Synchronization Of Chaotic Oscillators Research Articles

Complete synchronization of chaos in systems with nonlinear inertial coupling

Abstract The effect of complete synchronization of chaotic oscillations and complex spatial structures in systems with nonlinear inertial coupling of the memristive type is studied. Three models of interacting systems with complex dynamics are considered: a system of two radio engineering chaotic self-oscillators, a system of two logistic maps, and a two-layer multiplex network of logistic maps with a dissipative nonlocal intra-layer coupling and a nonlinear inertial local coupling between the layers. For all three models, the influence of memristive coupling is shown. It consists in the dependence of the synchronization boundary on the initial state of the coupling elements. It has been established that this property is retained with inertial nonlinear coupling close to memristive (with strong inertiality).

Permutation Entropy of State Transition Networks to Detect Synchronization

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.

Ring of map-based neural oscillators: From order to chaos and back

• A ring of map-based Rulkov neural models is considered. • Corporate effects “chaos-order” are demonstrated. • A synchronization of chaotic oscillators is studied. We study transitions from regular to chaotic dynamics and back of a ring of three map-based neurons with parameter mismatch. First, we consider the case when each neuron being isolated is in a stable equilibrium and show that an increase in the coupling strength can lead to chaotic dynamics following a quasiperiodic behavior. Then, we consider the case when the uncoupled neurons are in a chaotic state and demonstrate the route from chaos to periodicity as the coupling strength is increased. We show that this interesting effect results from lag synchronization of the coupled neural oscillators. The system stability is characterized by the largest Lyapunov exponents in the space of the coupling strength and parameter mismatch, while lag synchronization is measured using the similarity function.

An Electronic Implementation of Lorenz Chaotic Oscillator Synchronization for Bistatic Radar Applications

The authors report the simulation and actual implementation of chaotic oscillator synchronization for a pair of three-state variable Lorenz systems. The first oscillator performs as a driver (transmitter), while the second system is made to respond the received waveform. Specifically, one of the state variables of the driver circuit is judiciously selected to generate an RF waveform. This waveform is transmitted through a noisy channel and is used to drive the response system until both the systems of all three state variables achieve complete replacement synchronization. It is known that this synchronization is sensitive to changes in received power. To overcome this, a technique hereon called generalized projective synchronization is employed to accept an attenuated version of the transmitted waveform. A sensitivity analysis reveals that generalized projective synchronization is robust with respect to deviations in the control parameter values of the response system due to tolerances of electronic components. The potential of the generalized projective synchronization is illustrated for a bistatic radar system case, where synchronization is necessary to obtain the range-Doppler information of a moving target. In this instance, the cross correlation, short-time cross-correlogram and cross ambiguity surface of the transmitted and synchronized waveforms are of high quality as indicated by their entropy measures.

Fast Terminal Sliding-Mode Control With an Integral Filter Applied to a Van Der Pol Oscillator

This paper proposes a procedure for designing a fast terminal sliding-mode control involving an integral filter that guarantees a finite time of convergence to zero dynamics, robustness against polynomial bounded uncertainties, and external disturbances. It reduces the chattering effect by employing a state-dependent gain and adds an integral filter to smooth the input control signal. We suggest a simplified algorithm to design the controller. To test the performance of the algorithm, we use a Van der Pol chaotic oscillator synchronization with perturbations in the states. Finally, we compare the performance of the method by using the proposed controller with the filter turned-on and turned-off.

Enhancing synchronization in chaotic oscillators by induced heterogeneity

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.

Erratum to: “Analysis of phase synchronization of chaotic oscillations in terms of symbolic CTQ-analysis”

Erratum to: “Analysis of phase synchronization of chaotic oscillations in terms of symbolic CTQ-analysis”

Improved Time Response of Stabilization in Synchronization of Chaotic Oscillators Using Mathematica

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.

Analysis of phase synchronization of chaotic oscillations in terms of symbolic CTQ-analysis

The application of symbolic CTQ-analysis for studying synchronization of chaotic oscillations is considered. This approach differs substantially from its analogs since it makes it possible to diagnose and measure quantitatively the characteristics of intermittency regimes in synchronization of chaotic systems and, hence, to analyzer the temporal structure of synchronization. The application of the symbolic analysis apparatus based on the T alphabet to systems with phase locking and synchronization of time scales is demonstrated for the first time. As an example, a complex system of two mutually coupled nonidentical Rossler oscillators in the helical chaos regime with attractors having an ill-conditioned phase is considered. The results show that the method considered here makes it possible to reliably diagnose synchronism sooner than a phase locking and/or time-scale synchronization threshold is detected.

Mixed lag synchronization in chaotic oscillators and experimental observations

We report mixed lag synchronization in coupled counter-rotating oscillators. The trajectories of counter-rotating oscillators has opposite directions of rotation in uncoupled state. Under diffusive coupling via a scalar variable, a mixed lag synchronization emerges when a parameter mismatch is induced in two counter-rotating oscillators. In the state of mixed lag synchronization, one pair of state variables achieve synchronization shifted in time while another pair of state variables are in antisynchronization, however, they are too shifted by the same time. Transition point of mixed-phase synchronization is estimated analytically using approximation theory. Numerical example of the paradigmatic Rossler oscillator is presented and supported by electronic experiment.

Analysis of the time structure of synchronization in multidimensional chaotic systems

A new approach is proposed to the integrated analysis of the time structure of synchronization of multidimensional chaotic systems. The method allows one to diagnose and quantitatively evaluate the intermittency characteristics during synchronization of chaotic oscillations in the T-synchronization mode. A system of two identical logistic mappings with unidirectional coupling that operate in the developed chaos regime is analyzed. It is shown that the widely used approach, in which only synchronization patterns are subjected to analysis while desynchronization areas are considered as a background signal and removed from analysis, should be considered as methodologically incomplete.

Adaptive mechanism for synchronization of chaotic oscillators with interval time-delays

This paper addresses the adaptive feedback controller design for the synchronization of chaotic systems with interval time-delays in their state vectors by exploiting a lower and an upper bound on time-delays. Simple control and adaptation laws are developed for chaos synchronization, and linear matrix inequalities (LMIs) are derived to ensure asymptotic convergence of the synchronization error between the master–slave systems, using the proposed feedback control strategy, by employing a novel treatment of Lyapunov–Krasovskii functional. Further, the proposed strategy is strengthened by exploiting \(L_2 \) stability against disturbances and perturbations and corresponding LMIs for robust adaptive controller synthesis are derived. Furthermore, a novel delay-range-dependent robust adaptive synchronization control approach for dealing with locally Lipschitz non-delayed and delayed nonlinearities in the dynamics of chaotic oscillators is provided by employing an additional adaptation law for the nonlinearities. A numerical simulation example is provided to illustrate effectiveness of the proposed synchronization approach.

Complex behavior of chaotic synchronization under dual coupling channels

Most previous works on complete synchronization of chaotic oscillators focused on the one-channel interaction scheme where the oscillators are coupled through only one variable or a symmetric set of variables. Using the standard framework of master-stability function (MSF), we investigate the emergence of complex synchronization behaviors under all possible configurations of two-channel coupling, which include, for example, all possible cross coupling schemes among the dynamical variables. Utilizing the classic Rössler and Lorenz oscillators, we find a rich variety of synchronization phenomena not present in any previously extensively studied, single-channel coupling configurations. For example, in many cases two coupling channels can enhance or even generate synchronization where there is only weak or no synchronization under only one coupling channel, which has been verified in a coupled neuron system. There are also cases where the oscillators are originally synchronized under one coupling channel, but an additional synchronizable coupling channel can, however, destroy synchronization. Direct numerical simulations of actual synchronization dynamics verify the MSF-based predictions. Our extensive computation and heuristic analysis provide an atlas for synchronization of chaotic oscillators coupled through two channels, which can be used as a systematic reference to facilitate further research in this area.

Control of partial synchronization in chaotic oscillators

A design of coupling is proposed to control partial synchronization in two chaotic oscillators in a driver–response mode. A control of synchrony between one response variables is made possible (a transition from a complete synchronization to antisynchronization via amplitude death and vice versa without loss of synchrony) keeping the other pairs of variables undisturbed in their pre-desired states of coherence. Further, one of the response variables can be controlled so as to follow the dynamics of an external signal (periodic or chaotic) while keeping the coherent status of other variables unchanged. The stability of synchronization is established using the Hurwitz matrix criterion. Numerical example of an ecological foodweb model is presented. The control scheme is demonstrated in an electronic circuit of the Sprott system.

Static and adaptive feedback control for synchronization of different chaotic oscillators with mutually Lipschitz nonlinearities

This paper addresses the control law design for synchronization of two different chaotic oscillators with mutually Lipschitz nonlinearities. For analysis of the properties of two different nonlinearities, an advanced mutually Lipschitz condition is proposed. This mutually Lipschitz condition is more general than the traditional Lipschitz condition. Unlike the latter, it can be used for the design of a feedback controller for synchronization of chaotic oscillators of different dynamics. It is shown that any two different Lipschitz nonlinearities always satisfy the mutually Lipschitz condition. Applying the mutually Lipschitz condition, a quadratic Lyapunov function and uniformly ultimately bounded stability, easily designable and implementable robust control strategies utilizing algebraic Riccati equation and linear matrix inequalities, are derived for synchronization of two distinct chaotic oscillators. Furthermore, a novel adaptive control scheme for mutually Lipschitz chaotic systems is established by addressing the issue of adaptive cancellation of unknown mismatch between the dynamics of different chaotic systems. The proposed control technique is numerically tested for synchronization of two different chaotic Chua's circuits and for obtaining identical behavior between the modified Chua's circuit and the Rössler system.

A Theory on Noise-Induced Synchronization of Chaotic Oscillators

A Theory on Noise-Induced Synchronization of Chaotic Oscillators

Synchronization Attack to Chaotic Communication Systems

Synchronization of chaotic oscillators has an important application in cryptography. When two identical oscillators are coupled, they can be completely synchronized and the chaotic output of the transmitter oscillator can be used to mask a message. Although the oscillator parameters are usually used as secret keys, the sensitivity of such cryptosystems to parameter changes has never been systematically analyzed. To cryptanalyze a communication system based on synchro- nization of chaotic oscillators, we use a synchronization attack that allows estimating all unknown parameters by minimizing the synchronization error. Using this attack we cryptanalyze popular communication systems based on the Rossler and Chua chaotic electronic circuits. We suggest to include this attack as a standard security test for crypt-analysis of chaotic communication systems.

Synchronization in chaotic oscillators by cyclic coupling

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

Synchronization of chaotic Liouvillian systems: An application to Chua’s oscillator

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® and Matlab®-LMI programs. Simulation results are used to visualize and illustrate the effectiveness of Chua’s oscillator in synchronization.

Synchronization and anti-synchronization of chaotic oscillators under input saturation

This paper addresses the design of simple state feedback controllers for synchronization and anti-synchronization of chaotic oscillators under input saturation and disturbance. By employing sector condition, linear matrix inequality (LMI)-based sufficient conditions are derived to design (global or local) controllers for chaos synchronization. The proposed local synchronization strategy guarantees a region of stability in terms of difference between states of the master–slave systems. This region of stability can be enlarged by means of an LMI-based optimization algorithm, through which asymptotic synchronization of chaotic oscillators can be ensured for a large difference in their initial conditions. Further, a novel LMI-based robust control strategy is developed, for local synchronization of input-constrained chaotic oscillators, by providing an upper bound on synchronization error in terms of disturbance and initial conditions of chaotic systems. Moreover, the proposed robust state feedback control methodology is modified to provide an inaugural treatment for robust anti-synchronization of chaotic systems under input saturation and disturbance. The results of the proposed methodologies are verified through numerical simulations for synchronization and anti-synchronization of the master–slave chaotic Chua’s circuits under input saturation.