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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 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

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

We investigate the transition between oscillatory and amplitude death (AD) states and the existence of death islands in intrinsic time-delayed chaotic oscillators under the simultaneous presence of diffusive (direct) and environmental (indirect) coupling. Studies in two-parameter space reveal that depending upon parameters and intrinsic time delays the coupling can bring the oscillators to the AD state and again can revive the system to oscillatory states, thus creating death islands in parameter space; this observation is in sharp contrast to the death scenario of non-delayed oscillators under the same coupling scheme where no death islands are formed. Using a linear stability analysis, we derive the explicit conditions for different transition scenarios. We use a continuation package for the time-delay systems to precisely identify the zone of AD and its islands and their origin. We also extend our study to the network of oscillators and show that the observed results are general for a large number of oscillators, too. Finally, we demonstrate our results experimentally to verify the analytical and numerical findings.

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.

In this work, we show how "chimera states," namely, the dynamical situation when synchronized and desynchronized domains coexist in an oscillator ensemble, can be controlled through a linear augmentation (LA) technique. Specifically, in the networks of coupled chaotic oscillators, we obtain chimera states through induced multistability and demonstrate how LA can be used to control the size and spatial location of the incoherent and coherent populations in the ensemble. We examine basins of attraction of the system to analyze the effects of LA on its multistable behavior and thus on chimera states. Stability of the synchronized dynamics is analyzed through a master stability function. We find that these results are independent of a system's initial conditions and the strategy is applicable to the networks of globally, locally as well as nonlocally coupled oscillators. Our results suggest that LA control can be an effective method to control chimera states and to realize a desired collective dynamics in such ensembles.

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

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

We 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.

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

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.

In this paper, a novel algorithm based on the local projection noise reduction approach is applied to smooth noise for strongly contaminated chaotic oscillators. Specifically, one-dimensional time series are embedded into a high dimensional phase space and the noise level is defined through orthogonal projections of the data points within the neighbourhood of the reference point onto linear subspaces. The current vector of the phase space is denoised by performing two-dimensional discrete stationary wavelet transform (SWT)-based filtering in the neighbourhood of the phase point. Numerical results show that our algorithm effectively recovers continuous-time chaotic signals in heavy-noise environments and outperforms the classical local projection noise reduction approach for simulated data from the Rössler system and Duffing oscillator at signal-to-noise ratios (SNRs) from 15 to 0 dB, either for the real world data – human breath time series.

A class of non-diffusively coupled complex networks consisting of nodes of different dimensions is studied, in which the internal time delays are different from the coupling delays. Proper adaptive controllers are proposed for the stabilization and function matrix projective synchronization of such complex networks, respectively. The symmetric or diffusive conditions for the coupling matrices are not required. Finally, the results are applied to complex networks of chaotic oscillators showing the effectiveness of the proposed controllers.

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.

Summary Chaos behavior has been observed in various cellular and molecular processes. Here, we modeled reversible phosphorylation dynamics to elucidate a design principle for autonomous chaos generation that may arise from generic enzymatic reactions. A comprehensive parameter search demonstrated that the reaction system composed of a set of kinases and phosphatases and two substrates with two modification sites exhibits chaos behavior. All reactions are described according to the Michaelis-Menten reaction scheme without exotic functions being applied to enzymes and substrates. Clustering analysis of parameter sets that can generate chaos behavior revealed the existence of motif structures. These chaos motifs allow the two-substrate species to interact via enzyme availability and constrain the two substrates' dynamic changes in phosphorylation status so that they occur at different timescales. This chaos motif structure is found in several enzymatic reactions, suggesting that chaos behavior may underlie cellular autonomy in a variety of biochemical systems., Graphical Abstract, Highlights • Two substrates with reversible two-site phosphorylation can exhibit chaos behavior • The chaos does not require autocatalysis or allosteric regulation of enzymes • The chaos is a result of the coupling of two substrates via enzyme availability, Biochemical Mechanism; Molecular Biology; Enzyme Engineering; Biophysics

We consider the implementation of a weak feedback with two delay times for controlling the coherence of both deterministic chaotic and stochastic oscillators. This control strategy is revealed to allow one to decrease or enhance the coherence, which is quantified by the phase diffusion constant, by 2–3 orders of magnitude without destruction of the chaotic regime, which is by an order of magnitude more than one can achieve with a single delay time. Within the framework of the phase reduction, which is a rough approximation for the chaotic oscillators and rigorous for the stochastic ones, an analytical theory of the effect is constructed.

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

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.

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

We address the problem of constructing (globally) convergent, (reduced-order) observers for general nonlinear systems when the output measurements are subject to constant time-delays. Immersion and invariance (I&I) techniques are used to derive a general tool for constructing I&I observers in the presence of time-delays. We show that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space. In this manuscript, the observer may play two different roles. On the one hand, it may be used to reconstruct a delayed version of the unmeasured state from measurements of the available delayed outputs. We show that if the time-delay is known, standard I&I techniques can be directly applied to estimate the delayed unmeasured states. In this case, we refer to the observer as a retarded immersion and invariance observer. On the other hand, the observer may be used to reconstruct both the delay-free unmeasured states and the delay-free output from measurements of the delayed output. In this case, we refer to it as an immersion and invariance predictor. Two examples with chaotic oscillators are presented to show the performance of the observers.

A method for estimating the characteristics of the delayed coupling between the oscillatory systems, which is based on the empirical simulation of the phase dynamics with allowance for the phase-noise correlatedness, is proposed. The method efficiency is illustrated FOR the standard stochastic and chaotic oscillators in numerical experiments. Using this method for analyzing climatic time series, we confirm the presence of the delayed influence of the El Nino Southern oscillation on the North Atlantic Oscillation.

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.

This paper revisits the design of adaptive impulsive observers (AIOs) for nonlinear systems. The dynamics of observer state of the proposed AIO is modelled by an impulsive differential equation, by which the observer state is updated in an impulsive fashion. The parameter estimation law is modelled by an impulse-free time-varying differential equation associated with the impulse time sequence for determining when the observer state is updated. Unlike the previous work, the convergence analysis of the estimation error system is performed by applying a time-varying Lyapunov function based method, in conjunction with the application of a generalized version of Barbalat's Lemma. A sufficient condition for the existence of AIOs is also derived. For some special cases, it is shown that the sufficient condition can be formulated in terms of linear matrix inequalities (LMIs), and the observer matrices can be attained by solving a set of LMIs. Furthermore, with an additional persistence-of-excitation-type constraint, it is proved that the sufficient condition can guarantee the convergence of parameter estimation. Two examples of chaotic oscillators are provided to illustrate the design procedure of the proposed AIOs.

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.

Telemedicine is developing at high speed. In this context, patient’s privacy and security is of great importance. Therefore any physiological signal, needs to be encrypted before their transmission over any channel. In this paper, we have developed an encryption system using chaotic synchronization to encrypt and decrypt information. The system was used for secure transmission of electrocardiograms signals as example.

Recent years have witnessed an increasing attention on the study of distributed coordination for multiple agents due to its broad applications in many areas including formation control, distributed sensor network, flocking, distributed computation, and synchronization of coupled chaotic oscillators. A fundamental requirement on this topic is that all the agents should reach an agreement (consensus) using the shared data through local communications, which is determined by an underlying network topology. Toward this objective, a key step is to design a network-based control protocol such that as time goes on, all the agents asymptotically agree on some variable of interest in an appropriate sense. This chapter focuses on the problems of distributed control design for the consensus and formation of the discrete-time multiagent systems.

The problem of anti-synchronization is another phenomenon of interest that occurs in chaotic oscillators. This problem has appeared in modern repetitions of Huygens’ experiments (Bennett et al., Proc: Math Phys Eng Sci 458:563–579, 2002, [2]), lasers (Uchida et al., Phys Rev A 64:023801-1–023801-6, 2001, [2]), (Wedekind and Parlitz, Int J Bifurc Chaos 11(4):1141–1147, 2001, [3]), saltwater oscillators (Nakata et al., Phys D 115:313–320, 1998, [4]), and some biological systems where a nonchaotic signal is generated (Kim et al., Phys Lett A 320:39–46, 2003, [5]). Anti-synchronization has been treated as a direct modification of synchronization, simply with a sign change in the condition required for the error, and has been attacked with methods such as the active control (Emadzadeh and Haeri, Int J Electr Comput Energ Electron Commun Eng 1(6):898–901, 2007, [6]) (Guo-Hui, Chaos, Solitons Fractals 26:87–93, 2005, [7]) and the sliding mode control (Chen et al., Nonlinear Dyn 69:35–55, 2012, [8]). It can also be induced by noise (Kawamura, Phys D 270(1):20–29, 2014, [9]).