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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.

In recent years designing new multistable chaotic oscillators has been of noticeable interest. A multistable system is a double-edged sword which can have many benefits in some applications while in some other situations they can be even dangerous. In this paper, we introduce a new multistable two-dimensional oscillator. The forced version of this new oscillator can exhibit chaotic solutions which makes it much more exciting. Also, another scarce feature of this system is the complex basins of attraction for the infinite coexisting attractors. Some initial conditions can escape the whirlpools of nearby attractors and settle down in faraway destinations. The dynamical properties of this new system are investigated by the help of equilibria analysis, bifurcation diagram, Lyapunov exponents’ spectrum, and the plot of basins of attraction. The feasibility of the proposed system is also verified through circuit implementation.

In the present work, we present a new method for assessing causality in uni-directionally coupled chaotic oscillators with small frequency mismatch. This method is based on the correlation between changes in the phase dynamics of the slave oscillator and the dynamics of the phase difference between the oscillators. Application of the proposed approach to master-slave R¨ossler systems showed that the new algorithm is well-suited for assessing the presence and direction of coupling, especially in the case of weak coupling.

In this chapter, non-autonomous oscillators are considered as mappings from input functions of time to a circle’s worth of functions giving the state as a function of time. This view is justified using the theory of normally hyperbolic submanifolds of dynamical systems. It illuminates the phenomena of phase-locking, synchronisation and chimera; it allows an extension of the concept of coupling; and it allows a hierarchical aggregation treatment of synchronisation in networks of oscillators. The view extends to excitable and chaotic oscillators.

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.

In some of the real-world networks, the interacting agents synchronize in groups or clusters, not in a complete way; and also, many networks can be modeled by multi-layer networks where nodes are coupled in several independent networks. In this work, we deal with the problem of synchronizing a pair of layers of networks via a double distributed control. Specifically, these controllers depends on the degree of nodes. Our first aim is synchronizing just some nodes of each layer, and then as a second goal, we apply a control to the already synchronized nodes of each layer to assure that they follow the same dynamics in both layers.

Graphical abstract, Memristor is a non-linear circuit element in which voltage-current relationship is determined by the previous values of the voltage and current, generally the history of the circuit. The nonlinearity in this component can be considered as a fractional-order form, which yields a fractional memristor (fracmemristor). In this paper, a fractional-order memristor in a chaotic oscillator is applied, while the other electronic elements are of integer order. The fractional-order range is determined in a way that the circuit has chaotic solutions. Also, the statistical and dynamical features of this circuit are analyzed. Tools like Lyapunov exponents and bifurcation diagram show the existence of multistability and antimonotonicity, two less common properties in chaotic circuits.

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

Chaotic oscillators are known for their multiparametric sensitivity meaning their different responses to different impacts. In this paper, we explore the possibility of the chaotic oscillator to be used as analog sensors affected by changes in multiple parameters simultaneously, such as inductance and resistance of sensing element. A number of methods are applied for analyzing chaotic oscillations, including calculating the recurrence density and the maximal Lyapunov exponent. As a result, we show the ability of a single chaotic circuit to measure two parameters simultaneously.

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 propose a method of controlling an ensemble of mobile agents with variable coupling topology that is based on the principles of phase synchronization in a system of regular and chaotic oscillators. Results of modeling of the controlled motion of mobile agents in systems with serial, parallel, and strictly preset motion are presented.

The dynamic behavior of many physical, biological, and other systems, are organized according to the synchronization of chaotic oscillators. In this paper, we have proposed a new method with low sensitivity to noise for detecting synchronization by mapping time series to complex networks, called the ordinal partition network, and calculating the permutation entropy of that structure. We show that this method can detect different kinds of synchronization such as complete synchronization, phase synchronization, and generalized synchronization. In all cases, the estimated permutation entropy decreases with increased synchronization. This method is also capable of estimating the topology of the network graph from the time series, without knowledge of the dynamical equations of individual nodes. This approach has been applied for the two identical and nonidentical coupled Rössler systems, two nonidentical coupled Lorenz systems, and a ring of coupled Lorenz96 oscillators.

Natural and engineered networks, such as interconnected neurons, ecological and social networks, coupled oscillators, wireless terminals and power loads, are characterized by an appreciable heterogeneity in the local connectivity around each node. For instance, in both elementary structures such as stars and complex graphs having scale-free topology, a minority of elements are linked to the rest of the network disproportionately strongly. While the effect of the arrangement of structural connections on the emergent synchronization pattern has been studied extensively, considerably less is known about its influence on the temporal dynamics unfolding within each node. Here, we present a comprehensive investigation across diverse simulated and experimental systems, encompassing star and complex networks of Rössler systems, coupled hysteresis-based electronic oscillators, microcircuits of leaky integrate-and-fire model neurons, and finally recordings from in-vitro cultures of spontaneously-growing neuronal networks. We systematically consider a range of dynamical measures, including the correlation dimension, nonlinear prediction error, permutation entropy, and other information-theoretical indices. The empirical evidence gathered reveals that under situations of weak synchronization, wherein rather than a collective behavior one observes significantly differentiated dynamics, denser connectivity tends to locally promote the emergence of stronger signatures of nonlinear dynamics. In deterministic systems, transition to chaos and generation of higher-dimensional signals were observed; however, when the coupling is stronger, this relationship may be lost or even inverted. In systems with a strong stochastic component, the generation of more temporally-organized activity could be induced. These observations have many potential implications across diverse fields of basic and applied science, for example, in the design of distributed sensing systems based on wireless coupled oscillators, in network identification and control, as well as in the interpretation of neuroscientific and other dynamical data.

Multiscroll chaotic attractors generated by irregular saturated nonlinear functions with optimized positive Lyapunov exponent are designed and implemented. The saturated nonlinear functions are designed in an irregular way by modifying their parameters such as slopes, delays between slopes, and breakpoints. Then, the positive Lyapunov exponent is optimized using the differential evolution algorithm to obtain chaotic attractors with 2 to 5 scrolls. We observed that the resulting chaotic attractors present more complex dynamics when different patterns of irregular saturated nonlinear functions are considered. After that, the optimized chaotic oscillators are physically implemented with an analog discrete circuit to validate the use of proposed irregular saturated functions. Experimental results are consistent with MATLAB™ and SPICE circuit simulator. Finally, the synchronization between optimized and nonoptimized chaotic oscillators is demonstrated.

In this paper, the dynamic modeling of two identical oscillators which are coupled through threshold controller is proposed. Until now, most of the synchronization of chaotic systems found in literature is based on common coupling methods (unidirectional and bidirectional) that attracted the attention of researchers. To strengthen this, the idea illustrated here is to show the effectiveness of a new kind of coupling called threshold controller coupling. Using this, complete and anticipatory synchronization could be achieved. The system used is of second-order non-autonomous type. The coupled system is investigated using MATLAB–Simulink technique. The result shows that based on coupling strength, coupled system is switched among the basic synchronization, viz. lead and complete.

Chaotic dynamics has been shown in the dynamics of neurons and neural networks, in experimental data and numerical simulations. Theoretical studies have proposed an underlying role of chaos in neural systems. Nevertheless, whether chaotic neural oscillators make a significant contribution to network behaviour and whether the dynamical richness of neural networks is sensitive to the dynamics of isolated neurons, still remain open questions. We investigated synchronization transitions in heterogeneous neural networks of neurons connected by electrical coupling in a small world topology. The nodes in our model are oscillatory neurons that – when isolated – can exhibit either chaotic or non-chaotic behaviour, depending on conductance parameters. We found that the heterogeneity of firing rates and firing patterns make a greater contribution than chaos to the steepness of the synchronization transition curve. We also show that chaotic dynamics of the isolated neurons do not always make a visible difference in the transition to full synchrony. Moreover, macroscopic chaos is observed regardless of the dynamics nature of the neurons. However, performing a Functional Connectivity Dynamics analysis, we show that chaotic nodes can promote what is known as multi-stable behaviour, where the network dynamically switches between a number of different semi-synchronized, metastable states.

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.

We present new experimental results on a recently developed set-up, implementing a dynamically configurable network of chaotic oscillators with Chua’s circuits as nodes. The set-up has been designed and tailored to easily perform real time experiments on complex networks with arbitrary topology . We focus here on the emergence of symmetry related synchronization patterns, as well as on the switching among different clusters due to modification of the network structure and/or coupling strength, that are experimentally analyzed for the first time in such type of networks. The observed behavior confirms basic theoretical expectations on small networks, as recently appeared in literature. Moreover the scalability to higher complexity network, as allowed by the considered set-up, is briefly discussed.

This chapter recalls a collection of experimental results on reconfigurable topology networks of oscillators, with Chua's circuits as chaotic dynamical nodes. Collective behaviours arising from network node interactions, such as global synchronization, emergence of clusters, patterns and waves formation, have been observed in a setup expressly developed [1-7].

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.

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.

We investigate synchronization in an array network of nearest-neighbor coupled chaotic oscillators. By using of the Lyapunov stability theory and matrix theory, a criterion for stability of complete synchronization is deduced. Meanwhile, an estimate of the critical coupling strength is obtained to ensure achieving chaos synchronization. As an example application, a model of coupled Chua's circuits with linearly bidirectional coupling is studied to verify the validity of the criterion.

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.

In this paper, a new method of achieving anticipatory and complete synchronization through threshold coupling of two threshold controllers based chaotic oscillators is proposed. Until now, most of the synchronization of chaotic systems stated in the literature are based on common coupling methods like unidirectional and bidirectional coupling but rarely, cascaded and delay coupling. In this direction to add on a new kind of coupling known as threshold coupling is employed in this work. This coupled systems exhibit complete (CS) and anticipatory synchronizations (AS) for decreasing coupling resistor value without parameter mismatch and delay line. The proposed circuit is investigated experimentally and circuit behavior is also simulated using Multisim.

In the present paper, we briefly discuss the construction of mapping model of coupled oscillator systems for limit-cycle oscillators and chaotic oscillators. A comparison of the proposed mapping model and other models, i.e., the Ikeda map and the phase map model is made. Furthermore, the critical dynamics associated with the breakdown of chaotic phase synchronization based on the present mapping model is discussed.

The phenomenon of generalized chaotic synchronization has been studied in a system of two unidirectionally coupled chaotic oscillators modeling two-sresonator klystron autooscillators. A mechanism explaining the observed behavior is presented.

Coupled chaotic oscillators are found to be able to achieve the partial synchronization state in some networks. The scaling relation between the synchronization time and the coupling at the synchronous threshold is found and explained. In the studies, it was found that the coupling of the synchronization properties of networks is not homogeneous.

We propose a method for determining the range of the coupling parameter for which the network of slightly nonidentical chaotic oscillators demonstrates stable synchronous behavior. As an example of using this method, we study the complete-synchronization regime of a network of nonidentical Rossler oscillators.

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 behavior of two unidirectionally coupled chaotic oscillators near the boundary of their generalized synchronization is analyzed. Using the modified system method, the position of this boundary on the plane of control parameters is considered and the physical mechanisms leading to the establishment of the generalized synchronization regime are elucidated.

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.

Chaotic oscillators with a negative-differential-resistance (NDR) subsystem having a multibranch piecewise-linear i–v characteristic (IVC) are considered. The problem of switching between chaotic oscillation regimes by changing the IVC electronically is addressed. New circuit realizations of the NDR subsystem are described in detail. It is shown that they provide highly stable IVCs.

Small-world networks are highly clustered networks with small average distance among the vertices. There are many natural and technological networks that present this kind of connections. We study the phase synchronization of small-world chaotic oscillator networks in this paper. We find that for Rossler oscillator networks, in the synchronous regime, the oscillation phases are locked, while the amplitudes vary chaotically. We further show the dependence of phase synchronization on the network coupling strength, the product of the shortcuts-adding probability and the number of chaotic oscillators, as well as the maximal frequency mismatch.

Phase synchronization of a chaotic oscillator that is driven by two chaotic signals is investigated. The anti-biased PS in the presence of biased coupling is found, i.e., the response oscillator can be phase synchronized by the drive with a weaker coupling rather than the stronger driver. The mechanism for this behavior is explored. In the non-PS region, alternating phase-locking is observed.

A chaotic system can be used to transform a periodic input sequence of events, e.g. impulses, into a sequence of output events. Even after transient, the output sequence is often aperiodic. In particular, it might happen that the output events are basically synchronized with the input events, but that some of them are randomly skipped from the output time series (imperfect phase synchronization [Zacks et al., 1999, 2000]). This phenomenon has been neatly observed by Schimz and Hildebrand [1992] in a rich series of experiments on the locomotor behavior of Halobacterium salinarium, where the input event was a light impulse and the output event was the reversal in the swimming direction of the bacterium. In this paper, we show that the same phenomenon occurs when classical, low dimensional chaotic oscillators are forced by a periodic sequence of impulses. This proves, on one side, that imperfect phase synchronization is a common phenomenon in chaotic oscillators subjected to periodic sequences of impulses, and, on the other side, that there are high chances that the locomotor behavior of Halobacterium salinarium can be modeled by a low dimensional deterministic model.

A novel Lorenz-type system of nonlinear differential equations is proposed. Unlike the original Lorenz system, where the chaotic dynamics remain confined to the positive half-space with respect to the Z state variable due to a limiting threshold effect, the proposed system enables bipolar swing of this state variable. In addition, the classical set of parameters (a, b, c) controlling the behavior of the Lorenz system are reduced to a single parameter, namely a. Two possible modes of operation are admitted by the system; switching between these two modes results in the creation of a complex butterfly chaotic attractor. Numerical simulations and results from an experimental setup are presented.

SUMMARY In the state of generating a double-scroll attractor, an oscillator switches between two subspaces without external force. This switching phenomenon is called the iselfswitching phenomenon.i We recognize that investigations of the behavior of self-switching phenomena on full-coupled chaotic oscillators are important problems in order to clarify spatiotemporal phenomena in real physical systems. In this paper, we investigate the behavior of self-switching phenomena on full-coupled chaotic oscillators. In the system, we can observe that all chaotic oscillators switch the subspace synchronously. We call this isynchronization of self-switching phenomena.i Synchronization of selfswitching phenomena is generated when oscillators are asynchronized. This synchronization of self-switching phenomena is the first observation of the existence of order in the asynchronization of chaos, which means disorder of states. ' 2001 Scripta Technica, Electron Comm Jpn Pt 3, 85(1): 14n23, 2002

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.

A system of two inductively coupled chaotic oscillators is considered. Normalized equations of motion for oscillators with arbitrary parameters are derived in the case of a cubic approximation of the nonlinear elements. The results of a numerical analysis of the features of interaction between chaotic coupled nonidentical subsystems are presented and a realistic experiment is described.