Search

We show that interactions of inherently chaotic oscillators can lead to coexistence of regular oscillatory regimes and chaotic oscillations in the rings of coupled oscillators provided that the level of interaction between the oscillators exceeds a threshold value. The transformation of the initially chaotic dynamics into the regular dynamics in a number of the coupled oscillators is shown to result from suppression of chaos by separation of certain oscillation periods from the continuous spectra, which are characteristic of chaotic oscillations.

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.

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.

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.

This paper presents a digitally programmable multi-direction fully integrated chaotic oscillator. Unlike the conventional chaotic oscillators, the proposed digitally programmable multi-direction chaotic oscillator is fully integrated in one single chip, and it achieves lower supply voltage, lower power dissipation and smaller chip area. Moreover, by controlling the digitally programmable MOS switches turning on and off, the presented chaotic oscillator can provide chaotic oscillation in three different directions. The proposed digitally programmable multi-direction fully integrated chaos oscillator is verified with Cadence IC Design Tools. The post-layout simulation results demonstrate that the chaotic oscillator consumes 99.5 mW from ±2.5 V supply voltage, and it takes a compact chip area of 0.177 mm2. The integrated chaos oscillator has a wide range of practical application prospects in chaotic communications or other applications demanding portable chaos systems.

In this paper we address the problem of parameter estimation performance enhancement for Duffing-like chaotic oscillator with parametric uncertainties. The Dynamic Regressor Extension and Mixing (DREM) approach, which has been recently proposed by the authors, is proven to be a powerful tool in transients improvement; however, it was never applied before for chaotic systems. Here we elaborate on how the DREM approach can be applied for the parameter reconstruction problem of chaotic signal generator used in the data transmitting scenario. The set of numerical examples illustrates the benefits of the proposed approach.

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.

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.

The most important part of any information transmission system based on dynamic chaos is a chaotic oscillator. The paper deals with issues of construction and gives a comparative analysis of two chaotic oscillators: one on time delay integrators and another one with the inertial circuit of the second order with a frequency-independent shaper in the feedback circuit. The simulations developed in PSIM-9 allow to do research of various types of generated chaotic oscillations reliant on changes in control parameters.

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 investigate analytically and numerically coupled lattices of chaotic maps where the interaction is non-local, i.e., each site is coupled to all the other sites but the interaction strength decreases exponentially with the lattice distance. This kind of coupling models an assembly of pointlike chaotic oscillators in which the coupling is mediated by a rapidly diffusing chemical substance. We consider a case of a lattice of Bernoulli maps, for which the Lyapunov spectrum can be analytically computed and also the completely synchronized state of chaotic Ulam maps, for which we derive analytically the Lyapunov spectrum.

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.

The hardware realization of a chaotic communication system from the description of continuous-time multi-scroll chaotic oscillators is introduced herein by using field-programmable gate arrays (FPGA). That way, two multi-scroll chaotic oscillators generating 2 and 6 scrolls are synchronized by applying Hamiltonian forms and observer approach. The synchronized master-slave topology is used to implement a secure communication system by adding chaos to an image at the transmission stage and by subtracting chaos at the recover stage. The FPGA realization begins by applying numerical methods to solve the system of equations describing the whole chaotic communication system. Further, the replacement of multipliers by single constant multiplication blocks reduces the use of hardware resources and accelerates the processing time as well. Using chaotic oscillators with 2 and 6 scrolls, three kinds of images are processed: one in black and white and two in gray tones. Finally, the experimental results confirm the appropriateness on realizing chaotic communication systems for image processing by using FPGAs.

The intermittent behavior at the boundary of phase synchronization in the presence of noise is investigated. It is shown that in a certain range of the coupling parameter and noise intensity, the system experiences the intermittency of needle’s eye- and ring-type intermittencies. The basic results are demonstrated with two unidirectionally coupled Ressler chaotic oscillators.

Ensembles of several Rossler chaotic oscillators are considered. It is shown that a typical phenomenon for such systems is the emergence of different and sufficiently high dimensional invariant tori. The possibility of a quasi-periodic Hopf bifurcation and a cascade of such bifurcations based on tori of increasing dimension is demonstrated. The domains of resonance tori are revealed. Boundaries of these domains correspond to the saddle-node bifurcations. Inside the domains of resonance modes, torus-doubling bifurcations and destruction of tori are observed.

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.

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.

The problem of obtaining chaotic synchronous response in a drive — response system is considered for the case of the filter between drive and response systems which is equivalent to multipath channel. The proposed scheme is based on ring-structure chaotic oscillators. The establishment of the synchronous chaotic regime is shown.

In this paper we consider the problem of characterizing chaotic dynamics from noisy sequences of return times. We discuss features of computing the largest Lyapunov exponent and restrictions of the reliable estimation of the second exponent. We illustrate the ability of characterizing dynamics of small networks of chaotic oscillators for the case of under-threshold input signals.

Recently, chaotic systems with hidden attractors and multistability have been of great interest in the field of chaos and nonlinear dynamics. Two special categories of systems with multistability are systems with extreme multistability and systems with megastability. In this paper, the simplest (yet) megastable chaotic oscillator is designed and introduced. Dynamical properties of this new system are completely investigated through tools like bifurcation diagram, Lyapunov exponents, and basin of attraction. It is shown that between its countable infinite coexisting attractors, only one is self-excited and the rest are hidden.

A class of RC-OTA hysteretic-chaotic os- cillators has been previously reported using electronics; therefore, hysteresis is realized by an electronic circuit. To obtain a mathematical model of this RC-OTA chaotic-electronic device, hysteresis modeling turns an important issue. Here, we develop a new mathemat- ical hysteretic model proposing a new jounce-chaotic oscillator. Chaosity test is proved using Poincare theory. After that, a synchronization scheme is granted to synchronize our new jounce-chaotic oscillator (the transmitter) to a dynamics second-order system (the receiver).

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.

SundarapandianPehlivan chaotic system (SPCS) has been modeled and simulated in three distinct platforms (Numerical, Pspice and FPGA-based).SPCS has been modeled in VHDL language by using RK4 algorithm and has been synthesized for Xilinx Virtex-6 FPGA chip in development environment.The maximum operation frequency of FPGA-based chaotic system is 293.815MHz and the system can calculate 1,000,000 data in 0.201s.The high speed TRNG model has been implemented on an FPGA using SPCS and the maximum operating frequency has been achieved as 293MHz with a speed of 58.76Mbit/s.Proposed new TRNG can be run as fast as been verified by two statistical based standards, FIPS-140-1 and NIST-800-22. Chaotic systems and chaos-based applications have been commonly used in the fields of engineering recently. The most essential part of them is the chaotic oscillator that has very critical role in some applications such as chaotic communications and cryptography. In this study, SundarapandianPehlivan chaotic system has been modeled and simulated in three distinct platforms to show the advantages of FPGA-based chaotic oscillator with respect to alternative solutions. In the first stage, the chaotic system has been modeled numerically by the help of fourth order of RungeKutta (RK4) method. Additionally, phase portraits of the system have been obtained and Lyapunov exponents have been examined. Secondly, the system has been modeled by using PSpice for the implementation of the chaotic system with analog circuit elements. Then, Pspice simulation results have been compared with the numerical outcome to justify the designed model. Furthermore, the chaotic system has been physically confirmed with real analog circuit elements. Signals obtained from the physical system have been verified with both numerical and PSpice results. It has been also modeled by the help of method of RK4 in a hardware description language (VHDL) and the model further has been synthesized and tested for Xilinx Virtex-6 FPGA chip. Finally, the chaotic oscillator designed has been tested for True Random Number Generators (TRNG) and the maximum operating frequency has been achieved as 293MHz with a speed of 58.76Mbit/s. Besides, the random bit sets produced by TRNG have been further verified by FIPS-140-1 and NIST-800-22 statistical standards and it has been proved that the proposed design can be used in embedded cryptologic applications.

We present a review on the electronic design of chaotic oscillators. Multi-scroll chaotic oscillators are listed according to their electronic implementations. A 3-scrolls oscillator is analyzed from its mathematical description, and designed with current-feedback operational amplifiers. Finally, we list the integrated realizations, and discuss key points for future research on the design of multi-scroll chaotic oscillators.

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].

Chimeras, namely coexisting desynchronous and synchronized dynamics, are formed in an ensemble of identically coupled identical chaotic oscillators when the coupling induces multiple stable attractors, and further when the basins of the different attractors are intertwined in a complex manner. When there is coupling-induced multistability, an ensemble of identical chaotic oscillators-with global coupling, or also under the influence of common noise or an external drive (chaotic, periodic, or quasiperiodic)-inevitably exhibits chimeric behavior. Induced multistability in the system leads to the formation of distinct subpopulations, one or more of which support synchronized dynamics, while in others the motion is asynchronous or incoherent. We study the mechanism for the emergence of such chimeric states, and we discuss the generality of our results.

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.

In this paper, by modifying a known two-dimensional oscillator, we obtain an interesting new oscillator with coexisting limit cycles and point attractors. Then by changing this new system to its forced version and choosing a proper set of parameters, we introduce a chaotic system with some very interesting features. In this system, not only can we see the coexistence of different types of attractors, but also a fascinating phenomenon: some initial conditions can escape from the gravity of nearby attractors and travel far away before being trapped in an attractor beyond the usual access.

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.

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.

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.

Nowadays, many works have been presented regarding the modeling, simulation and circuit realization of different kinds of continuous-time multi-scroll chaotic attractors. However, very few works describe the experimental realization of attractors having high maximum Lyapunov exponent (MLE) and high entropy, which are desirable characteristics to guarantee better chaotic unpredictability. For instance, two chaotic oscillators having the same MLE values can behave in a very different way, e.g. showing different entropy values. That way, we describe the experimental realization of an optimized multi-scroll chaotic oscillator with both high MLE and entropy. First, the MLE is optimized by applying an evolutionary algorithm, which provides a set of feasible solutions. Second, the associated entropy is evaluated for each feasible solution. In this chapter, experimental results are shown for the electronic implementation of a chaotic oscillator generating 2-, 5- and 10-scrolls. Finally, the experimental results show that by increasing the number of scrolls both the MLE and its associated entropy increase in a similar proportion, thus guaranteeing better unpredictability.

In this work we analyse nonlocally coupled networks of identical chaotic oscillators. We study both time-discrete and time-continuous systems (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 chaotic non-hyperbolic systems and cannot be found in networks of hyperbolic systems. This hypothesis is supported by numerical simulations for hyperbolic and non-hyperbolic cases.

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.

A method for the estimation of the Lyapunov exponent corresponding to enslaved phase dynamics from time series has been proposed. It is valid for both nonautonomous systems demonstrating periodic dynamics in the presence of noise and coupled chaotic oscillators and allows us to estimate precisely enough the value of this Lyapunov exponent in the supercritical region of the control parameters. The main results are illustrated with the help of the examples of the noised circle map, the nonautonomous Van der Pole oscillator in the presence of noise, and coupled chaotic Rössler systems.

Intermittent behavior near the boundary of phase synchronization in the presence of noise is studied. In certain range of the coupling parameter and noise intensity the intermittency of eyelet and ring intermittencies is shown to take place. Main results are illustrated using the example of two unidirectional coupled Rössler systems. Similar behavior is shown to take place in two hydrodynamical models of Pierce diode coupled unidirectional., {"references":["P. Berge, Y. Pomeau, and C. Vidal, Order within Chaos, John Wiley and\nSons, New York, 1984.","C. M. Kim, G. Yim, J. Ryu, and Y. Park, \"Characteristic relations of\ntype III intermittency in an electronic circuit,\" Phys. Rev. Lett., vol. 80,\nno. 24, pp. 5317–5320, June 1998.","J. L. Perez Velazquez and et al., \"Type III intermittency in human partial\nepilepsy,\" European Journal of Neuroscience, vol. 11, pp. 2571–2576,\n1999.","I. Z. Kiss and J. L. Hudson, \"Phase synchronization and suppression of\nchaos through intermittency in forcing of an electrochemical oscillator,\"\nPhys. Rev. E, vol. 64, no. 4, pp. 046215, 2001.","S. Boccaletti, E. Allaria, R. Meucci, and F. T. Arecchi, \"Experimental\ncharacterization of the transition to phase synchronization of chaotic\nCO2 laser systems,\" Phys. Rev. Lett., vol. 89, no. 19, pp. 194101, 2002.","J. L. Cabrera and J. Milnor, \"On-off intermittency in a human balancing\ntask,\" Phys. Rev. Lett., vol. 89, no. 15, pp. 158702, 2002.","A. E. Hramov, A. A. Koronovskii, I. S. Midzyanovskaya, E. Sitnikova,\nand C. M. Rijn, \"On-off intermittency in time series of spontaneous\nparoxysmal activity in rats with genetic absence epilepsy,\" Chaos, vol.\n16, pp. 043111, 2006.","E. Sitnikova, A. E. Hramov, V. V. Grubov, A. A. Ovchinnkov, and A.\nA. Koronovsky, \"Onoff intermittency of thalamo-cortical oscillations in\nthe electroencephalogram of rats with genetic predisposition to absence\nepilepsy,\" Brain Research, vol. 1436, pp. 147–156, 2012.","M. Dubois, M. Rubio, and P. Berg´e, \"Experimental evidence of\nintermiasttencies associated with a subharmonic bifurcation,\" Phys. Rev.\nLett., vol. 51, pp. 1446–1449, 1983.\n[10] J. F. Heagy, N. Platt, and S. M. Hammel, \"Characterization of on–off\nintermittency,\" Phys. Rev. E, vol. 49, no. 2, pp. 1140–1150, 1994.\n[11] A. S. Pikovsky, G. V. Osipov, M. G. Rosenblum, M. Zaks, and J.\nKurths, \"Attractor–repeller collision and eyelet intermittency at the\ntransition to phase synchronization,\" Phys. Rev. Lett., vol. 79, no. 1, pp.\n47–50, 1997.\n[12] A. E. Hramov, A. A. Koronovskii, M. K. Kurovskaya, and S. Boccaletti,\n\"Ring intermittency in coupled chaotic oscillators at the boundary of\nphase synchronization,\" Phys. Rev. Lett., vol. 97, pp. 114101, 2006.\n[13] A. E. Hramov, A. A. Koronovskii, O. I. Moskalenko, M. O. Zhuravlev,\nV. I. Ponomarenko, and M. D. Prokhorov, \"Intermittency of\nintermittencies,\" CHAOS, vol. 23, no. 3, pp. 033129, 2013.\n[14] N. N. Nikitin, S. V. Pervachev, and V. D. Razevig, \"About solution of\nstochastic diferential equations of follow-up systems,\" Automation and\ntelemechanics, vol. 4, pp. 133–137, 1975, in Russian.\n[15] M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, \"From phase to lag\nsynchronization in coupled chaotic oscillators,\" Phys. Rev. Lett., vol. 78,\nno. 22, pp. 4193–4196, 1997.\n[16] A. E. Hramov, A. A. Koronovskii, and M. K. Kurovskaya, \"Two types\nof phase synchronization destruction,\" Phys. Rev. E, vol. 75, no. 3, pp.\n036205, 2007.\n[17] B. B. Godfrey, \"Oscillatory nonlinear electron flow in Pierce diode,\"\nPhys. Fluids, vol. 30, pp. 1553, 1987.\n[18] H. Matsumoto, H. Yokoyama, and D. Summers, \"Computer simulations\nof the chaotic dynamics of the Pierce beam–plasma system,\" Phys.\nPlasmas, vol. 3, no. 1, pp. 177, 1996.\n[19] R. A. Filatov, A. E. Hramov, and A. A. Koronovskii, \"Chaotic\nsynchronization in coupled spatially extended beam-plasma systems,\"\nPhys. Lett. A, vol. 358, pp. 301–308, 2006.\n[20] P. J. Rouch, Computational fluid dynamics. Hermosa publishers,\nAlbuquerque, 1976.\n[21] O. I. Moskalenko, A. A. Koronovskii, A. E. Hramov, and S. Boccaletti,\n\"Generalized synchronization in mutually coupled oscillators and\ncomplex networks,\" Phys. Rev. E, vol. 86, pp. 036216, 2012.\n[22] A.E. Hramov, A.A. Koronovskii, \"Detecting unstable periodic spatiotemporal\nstates of spatial extended chaotic systems\", Europhysics\nLetters, vol. 80, pp. 10001, 2007.\n[23] O. I. Moskalenko, A. A. Koronovskii, A. E. Hramov, M. O. Zhuravlev,\nYu. I. Levin, \"Cooperation of deterministic and stochastic mechanisms\nresulting in the intermittent behavior\", Chaos, Solitons & Fractals, vol.\n68, pp. 58-64, 2014."]}

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.

A method is proposed for designing chaotic oscillators. Mathematically, three so-called partial oscillators S j (j = 1, 2, 3) are chosen, each of which is modeled by a nonlinear system of ordinary differential equations with a single attractor—an equilibrium or a cycle (the case S 1 = S 2 = S 3 is not excluded). It is shown that, when unidirectionally connected in a circle of the form with suitably chosen parameters, these oscillators can exhibit a joint chaotic behavior.

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.

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.

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.

This paper presents a chaotic oscillator based on the nonlinearity of the typical transconductance (Gm). This chaotic circuit only consists of 13 MOS transistors and three grounded capacitors, which is one of the smallest chaotic oscillators. This circuit operates on low voltage supply (± 1.5 V). The dimensionless form of the circuit is also introduced to confirm the circuit simulation.

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.