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Compared to the traditional single scroll and double scroll chaotic systems, multi-scroll chaotic systems present more complex structure and dynamic behavior, possess good application prospect in information security and secure communications. Therefore, theoretical research and circuit implementation of multi-scroll chaotic attractor generation has become a hot spot in the research field of chaos at present domain. In this paper, we briefly overview the recent progress that has been reported in the study of multi-scroll chaotic oscillators based on current-mode devices. Multi-scroll chaotic oscillators are listed according to their electronic implementations. Finally, we list multi-scroll chaotic oscillators based on current-mode devices, and prospect development trends of multi-scroll chaotic oscillators based on current-mode devices in the future.

Akgul, Akif/0000-0001-9151-3052; istanbullu, ayhan/0000-0002-7066-4238 WOS: 000372543600003 There has recently been an increase in the number of new chaotic system designs and chaos-based engineering applications. In this study, since homoclinic and heteroclinic orbits did not exist and analyses like Shilnikov method could not be used, a 3D chaotic system without equilibrium points was included and thus different engineering applications especially for encryption studies were realized. The 3D chaotic system without equilibrium points represents a new different phenomenon and an almost unexplored field of research. First of all, chaotic system without equilibrium points was examined as the basis and electronic circuit application of the chaotic system was realized and oscilloscope outputs of phase portraits were obtained. Later, chaotic system without equilibrium points was modelled on Labview Field Programmable Gate Array (FPGA) and then FPGA chip statistics, phase portraits and oscilloscope outputs were derived. With another study, VHDL and RK-4 algorithm were used and a new FPGA-based chaotic oscillators design was achieved. Results of Labview-based design on FPGA- and VHDL-based design were compared. Results of chaotic oscillator units designed here were gained via Xilinx ISE Simulator. Finally, a new chaos-based RNG design was achieved and internationally accepted FIPS-140-1 and NIST-800-22 randomness tests were run. Furthermore, video encryption application and security analyses were carried out with the RNG designed here.

In this paper we deal with the synchronization of chaotic oscillators with Liouvillian properties (chaotic Liouvillian system) based on nonlinear observer design. The strategy consists of proposing a polynomial observer (slave system) which tends to follow exponentially the chaotic oscillator (master system). The proposed technique is applied in the synchronization of Chua's circuit using Matlab-Simulink(R) and Matlab(R)-LMI programs. Simulation results are used to visualize and illustrate the effectiveness of Chua's oscillator in synchronization.

The paper describes a portable didactic experiment, realizing a complex network of Chua's circuits, along with its use for introducing collective dynamics in networks of nonlinear oscillators to students. The equipment enables the full configurability of the node's parameters and the network structure (topology and link impedances). An 8 nodes portable version is realized as a demonstrator, however the corresponding laboratory experiment is designed to be easily scalable to a high number of nodes. The network's control and the data acquisition are automated thanks to a USB interface board and a LabVIEW control interface. The experiment allows the student to perform some direct real time analysis of a set of networked (coupled) Chua's circuits, with the goal to give immediate impact to the discovery of collective behaviors. In particular, the proposed experiences range from the characterization of the complex dynamics of the isolated nodes to the complete synchronization, as well as the emergence of clustering and waves. It is adopted within some course on Advanced Circuit Theory at the Master Degree level.

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.

The problem of synchronization of chaotic oscillators, using state observers, is handled through the use of a simplified perturbed linear integration model of the chaotic system output dynamics. The linear simplified model of the chaotic system does not entitle approximate linearizations, nor state coordinate transformations, but, simply, a pure linear integration model with additive unknown but bounded perturbation inputs lumping all the output dynamics nonlinearities. An extended linear state observer (here addressed as Generalized Proportional Integral (GPI) observer) is proposed for the accurate estimation of the phase variables and the perturbation input of the nonlinear output dynamics. The effectiveness of the approach is tested in the synchronization of two study cases: The Genesio–Tesi chaotic system and the Rossler oscillator. As an application of the estimation process, a coding–decoding process involving encrypted messages, in transmitted phase variables, is implemented using a Rossler chaotic system.

When two different chaotic oscillators are coupled, generalized synchronization can occur. It may imply a very complicated relation between the states of drive and response systems. We propose a method that can be used to detect and characterize the generalized synchronization in modulated time-delayed systems. Using Krasovskii–Lyapunov theory, sufficient condition for generalized synchronization is derived. The proposed technique has been applied to synchronize prototype and Ikeda models by numerical simulation.

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.

A system for generating a multi-butterfly chaotic attractor using the multi-level-logic pulse-excitation technique is proposed. Two-butterfly, three-butterfly and four-butterfly attractors are demonstrated. Results from an experimental setup are also shown.

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 behavior of two unidirectionally coupled chaotic oscillators exhibited at the threshold of generalized chaotic synchronization is considered. The modified-system approach is applied to explain physical mechanisms of formation of this regime in the cases of large and small mismatches of interacting systems.

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 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.

Coupled chaotic oscillators show phase diffusion phenomenon in association with the breakdown of chaotic phase sychronization. In the present paper, the diffusion coefficient of the phase diffusion near the transition point of chaotic phase synchronization is discussed. The diffusion coefficient shows enhancement just after the desynchronization point and obeys certain scaling laws. To confirm the universality of this phenomenon, we study coupled Rossler oscillators and coupled chaotic maps recently proposed.

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.

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.

The phenomenon of ideal synchronization of a pair of identical dynamical systems coupled by a one-to-one negative feedback mechanism is described and explained. The theoretical analysis supported by numerical simulations, which is presented in this paper, confirms the properties of chaotic synchronization (in the case of full coupling) that have been observed earlier. Additionally, this analysis develops a description of the problem under consideration for partial coupling between analyzed dynamical systems. Two classical Lorenz systems and Duffing oscillators coupled by a one-to-one negative feedback mechanism are considered as numerical examples.

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.

A new equation system (based on synchronization of coupled chaotic oscillators) is proposed for bringing autonomous mobile agents into formation. Chaotic itinerancy of the oscillators is used to generate the re-formation. Sannomiya's fish simulator is applied to test the system behavior and the re-formation performance was verified using trajectory and quantitative measures.

The paper combines theoretical analyses with computer simulation studies of spatiotemporal synchronization regimes arising in a two-dimensional (2D) lattice of diffusively coupled identical oscillators with complicated individual dynamics. The existence of linear invariant manifolds, defining different modes of spatiotemporal synchronization, is examined. The set of possible modes of cluster synchronization is stated. The appearance and order of stabilization of the cluster synchronization regimes with increasing coupling between the oscillators are revealed for 2D lattices of coupled Lur'e systems and of coupled Rossler oscillators.

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

A simple chaotic system with a single nonquadratic term is developed to be a switchable chaotic signal generator in this paper. Additional nonlinearity of the absolute value function is introduced for reforming the structure without damaging the basic dynamics but yielding a new independent amplitude–frequency controller. A switchable chaotic experimental oscillator is designed afterwards, where two coefficients corresponding to two independent rheostats rescale the amplitude and frequency of the chaotic signals smoothly. To our knowledge, this has never been found in other chaotic oscillators.

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.

This paper studies a network of N chaotic oscillators connected by intermittently coupled capacitors (ICCs). The unit oscillator includes a capacitor and a binary hysteresis element with periodic thresholds. ICCs imply periodic and impulsive connection of the capacitors selected one by one from each cell. In order to analyze the ICC network, we provide the hybrid return map (HRM) with one real and N binary states. Using the HRM, we can clarify that the ICC network has co-existence states of various chaotic and periodic synchronous phenomena, and exhibits one of them depending on the initial states. The number of the co-existing phenomena increases illimitably as N increases. Some of the typical phenomena are confirmed in the laboratory.

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.

Rings of chaotic oscillators coupled unidirectionally through driving are studied. While synchronization is observed for small sizes of the ring, beyond a certain critical size a desynchronizing transition occurs. In the two examples studied here the system exhibits a transition to periodic rotating waves for rings of Lorenz systems, while one finds a sort of chaotic rotating waves when Chua's circuit is used.

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.

In this paper, we report mixed synchronization between two counter rotating chaotic oscillators. We describe a procedure how to obtain a counter rotating oscillator for generalized oscillators. We elaborate the method with numerical examples of the Sprott system, Pikovsky-Rabinovich (PR) circuit model. Noise-induced mixed synchronization is also reported in PR circuit model. The physical realization of mixed synchronization in an electronic circuit of two counters-rotating Sprott systems also shown., Comment: 15 pages(single column), 9 figures

Chaotic dynamics of numerous memristor-based circuits is widely reported in literature. Recently, some works have appeared which study the problem of synchronization control of these systems in a master-slave configuration. In the present paper, the spontaneous dynamic behavior of two chaotic memristor-based Chua’s circuits, mutually interacting through a coupling resistance, was studied via computer simulations in order to study possible self-organized synchronization phenomena. The used memristor is a flux controlled memristor with a cubic nonlinearity, and it can be regarded as a time-varying memductance. The memristor, in effect, retains memory of its past dynamic and any difference in the initial conditions of the two circuits results in different values of the corresponding memductances. In this sense, due to the memory effect of the memristor, even if coupled circuits have the same parameters they do not constitute two completely identical chaotic oscillators. As is known, for nonidentical chaotic systems, in addition to complete synchronizations (CS) other weaker forms of synchronization which provide correlations between the signals of the two systems can also occur. Depending on initial conditions and coupling strength, both chaotic and nonchaotic synchronization are observed for the system considered in this work.

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 quite universal mechanism of establishing chaotic synchronization regime in coupled dynamic systems is found. It is shown that the synchronous regime arises due to the phase coupling between the Fourier-spectrum components of the interacting chaotic oscillators.

In a world of an ever increasing information-trafficking, ultra wideband information transmission, as well as security are main issues. Chaotic oscillators inherently possess these two properties. Consequently, chaotic transmitter-receiver synchronization emerges as a key-topic in networked communication. The issues of synchronization robustness and synchronization stability are of great importance, if one means to turn in advantage properties of chaotic operating circuits and systems. As a result, not only synchronization accomplishment but also the ways a system desynchronizes is of great importance, if not crucial; however, these are not, usually, thoroughly studied. In this contribution a short overview of experimental work dedicated to the ways chaotic-synchronized communication systems desynchronize, is provided.

Chaotic dynamics are an interesting topic in nonlinear science that has been intensively studied during the last three decades due to its wide availability. Motivated by much researches on synchronization, the authors of this study have improved the time response of stabilization when parametrically excited Φ6—Van der Pol Oscillator (VDPO) and Φ6—Duffing Oscillator (DO) are synchronized identically as well as non-identically (with each other) using the Linear Active Control (LAC) technique using Mathematica. Furthermore, the authors have synchronized the same pairs of the oscillators using a more robust synchronization with faster time response of stability called Robust Adaptive Sliding Mode Control (RASMC). A comparative study has been done between the previous results of Njah’s work and our results based on Mathematica via LAC. The time response of stabilization of synchronization using RASMC has been discussed.

This work deals with the master-slave synchronization of chaotic oscillators with Liouvillian properties (chaotic Liouvillian system) based on nonlinear observer design. The strategy consists of proposing a polynomial observer (slave system) which tends to follow exponentially the chaotic oscillator (master system). The proposed technique is applied in the synchronization of Chua's circuit using Matlab-Simulink® and Matlab®-LMI programs. Experimental results are shown to visualize and illustrate the effectiveness of the proposed observer.

Experimental evidence is provided for the synchronization of chaotic self-excited oscillators and chaotic synchronous response. With unidirectional coupling between two chaotic nonlinear circuits, synchronous chaotic oscillations can result from either forced synchronization or the response phenomenon. With mutual coupling, two originally chaotic oscillators can become time periodic, a result that we have shown to persist for values of the coupling down to noise level.

In this paper, pinning control in a system of moving agents (each one associated to a chaotic dynamical system) has been investigated. In particular, we have studied and compared two different strategies for pinning control and discussed the not trivial relation between synchronization and control of the chaotic agents. Our results show how system parameters like agent density are critical in order to reach a synchronous behavior of the agents as well as to reach global control of the system by only pinning a reduced set of agents.

In this chapter, the author reviews the variety of forms that the synchronization of the dynamics of mutually coupled or unidirectionally driven chaotic oscillators can display. The aim is to provide a presentation of a background of knowledge on fundamental physics and mathematics that the author expects to be useful in the development of the application of chaos synchronization to telecommunications and cryptography.