Based on the invariance principle of differential equations, a simple adaptive-feedback scheme is proposed to strictly synchronize almost all chaotic systems. Unlike the usual linear feedback, the variable feedback strength is automatically adapted to completely synchronize two almost arbitrary identical chaotic systems, so this scheme is analytical, and simple to implement in practice. Moreover it is quite robust against the effect of noise. The famous Lorenz and R$\ddot{\textrm{o}}$ssler hyperchaos systems are used as illustrative examples.
We show the first solvable chaotic synchronization model of unidirectionally coupled dynamical systems. We establish a new interpretation of the conditional Lyapunov exponent that characterizes chaotic synchronization completely. Moreover, we newly show how the conditional Lyapunov exponent relates to common noise-induced synchronization phenomena by the new interpretation.
Synchronization of fractional-order chaotic systems is a hot topic in the field of nonlinear study. The co-coupled synchronization between two fractional-order chaotic systems with different initial conditions is investigated in this paper. Based on Lyapunov stability principle and Gerschgorin theorem, the co-coupled synchronization theorem of fractional-order chaotic systems is deduced, and the range of coupling coefficients is confirmed for synchronization of fractional-order unified chaotic systems. By building up the synchronization simulation model on Simulink, the co-coupled synchronizat…
This paper describes the security weaknesses of a recently proposed secure communication method based on chaotic masking using projective synchronization of two chaotic systems. We show that the system is insecure and how to break it in two different ways, by high-pass filtering and by generalized synchronization.
This paper deals with two types of synchronous behavior of chaotic oscillators -- generalized synchronization and noise--induced synchronization. It has been shown that both these types of synchronization are caused by similar mechanisms and should be considered as the same type of the chaotic oscillator behavior. The mechanisms resulting in the generalized synchronization are mostly similar to ones taking place in the case of the noise-induced synchronization with biased noise.
Synchronization of identical chaotic systems subjected to common noise has been the subject of recent research. Studies on several chaotic systems have shown that, the synchronization is actually induced by the non-zero mean of the noise, and symmetric noise with zero-mean cannot lead to synchronization. Here it is presented that synchronization can be achieved by {\sl zero-mean} noise in some chaotic maps with large convergence regions.
We present the first report on inverse chaos synchronization between bi-directionally non-linearly and linearly coupled variable multiple time delay Ikeda systems. The results are of certain importance in secure chaos-based communication systems.
We demonstrate that a unidirectional coupling between a pattern forming system and its replica induces complete synchronization of the slave to the master system onto a spatiotemporal chaotic state.
We present the first report on inverse chaos synchronization where a driven modulated multiple time-delay chaotic system synchronizes to the inverse state of the driver system. Numerical simulations fully support the analytical approach.
The chaotic dynamics of fractional order systems begin to attract much attentions in recent years. In this brief report, we study the master-slave synchronization of fractional order chaotic systems. It is shown that fractional order chaotic systems can also be synchronized.
Small networks of chaotic units which are coupled by their time-delayed variables, are investigated. In spite of the time delay, the units can synchronize isochronally, i.e. without time shift. Moreover, networks can not only synchronize completely, but can also split into different synchronized sublattices. These synchronization patterns are stable attractors of the network dynamics. Different networks with their associated behaviors and synchronization patterns are presented. In particular, we investigate sublattice synchronization, symmetry breaking, spreading chaotic motifs, synchronizatio…
Synchronization dynamics of mutually coupled chaotic semiconductor lasers are investigated experimentally and compared to identical synchronization of unidirectionally coupled lasers. Mutual coupling shows high quality synchronization in a broad range of self-feedback and coupling strengths. It is found to be tolerant to significant parameter mismatch which for unidirectional coupling would result in loss of synchronization. The advantages of mutual coupling are emphasized in light of its potential use in chaos communications.
The conditions for synchronization in unidirectionally coupled chaotic oscillators are revisited. We demonstrate with typical examples that the conditional Lyapunov exponents (CLEs) play an important role in distinguishing between intermittent and permanent synchronizations, when the analytic conditions for chaos synchronization are not uniformly obeyed. We show that intermittent synchronization can occur when CLEs are very small positive or negative values close to zero while permanent synchronization occurs when CLEs take sufficiently large negative values. There is also strong evidence for …
In this paper, an electronic circuit implementation of a robustly chaotic two-dimensional map is presented. Two such electronic circuits are realized. One of the circuits is configured as the driver and the other circuit is configured as the driven system. Synchronization of chaos between the driver and the driven system is demonstrated.
This paper proposes an Extended-Kalman-Filter-like observer for parameter estimation during synchronization of chaotic systems. The exponential stability of the observer is guaranteed by a persistent excitation condition. This approach is shown to be suitable for various classical chaotic systems and several simulations are presented accordingly.
The chaotic dynamics of fractional (non-integer) order systems have begun to attract much attention in recent years. In this paper, we study the projective synchronization in two coupled fractional order chaotic oscillators. It is shown that projective synchronization can also exist in coupled fractional order chaotic systems. A simple feedback control method for controlling the scaling factor onto a desired value is also presented.
The phenomenon of synchronization occurring in a locally coupled map lattice subject to an external drive is compared to the synchronization process in an autonomous coupled map system with similar local couplings plus a global interaction. It is shown that chaotic synchronized states in both systems are equivalent, but the collective states arising after the chaotic synchronized state becomes unstable can be different in these two systems. It is found that the external drive induces chaotic synchronization as well as synchronization of unstable periodic orbits of the local dynamics in the dri…
We investigate the phenomenon of chaos synchronization in systems subject to coexisting autonomous and external global fields by employing a simple model of coupled maps. Two states of chaos synchronization are found: (i) complete synchronization, where the maps synchronize among themselves and to the external field, and (ii) generalized or internal synchronization, where the maps synchronize among themselves but not to the external global field. We show that the stability conditions for both states can be achieved for a system of minimum size of two maps. We consider local maps possessing rob…
This paper describes the security weakness of a recently proposed secure communication method based on discrete-time chaos synchronization. We show that the security is compromised even without precise knowledge of the chaotic system used. We also make many suggestions to improve its security in future versions.
In this paper we investigate dual synchronization and dual-cross synchronization between the transmitter and receiver, each one consisting of two master and slave systems. These findings are considered to be of interest in multichannel communication schemes.
This paper presents the result of the investigation of chaotic oscillator synchronization. A new approach for detecting of synchronized behaviour of chaotic oscillators has been proposed. This approach is based on the analysis of different time scales in the time series generated by the coupled chaotic oscillators. This approach has been applied for the coupled Rossler and Lorenz systems.
We theoretically study analytic-phase synchronization in strongly-competing oscillator systems. Using the example of composite-cavity modes coupled via a class-B laser active medium, we discover that inherent chaotic phase synchronization can arise concurrently at two different chaotic attractors, leading to bistable phase-synchronized solutions. In our example, the underlying mechanism for bistability and inherent phase synchronization is population pulsation within the active medium.
We have found a synchronization behavior between two identical chaotic systems^M when their delay times are modulated by a common irregular signal. ^M This phenomenon is demonstrated both in two identical chaotic maps whose delay times are driven by a common^M chaotic or random signal and in two identical chaotic oscillators whose delay times are driven by^M a signal of another chaotic oscillator. We analyze the phenomenon by using^M the Lyapunov exponents and discuss it in relation with generalized synchronization.^M
Generalized synchronization is analyzed in unidirectionally coupled oscillatory systems exhibiting spatiotemporal chaotic behavior described by Ginzburg-Landau equations. Several types of coupling betweenthe systems are analyzed. The largest spatial Lyapunov exponent is proposed as a new characteristic of the state of a distributed system, and its calculation is described for a distributed oscillatory system. Partial generalized synchronization is introduced as a new type of chaotic synchronization in spatially nonuniform distributed systems. The physical mechanisms responsible for the onset o…
This paper addresses the problem of finite-time synchronization of tunnel diode based chaotic oscillators. After a brief investigation of its chaotic dynamics, we propose an active adaptive feedback coupling which accomplishes the synchronization of tunnel diode based chaotic systems with and without the presence of delay(s), basing ourselves on Lyapunov and on Krasovskii-Lyapunov stability theories. This feedback coupling could be applied to many other chaotic systems. A finite horizon can be arbitrarily established by ensuring that chaos synchronization is achieved at a pre-established time.…
We elucidate conditions for chaos anti-synchronization between two uni-directionally coupled multiple time delay power systems. The results are of some importance to prevent power black-out in the entire power grid.
The problem of projective synchronization(ps) and control are studied in modified unified chaotic system which possess partially linearity property. The desired ratio factor of corresponding subsystem variable could be obtained by state feedback control. Theoretical analysis and numerical simulations are provided to illustrate the projective synchronization and the feasibility of the proposed control method. The effect on projective synchronization caused by channel noise and parameter mismatch are investigated in detail, the results showed that parameter mismatch has more effect on projective…
Dynamical networks are important models for the behaviour of complex systems, modelling physical, biological and societal systems, including the brain, food webs, epidemic disease in populations, power grids and many other. Such dynamical networks can exhibit behaviour in which deterministic chaos, exhibiting unpredictability and disorder, coexists with synchronization, a classical paradigm of order. We survey the main theory behind complete, generalized and phase synchronization phenomena in simple as well as complex networks and discuss applications to secure communications, parameter estima…
Two types of phase synchronization (accordingly, two scenarios of breaking phase synchronization) between coupled stochastic oscillators are shown to exist depending on the discrepancy between the control parameters of interacting oscillators, as in the case of classical synchronization of periodic oscillators. If interacting stochastic oscillators are weakly detuned, the phase coherency of the attractors persists when phase synchronization breaks. Conversely, if the control parameters differ considerably, the chaotic attractor becomes phase-incoherent under the conditions of phase synchroniza…
We report a new type of chaos synchronization:inverse anticipating synchronization, where a time delay chaotic system can drive another system in such a way that the driven system anticipates the driver by synchronizing with its inverse future state. We extend the concept of inverse anticipating chaos synchronization to cascaded systems. We propose means for the experimental observation of inverse anticipating chaos synchronization in external cavity lasers.
Synchronization of chaotic units coupled by their time delayed variables are investigated analytically. A new type of cooperative behavior is found: sublattice synchronization. Although the units of one sublattice are not directly coupled to each other, they completely synchronize without time delay. The chaotic trajectories of different sublattices are only weakly correlated but not related by generalized synchronization. Nevertheless, the trajectory of one sublattice is predictable from the complete trajectory of the other one. The spectra of Lyapunov exponents are calculated analytically in…
A security analysis of a recently proposed secure communication scheme based on the phase synchronization of chaotic systems is presented. It is shown that the system parameters directly determine the ciphertext waveform, hence it can be readily broken by parameter estimation of the ciphertext signal.
We have studied the discrete nonlinear Schroedinger equation (DNLSE) with on-site defects and periodic boundary conditions. When the array dynamics becomes chaotic, the otherwise quasiperiodic amplitude correlations between the oscillators are destroyed. However, we show that synchronization of symbolic information of suitable amplitude signals is possible in this hamiltonian system.
We present in this paper, the synchronization dynamics observed in a network of mutually coupled simple chaotic systems. The network consisting of chaotic systems arranged in a square matrix network is studied for their different types of synchronization behavior. The chaotic attractors of the simple $2 \times 2$ matrix network exhibiting strange non-chaotic attractors in their synchronization dynamics for smaller values of the coupling strength is reported. Further, the existence of islands of unsynchronized and synchronized states of strange non-chaotic attractors for smaller values of coupl…
We investigate chaos synchronization between Josephson junctions coupled uni-directionally with time-delay. We demonstrate the possibility of high quality synchronization with numerical simulations of such systems. The results are of certain importance for obtaining the high power system of Josephson junctions, which is promising for practical applications.
We prove a sufficient condition for synchronization for coupled one-dimensional maps and estimate the size of the window of parameters where synchronization takes place. It is shown that coupled systems on graphs with positive eigenvalues (EVs) of the normalized graph Laplacian concentrated around 1 are more amenable for synchronization. In the light of this condition, we review spectral properties of Cayley, quasirandom, power-law graphs, and expanders and relate them to synchronization of the corresponding networks. The analysis of synchronization on these graphs is illustrated with numerica…
A coupled map model for the chaotic phase synchronization and its desynchronization phenomenon is proposed. The model is constructed by integrating the coupled kicked oscillator system, kicking strength depending on the complex state variables. It is shown that the proposed model clearly exhibits the chaotic phase synchronization phenomenon. Furthermore, we numerically prove that in the region where the phase synchronization is weakly broken, the anomalous scaling of the phase difference rotation number is observed. This proves that the present model belongs to the same universality class foun…
In this paper, an adaptive fractional controller has been designed to control chaotic systems. In fact, this controller is a fractional PID controller, which the coefficients will be tuned according to a proper adaptation mechanism. The adaptation law will be constructed from a sliding surface via gradient method. The adaptive fractional controller is implemented on a gyro system to signify the performance of the proposed technique.
This work analyzes the security weakness of a recently proposed communication method based on chaotic modulation and masking using synchronization of two chaotic systems with different orders. It is shown that its application to secure communication is unsafe, because it can be broken in two different ways, by high-pass filtering and by reduced order system synchronization, without knowing neither the system parameter values nor the system key.
A chaotic dynamics is typically characterized by the emergence of strange attractors with their fractal or multifractal structure. On the other hand, chaotic synchronization is a unique emergent self-organization phenomenon in nature. Classically, synchronization was characterized in terms of macroscopic parameters, such as the spectrum of Lyapunov exponents. Recently, however, we attempted a microscopic description of synchronization, called topological synchronization, and showed that chaotic synchronization is, in fact, a continuous process that starts in low-density areas of the attractor.…
In this paper, fractional order Coullet system is studied. An active control technique is applied to control this chaotic system. This type of controller is also applied to synchronize chaotic fractional-order systems in master-slave structure. The synchronization procedure is shown via simulation. The boundary of stability is obtained by both of theoretical analysis and simulation result. The numerical simulations show the effectiveness of the proposed controller.
We investigate synchronization in the presence of delay time modulation for application to communication. We have observed that the robust synchronization is established by a common delay signal and its threshold is presented using Lyapunov exponents analysis. The influence of the delay time modulation in chaotic oscillators is also discussed.
A method of targeting engineering synchronization states in two identical and mismatch chaotic systems is explained in details. The method is proposed using linear feedback controller coupling for engineering synchronization such as mixed synchronization, linear and nonlinear generalized synchronization and targeting fixed point. The general form of coupling design to target any desire synchronization state under unidirectional coupling with the help of Lyapunov function stability theory is derived analytically. A scaling factor is introduced in the coupling definition to smooth control withou…
By studying the autocorrelation function of the optoelectronic feedback semiconductor laser output we establish that the signatures of time delays can be erased in systems incorporating modulated feedback time delays. This property is of importance for the suitability of such laser systems for secure chaos-based communication systems. We also make the first report on chaos synchronization in both unidirectionally and bidirectionally coupled multiple time delay chaotic semiconductor lasers with modulated optoelectronic feedbacks.
Complete chaotic synchronization of end lasers has been observed in a line of mutually coupled, time-delayed system of three lasers, with no direct communication between the end lasers. The present paper uses ideas from generalized synchronization to explain the complete synchronization in the presence of long coupling delays, applied to a model of mutually coupled semiconductor lasers in a line. These ideas significantly simplify the analysis by casting the stability in terms of the local dynamics of each laser. The variational equations near the synchronization manifold are analyzed, and use…
The main aim of this comment is to emphasize that the conditional Lyapunov exponents play an important role in distinguishing between intermittent and persistent synchronization, when the analytic criteria for asymptotic stability are not uniformly obeyed.
We report on all- to- all chaos synchronization in a network of networks based on the Ikeda model. We consider one of the simplest cases. We find the existence and stability conditions for such a synchronization regime. Numerical simulations validate the analytical findings. The results can be of certain importance in achieving high level output for the coupled systems and information processing.
The properties of functional relation between a non-invertible chaotic drive and a response map in the regime of generalized synchronization of chaos are studied. It is shown that despite a very fuzzy image of the relation between the current states of the maps, the functional relation becomes apparent when a sufficient interval of driving trajectory is taken into account. This paper develops a theoretical framework of such functional relation and illustrates the main theoretical conclusions using numerical simulations.
We introduce a technique to detect and quantify local functional dependencies between coupled chaotic systems. The method estimates the fraction of locally syncronized configurations, in a pair of signals with an arbitrary state of global syncronization. Application to a pair of interacting Rossler oscillators shows that our method is capable to quantify the number of dynamical configurations where a local prediction task is possible, also in absence of global synchronization features.
In this paper, we report the enhanced stability of induced synchronization by transient uncoupling observed in certain unidirectionally coupled second-order chaotic systems. The stability of synchronization observed in the coupled systems subjected to transient uncoupling is analyzed using the {\emph{Master Stability Function}}. The existence of the coupled systems in stable synchronized states over a certain range of the clipping fraction of the driven system is identified. The enhanced stable synchronized states are obtained for fixed values of clipping fraction in certain second-order chaot…
We propose a simple and new unified method to achieve lag, complete and anticipatory synchronizations in coupled nonlinear systems. It can be considered as an alternative to the subsystem and intentional parameter mismatch methods. This novel method is illustrated in a unidirectionally coupled RC phase shift network based Chua's circuit. Employing feedback coupling, different types of chaos synchronization are observed experimentally and numerically in coupled identical chaotic oscillators {\emph{without using time delay}}. With a simple switch in the experimental set up we observe different k…
We derive rigorous conditions for the synchronization of all-optically coupled lasers. In particular, we elucidate the role of the optical coupling phases for synchronizability by systematically discussing all possible network motifs containing two lasers with delayed coupling and feedback. Hereby we explain previous experimental findings. Further, we study larger networks and elaborate optimal conditions for chaos synchronization. We show that the relative phases between lasers can be used to optimize the effective coupling matrix.
Networks of nonlinear units with time-delayed couplings can synchronize to a common chaotic trajectory. Although the delay time may be very large, the units can synchronize completely without time shift. For networks of coupled Bernoulli maps, analytic results are derived for the stability of the chaotic synchronization manifold. For a single delay time, chaos synchronization is related to the spectral gap of the coupling matrix. For networks with multiple delay times, analytic results are obtained from the theory of polynomials. Finally, the analytic results are compared with networks of iter…
We investigate phase synchronization between two identical or detuned response oscillators coupled to a slightly different drive oscillator. Our result is that phase synchronization can occur between response oscillators when they are driven by correlated (but not identical) inputs from the drive oscillator. We call this phenomenon Generalized Phase Synchronization (GPS) and clarify its characteristics using Lyapunov exponents and phase difference plots.
We study chaos synchronization between Josephson junctions governed by a central junction with a time delay and demonstrate with numerical simulations the possibility of high quality synchronization. The results are important for obtaining high power systems of Josephson junctions, which are promising for practical applications.
We present a methodology for synchronization of chaotic oscillators with linear feedback control. The proposed method is based on analyzing the chaotic oscillator as a multi-mode linear system and deriving sufficient conditions for asymptotic stability. The oscillators are synchronized in a master-slave configuration, wherein a subset of the state variables for implementing the feedback control, enabling applications in cryptography for message encryption using the unused chaotic state variables. Controller stability is ensured through conventional root-locus technique for designing appropriat…
Two identical chaotic oscillators that are mutually coupled via time delayed signals show very complex patterns of completely synchronized dynamics including stationary states and periodic as well as chaotic oscillations. We have experimentally observed these synchronized states in delay-coupled electronic circuits and have analyzed their stability by numerical simulations and analytical calculations. We found that the conditions for longitudinal and transversal stability largely exclude each other and prevent e.g. the synchronization of Pyragas-controlled orbits. Most striking is the observat…
We show that the threshold of complete synchronization in a lattice of coupled non-smooth chaotic maps is determined by linear stability along the directions transversal to the synchronization subspace. As a result, the numerically determined synchronization threshold agree with the analytical results previously obtained [C. Anteneodo et al., Phys. Rev. E 68, 045202(R) (2003)] for this class of systems. We present both careful numerical experiments and a rigorous mathematical explanation confirming this fact, allowing for a generalization involving hyperbolic coupled map lattices.
We present the first report of the synchronization regimes in both unidirectionally and bidirectionally-coupled multiple time delay chaotic laser diodes subject to incoherent optical feedbacks and incoherent optical injection. We derive the existence conditions and numerically study the stability for lag, complete, and anticipating synchronization regimes and cascaded synchronization. We also study in detail the effect of parameter mismatches and noise on the synchronization quality. It is emphasized that sensitivity of the synchronization quality to parameter mismatches can lead to a high lev…
