider two types of synchronization: complete synchronization and effectively complete synchronization. The sufficient conditions that two different systems can be synchronized by this method is discussed. Furthermore, a hierarchical idea to synchronize multiple chaotic subsystems is proposed.

A nonlinear adaptive control strategy for a class of chaotic oscillators is proposed. The class of chaotic systems include both externally and parametrically excited systems, such as the Duffing and the Van de Pol oscillators. By assuming that the exact model of the system is not known, the controller employs a nonlinear observer to track the states and disturbances of the nonlinear system, and constructs a state-error-like feedback law. Simulations are provided to illustrate the effectiveness of the controller.

The effects of impurities introduced into a lattice of strongly coupled oscillators are studied. Contrary to our expectations, a single impurity can control the chaotic behavior of a long array of chaotic oscillators and bring about complete synchronization. This effect appears to be independent of the length of the chain, obtaining self-organization for chains as long as 512 lattice sites. By appropriately introducing more that one impurity, if needed, longer chains of oscillators can be tamed.

When two identical chaotic oscillators interact, one or more intervals of coupling parameters generally exist in which the synchronized state is weakly stable, and its basin of attraction is riddled with holes that are repelled from it. The paper discusses the role of the absorbing area for the emergence of local vs global riddling and for controlling the dynamics, once synchronization breaks down.

We investigate the collective behavior of a system of chaotic Rossler oscillators indirectly coupled through a common environment that possesses its own dynamics and which in turn is modulated by the interaction with the oscillators. By varying the parameter representing the coupling strength between the oscillators and the environment, we find two collective states previously not reported in systems with environmental coupling: (i) nontrivial collective behavior, characterized by a periodic evolution of macroscopic variables coexisting with the local chaotic dynamics; and (ii) dynamical clustering, consisting of the formation of differentiated subsets of synchronized elements within the system. These states are relevant for many physical and biological systems where interactions with a dynamical environment are frequent., Comment: 4 pages, 4 figs

The synchronization of two different chaotic oscillators is studied. First, this problem is understood in terms of chaos suppression. Next, a robust asymptotic controller is used in the chaos control. The robust controller comprises an uncertainty estimator and a linearizing control law. In this way, synchronization is obtained despite the master model being strictly different from the slave model. In fact, the controller does not know the master oscillator model.

A time delay coordinates (TDC) strategy to control a class of chaotic oscillators is studied. The class of chaotic systems includes both externally and parametrically excited systems, such as the Duffing and the Coulombic oscillators. By assuming that the exact model of the system is not known and that position is the only state available for measurements, the controller comprises a linearizing-like feedback and an uncertainty estimator. Simulations are provided to illustrate the performance of the controller.

Synchronisation between coupled oscillatory systems is a common phenomenon in many natural as well as technical systems. Varying the strength of coupling often leads to qualitative changes in the complex dynamics of the mutually coupled systems including different types of synchronisation such as phase, lag, generalised, or even complete synchronisation. Here, we study the geometric signatures of coupling along with the onset of generalised synchronisation between two coupled chaotic oscillators by mapping the systems' individual as well as joint recurrences in phase space to a complex network. For a paradigmatic continuous-time model system, the transitivity properties of the resulting joint recurrence networks display distinct variations associated with changes in the structural similarity between different parts of the considered trajectories. They therefore provide a useful indicator for the emergence of generalised synchronisation. This paper is dedicated to the 25th anniversary of the introduction of recurrence plots by Eckmann et al. (Europhys. Lett. 4 (1987), 973)., Comment: 7 pages, 3 figures

The dynamical behavior of two coupled chaotic oscillators caused by current-driven ion acoustic instability is experimentally studied. It is found that when two oscillators are interacting with the mediation of a central mesh grid, and the DC potential applied to either of the two oscillators is varied gradually while maintaining coupling, the two oscillators synchronize in a certain region.

We analyze nonlocally coupled networks of identical chaotic oscillators with either time-discrete or time-continuous dynamics (Henon map, Lozi map, Lorenz system). We hypothesize that chimera states, in which spatial domains of coherent (synchronous) and incoherent (desynchronized) dynamics coexist, can be obtained only in networks of oscillators with nonhyperbolic chaotic attractors and cannot be found in networks of systems with hyperbolic chaotic attractors. This hypothesis is supported by analytical results and numerical simulations for hyperbolic and nonhyperbolic cases.

Low-order Duffing and high-order Rossler chaotic oscillator are connected together and new self-adaption frequency detection method is presented. The frequency difference control between unknown signal and the periodic driving force is realized in this paper and the self-adaption is obtained. Thus, the detection precision and speed are promoted. The limitation that there are too many chaotic oscillators in Duffing system is broken. Meanwhile the disadvantage that the detection speed is lower in Rossler chaotic control is overcome. The self-adaption choice of frequency difference control is realized using the Duffing and Rossler different chaotic oscillators to obtain unknown signal frequency. The simulation results show that the presented method is feasible and effective.