Narrow Results

This paper addresses the adaptive synchronization and parameters identification problem of a class of high-dimensional autonomous uncertain chaotic systems. It is proved that the controller and update rule can make the states of the drive system and the response system with unknown system parameters asymptotically synchronized, and identify the response system's unknown parameters. Chen system, coupled dynamos system and Rössler hyperchaotic system are used as examples for detailed description. The results of numerical simulations show the effectiveness of the adaptive controller.

By utilizing the fractional calculus techniques and spatiotemporal chaos theory, this paper brings Lorenz system to fractional-order spatiotemporal coupled differential equation for the first time, and proposes the fractional-order spatiotemporal coupled Lorenz system. Based on that, we study the problem of chaotic synchronization of fractional-order spatiotemporal coupled Lorenz systems, design the linear controller and nonlinear controller by utilizing the Lyapunov stability theory and prove the correctness in theory. The numerical simulation results demonstrate the validity of controllers in high-dimension fractional-order spatiotemporal coupled Lorenz system.

In this paper, we introduce the heterogeneity in the parameter $\sigma$ to three coupled Lorenz oscillators and investigate the effects of parameter heterogeneity on the coupling dynamics. In the presence of parameter heterogeneity, the complete synchronous state is replaced by lag synchronous state which owns the largest Lyapunov exponent exactly the same as that of the complete synchronous chaos. We find two types of oscillation quenching states induced by the parameter heterogeneity, homogeneous nontrivial equilibria and heterogeneous equilibria. In the homogeneous nontrivial equilibria, all oscillators fall onto a same nontrivial equilibrium of the isolated Lorenz oscillator, which requires low coupling strength. Depending on the coupling function, the heterogeneous equilibria may appear at intermediate coupling strength or large coupling strength. We numerically show that the transitions among lag synchronous state and different types of quenching states are always discontinuous ones. The stability diagram of the lag synchronous chaos is presented theoretically, which is compatible with those based on the synchronization error and Lyapunov exponents.

The synchronization problem of chaotic fractional-order Rucklidge systems is studied both theoretically and numerically. Three different synchronization schemes based on the Pecora–Carroll principle, the linear feedback control and the bidirectional coupling are proposed to realize chaotic synchronization. It is shown that such schemes can achieve the same aim for the same set of system parameter values (including fractional orders). This provides an alternate choice for applications of fractional-order dynamical systems in engineering fields.

Assuming the Rössler system as a reference, this paper studies two cases of chaotic synchronization of a pair of (master and slave) systems: one with fully uncertain parameters for both, the other where the master system has fixed given parameters while the slave system is endowed with uncertain parameters. The respective adaptive controller based on parameter identification is then designed, according to the Lyapunov stability theorem. Then, it is proved that the two controllers are capable of making the two (identical) Rössler systems asymptotically synchronized. Numerical simulation results further testify the efficiency of controllers.

This paper addresses the adaptive synchronization problem of a class of different uncertain chaotic systems. A general adaptive robust controller and parameters update rule are designed. It is proved theoretically that the controller and update rule can make the drive-response systems with different structures asymptotically synchronized, and change the unknown parameters to constants when noise exists. When the drive system is certain, the unknown parameters of the response system can be updated to the predicted values. The results of numerical simulations show the effectiveness of the adaptive controller.

This paper studies the hyperchaotic Rössler system and the state observation problem of such a system being investigated. Based on the time-domain approach, a simple observer for the hyperchaotic Rössler system is proposed to guarantee the global exponential stability of the resulting error system. The scheme is easy to implement and different from the other observer design that it does not need to transmit all signals of the dynamical system. It is proved theoretically, and numerical simulations show the effectiveness of the scheme finally.

This paper presents tracking control and synchronization strategies for Chen system. Two universal controllers, a tracking controller and a synchronization controller based on Backstepping design method were designed. It is proved theoretically that the tracking controller enables the error signal exponentially to converge to zero. The validity of Backstepping synchronization controller is also proved. Numerical simulations further validated the two controllers.

In this paper, the drive system and the response system can be in a state of linear generalized synchronization via transmitting single signal. By means of a transitional system, the response system is obtained by variable replacement method. Chen system and hyperchaotic Chen system are used as examples in numerical simulations. Simulation results show the effectiveness of the method.

A novel fractional-order adaptive non-singular terminal sliding mode control (FONTSMC) method is investigated for the synchronization of two nonlinear fractional-order chaotic systems in the presence of external disturbance. The proposed controller consists of a fractional-order non-singular terminal sliding mode surface and an adaptive gain adjusted with sliding surface. Based on Lyapunov stability theory and stability theorem for fractional-order dynamic systems, the controlled system’s stable synchronization is guaranteed. A dual-channel secure communication system is presented to transmit useful signals based on the proposed synchronization controller. Finally, numerical simulations and comparison with fractional-order PID controller, fractional-order PD sliding mode controller and adaptive terminal sliding mode controller are given to demonstrate the effectiveness and the robustness of the proposed FONTSMC control. The application of the proposed synchronization method is studied in the dual-channel secure communication.

This paper investigates novel adaptive observation control for synchronization of uncertain chaotic coronary artery system (CCAS). Novel adaptive observers are developed to obtain the estimated system states of uncertain CCAS which has unknown parameters. By utilizing the adaptive observers and controller we proposed, the adaptation of unknown parameters has been achieved and the asymptotic stability of the synchronization error as well as the estimation errors are guaranteed. For the purpose of further reducing the conservatism, new Lyapunov Krasovskii functions (LKFs) which maintain more system states have been constructed in terms of Wirtinger-based inequality, a new double integral inequality and improved reciprocally convex inequality. Using $H_{\infty }$ control to ensure robust performance of CCAS which has unknown parameters and external disturbance. The adaptive observer gain matrices and controller gain matrix could be obtained by employing a decoupling approach. The effectiveness of this control methodology has been illustrated by a numerical simulation result.

In this paper, a novel four-dimensional fractional-order chaotic system is proposed, and its dynamic characteristics are analyzed by equilibrium points, phase diagram, Lyapunov exponent spectrum and bifurcation diagram. Then, the electronic circuit of the chaotic system is designed and implemented by PSpice. The results of numerical simulation are in good agreement with the analog circuit simulation. In addition, the four-dimensional fractional-order system is implemented with field programmable gate array (FPGA). Finally, the synchronization of two four-dimensional chaotic systems with different initial values is realized by finite time control method, and it is also realized by FPGA. The design of fractional-order system based on hardware concerned in this paper will provide a certain theoretical basis for its application in the fields of secure communication, image encryption and so on.

In this study, a second-generation positive current conveyor (CCII+)-based analog circuit is proposed for the electronic implementation of a different dynamical system which is an adaptation of the chaotic Lorenz differential equation set. The proposed circuit is more cost-effective and contains less active and passive elements than the circuit obtained by applying the classical parallel synthesis method with opamps. Mathematical analyses and SPICE simulations are performed for chaotic phase portraits and bifurcation diagrams. The proposed dynamical circuit is implemented on the board by using commercially available active and passive elements on the market and an experimental study is conducted. In order to demonstrate the usability of this proposed circuit in secure communication studies, three different synchronization methods are applied and one of them is implemented. The obtained experimental results are in good agreement with the mathematical analysis and simulation results.

This Letter presents an experimental realization of a recently proposed method to anticipate future states of nonlinear time-delayed feedback systems. The electronic circuit allows for a real-time anticipation of even strongly irregular signals. It is found that synchronization of the driven circuit with chaotic future states of the driving circuit is insensitive to signal and system perturbations.

The work is devoted to the analysis of dynamics of traveling waves in a chain of self-oscillators with period-doubling route to chaos. As a model we use a ring of Chua's circuits symmetrically coupled via a resistor. We consider how complicated are temporal regimes with parameters changing influences on spatial structures in the chain. We demonstrate that spatial periodicity exists until transition to chaos through period-doubling and tori birth bifurcations of regular regimes. Temporal quasi-periodicity does not induce spatial quasi-periodicity in the ring. After transition to chaos exact spatial periodicity is changed by the spatial periodicity in the average. The periodic spatial structures in the chain are connected with synchronization of oscillations. For quantity researching of the synchronization we propose a measure of chaotic synchronization based on the coherence function and investigate the dependence of the level of synchronization on the strength of coupling and on the chaos developing in the system. We demonstrate that the spatial periodic structure is completely destroyed as a consequence of loss of coherence of oscillations on base frequencies.

In this Letter, we propose a new synchronization principle for a class of Lur'e systems. We design, using only a single scalar output, a possible class of observers to detect whether two dynamical systems exhibit identical oscillations. The proposed method is then applied to suggest a means to secure communications. The transmitter contains a chaotic oscillator with an input that is modulated by the information signal. The receiver is a copy of the transmitter driven by a synchronization signal. The advantage of this method over the existing one is that the synchronization time is explicitly computed. An illustrative example of the cubic Chua's circuit is given to show the effectiveness of the proposed approach.

A general methodology for designing chaotic and hyperchaotic cryptosystems has been developed using the control systems theory. Grassi et al. proposed a nonlinear-observer-based decrypter for the state of an encrypter. If we can design the decrypter without the knowledge of the parameters of the encrypter, the chaos-based secure communication systems are not secure. In this paper, we have designed an observer-based chaotic communication system, which allows us to assign the relative degree and the zeros of its encrypter system. Moreover, under some conditions, we have designed an adaptive decrypter using the adaptive parameter adjustment law based on a Riccati equation when the transfer function of the encrypter is of minimal-phase type. The simulations via MATLAB/Simulink suggest that the encrypter dynamics should be designed such that its relative degree is more than 2 and its zeros are unstable so as to fail to synchronize the cryptosystem for the intruders.

In this paper, we are concerned with the complete and generalized synchronization problem for some kind of PDE chaotic systems. The criteria for the existence of synchronization manifold is given in a look-like general framework by invariant manifold method. The corresponding optimization calculation problems and procedures are proposed.

This paper investigates different types of chaotic synchronization in a system of two coupled sine maps. Due to the bimodal nature of the individual map, there is a range of parameters in which two synchronized chaotic states coexist along the main diagonal. In certain parameter regions, various (regular or chaotic) asynchronous states coexist with the synchronized chaotic states, and the basins of attraction become quite complicated. We determine the regions of stability for the so-called principal cycles that arise through transverse period-doubling bifurcations of synchronized saddle cycles. Particular emphasis is paid to the occurrence of chaotic antisynchronization, the coexistence of antisynchronous chaotic states, and the presence of narrow regions of parameter space in which states of chaotic synchronization and antisynchronization exist simultaneously. For each of these cases we provide detailed pictures of the associated basin structures.

In this paper, we study synchronization and asynchronization in a Coupled Lorenz-type Map Lattice (CLML). Lorenz-type map forms a chaotic system with an appropriate discontinuous function. We prove that in a CLML with suitable coupling strength, there is a subset of full measure in the phase space such that chaotic synchronization occurs for any orbit starting from this subset and there is a dense subset of measure zero in the phase space such that synchronization will never occur. We also provide numerical observations to explain our results.