Synchronization between fractional-order chaotic systems and integer orders chaotic systems (fractional-order chaotic systems)

Based on the idea of tracking control and stability theory of fractional-order systems, a controller is designed to synchronize the fractional-order chaotic system with chaotic systems of integer orders, and synchronize the different fractional-order chaotic systems. The proposed synchronization approach in this paper shows that the synchronization between fractional-order chaotic systems and chaotic systems of integer orders can be achieved, and the synchronization between different fractional-order chaotic systems can also be realized. Numerical experiments show that the present method works very well.     

Mapping equivalent approach to analysis and realization of memristor-based dynamical circuit

A novel mapping equivalent approach is proposed in this paper, which can be used for analyzing and realizing a memristor-based dynamical circuit equivalently by a nonlinear dynamical circuit with the same topologies and circuit parameters. A memristor-based chaotic circuit and the corresponding Chua’s chaotic circuit with two output differentiators are taken as examples to illustrate this approach. Equivalent dynamical analysis and realization of the memristor-based chaotic circuit are performed by using Chua’s chaotic circuit. The results indicate that the outputs of memristor-based chaotic circuit and the corresponding outputs of Chua’s chaotic circuit have identical dynamics. The proposed approach verified by numerical simulations and experimental observations is useful in designing and analyzing memristor-based dynamical circuits.     

Adaptive synchronization of uncertain chaotic systems via switching mechanism

This paper deals with the problem of synchronization for a class of uncertain chaotic systems.The uncertainties under consideration are assumed to be Lipschitz-like nonlinearity in tracking error,with unknown growth rate.A logic-based switching mechanism is presented for tracking a smooth orbit that can be a limit cycle or a chaotic orbit of another system.Based on the Lyapunov approach,the adaptation law is determined to tune the controller gain vector online according to the possible nonlinearities.To demonstrate the efficiency of the proposed scheme,the well-known chaotic system namely Chua’s circuit is considered as an illustrative example.     

Extraction of periodic signals in chaotic secure communication using Duffing oscillators

This paper presents a novel approach to extract the periodic signals masked by a chaotic carrier. It verifies that the driven Duffing oscillator is immune to the chaotic carrier and sensitive to certain periodic signals. A preliminary detection scenario illustrates that the frequency and amplitude of the hidden sine wave signal can be extracted from the chaotic carrier by numerical simulation. The obtained results indicate that the hidden messages in chaotic secure communication can be eavesdropped utilizing Duffing oscillators.     

A new pseudorandom number generator based on a complex number chaotic equation

In recent years, various chaotic equation based pseudorandom number generators have been proposed. However, the chaotic equations are all defined in the real number field. In this paper, an equation is proposed and proved to be chaotic in the imaginary axis. And a pseudorandom number generator is constructed based on the chaotic equation. The alteration of the definitional domain of the chaotic equation from the real number field to the complex one provides a new approach to the construction of chaotic equations, and a new method to generate pseudorandom number sequences accordingly. Both theoretical analysis and experimental results show that the sequences generated by the proposed pseudorandom number generator possess many good properties.     

Modified adaptive controller for synchronization of incommensurate fractional-order chaotic systems

We investigate the synchronization of a class of incommensurate fractional-order chaotic systems,and propose a modified adaptive controller for fractional-order chaos synchronization based on the Lyapunov stability theory,the fractional order differential inequality,and the adaptive strategy.This synchronization approach is simple,universal,and theoretically rigorous.It enables the synchronization of0 fractional-order chaotic systems to be achieved in a systematic way.The simulation results for the fractional-order Qi chaotic system and the four-wing hyperchaotic system are provided to illustrate the effectiveness of the proposed scheme.     

Adaptive synchronization with nonlinear input

This paper considers the adaptive synchronization problem of the drive—response type chaotic systems. Using a transmitted scalar signal with an unknown time-delay, a response system is constructed. By appropriately selecting the observer gain and designing the controller, synchronization can be achieved in the presence of the drive system's disturbances and unknown parameters. A well-known chaotic system, Chua's circuit, is considered as an illustrative example to demonstrate the effectiveness of the proposed approach.     

Blind adaptive identification of FIR channel in chaotic communication systems

In this paper we study the problem of blind channel identification in chaotic communications. An adaptive algorithm is proposed, which exploits the boundness property of chaotic signals. Compared with the EKF-based approach, the proposed algorithm achieves a great complexity gain but at the expense of a slight accuracy degradation.However, our approach enjoys the important advantage that it does not require the a priori information such as nonlinearity of chaotic dynamics and the variances of measurement noise and the coefficient model noise. In addition,our approach is applicable to the ARMA system.     

The 0-1 test algorithm for chaos and its applications

To determine whether a given deterministic nonlinear dynamic system is chaotic or periodic,a novel test approach named zero-one (0-1) test has been proposed recently.In this approach,the regular and chaotic motions can be decided by calculating the parameter K approaching asymptotically to zero or one.In this study,we focus on the 0-1 test algorithm and illustrate the selection of parameters of this algorithm by numerical experiments.To validate the reliability and the universality of this algorithm,it is applied to typical nonlinear dynamic systems,including fractional-order dynamic system.     

Cascade adaptive control of uncertain unified chaotic systems

The chaos control of uncertain unified chaotic systems is considered. Cascade adaptive control approach with only one control input is presented to stabilize states of the uncertain unified chaotic system at the zero equilibrium point. Since an adaptive controller based on dynamic compensation mechanism is employed,the exact model of the unified chaotic system is not necessarily required. By choosing appropriate controller parameters,chaotic phenomenon can be suppressed and the response speed is tunable. Sufficient condition for the asymptotic stability of the approach is derived. Numerical simulation results confirm that the cascade adaptive control approach with only one control signal is valid in chaos control of uncertain unified chaotic systems.     

Synchronization of chaotic VCSELs by external chaotic signal parameter modulation

Synchronization of chaotic vertical-cavity surface-emitting lasers （VCSELs） is achieved by external chaotic signal modulation successfully. Simulation indicates that we can get chaos synchronization if the intensity of external chaotic signal is large enough. First of all, we use direct current modulation to achieve the chaos of VCSELs, and determine the laser＇s chaotic state by analyzing time series of the output and the corresponding power spectrum. And then we achieve synchronization of the two chaotic systems by external chaotic signal parameter modulation. We also find that the larger the modulation intensity is, the easier it is to achieve synchronization for chaotic VCSELs. This approach can also be applied to systems with a number of modulated lasers.     

A new output feedback synchronization theorem for a class of chaotic systems with a scalar transmitted signal

This paper proposes a new, simple and yet applicable output feedback synchronization theorem for a large class of chaotic systems. We take a linear combination of drive system state variables as a scale-driving signal. It is proved that synchronization between the drive and the response systems can be obtained via a simple linear output error feedback control. The linear feedback gain is a function of a free parameter. The approach is illustrated using the R\"{o}ssler hyperchaotic systems and Chua's chaotic oscillators.     

Chaotifying a stable linear controllable system by single input state feedback

In this paper, an approach for chaotifying a stable controllable linear system via single input state-feedback is presented. The overflow function of the system states is designed as the feedback controller, which can make the fixed point of the closed-loop system to be a snap-back repeller, thereby yields chaotic dynamics. Based on the Marotto theorem, it proves theoretically that the closed-loop system is chaotic in the sense of Li and Yorke. Finally, the simulation results are used to illustrate the effectiveness of the proposed method.     

A simple method to simultaneously achieve synchronization andanti-synchronization in chaotic systems

In this paper, a novel adaptive control approach is presented to simultaneously achieve synchronization and anti-synchronization in partially linear chaotic systems. Through appropriately separating state vectors of such systems, synchronization and anti-synchronization could be simultaneously realized in different subspaces, which may be strictly proven theoretically. Simulation results for a Lorenz chaotic system and a new hyper-chaotic system are provided to illustrate the effectiveness of the proposed method. Finally, a new secure communication scheme based on such a synchronization phenomenon of the hyper-chaotic system is demonstrated. Numerical results show success in transmitting a periodic signal with high security.     

Open-loop frequency response for a chaotic masking system

In this paper, a new numerical simulation approach is proposed for the study of open-loop frequency response of a chaotic masking system. Using Chua's circuit and the Lorenz system as illustrative examples, we have shown that one can employ chaos synchronization to separate the feedback network from a chaotic masking system, and then use numerical simulation to obtain the open-loop synchronization response, the phase response, and the amplitude response of a chaotic masking system. Based on the analysis of the frequency response, we have also proved that changing the amplitude of the exciting (input) signal within normal working domain does not influence the frequency response of the chaotic masking system. The new numerical simulation method developed in this paper can be extended to consider the open-loop frequency response of other systems described by differential or difference equations.     

An efficient chaotic source coding scheme with variable-length blocks

An efficient chaotic source coding scheme operating on variable-length blocks is proposed.With the source message represented by a trajectory in the state space of a chaotic system,data compression is achieved when the dynamical system is adapted to the probability distribution of the source symbols.For infinite-precision computation,the theoretical compression performance of this chaotic coding approach attains that of optimal entropy coding.In finite-precision implementation,it can be realized by encoding variable-length blocks using a piecewise linear chaotic map within the precision of register length.In the decoding process,the bit shift in the register can track the synchronization of the initial value and the corresponding block.Therefore,all the variable-length blocks are decoded correctly.Simulation results show that the proposed scheme performs well with high efficiency and minor compression loss when compared with traditional entropy coding.     

Exponential synchronization of stochastic impulsive perturbed chaotic Lur''e systems with time-varying delay and parametric uncertainty

This paper is devoted to investigating the scheme of exponential synchronization for uncertain stochastic impulsive perturbed chaotic Lur＇e systems. The parametric uncertainty is assumed to be norm bounded. Based on the Lyapunov function method, time-varying delay feedback control technique and a modified Halanay inequality for stochastic differential equations, several sufficient conditions are presented to guarantee the exponential synchronization in mean square between two identical uncertain chaotic Lur＇e systems with stochastic and impulsive perturbations. These conditions are expressed in terms of linear matrix inequalities （LMIs）, which can easily be checked by utilizing the numerically efficient Matlab LMI toolbox. It is worth pointing out that the approach developed in this paper can provide a more general framework for the synchronization of multi-perturbation chaotic Lur＇e systems, which reflects a more realistic dynamics. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.     

Synchronization of spatiotemporal chaos in complex networks via backstepping

A backstepping approach is proposed for the synchronization of chain networks of multi-spatiotemporal chaotic systems with topologically equivalent structures.The synchronization of multi-spatiotemporal chaotic systems is implemented by adding the control only to a terminal node,and the controller is designed via a corresponding update law.The control law is applied to spatiotemporal Gray-Scott systems.Numerical results demonstrate the effectiveness and the feasibility of the proposed approach.     

Generation of a novel spherical chaotic attractor from a new three-dimensional system

A new three-dimensional （3D） system is constructed and a novel spherical chaotic attractor is generated from the system. Basic dynamical behaviors of the chaotic system are investigated respectively. Novel spherical chaotic attractors can be generated from the system within a wide range of parameter values. The shapes of spherical chaotic attractors can be impacted by the variation of parameters. Finally, a simpler 3D system and a more complex 3D system with the same capability of generating spherical chaotic attractors are put forward respectively.     

Multiple mixed state variable incremental integration for reconstructing extreme multistability in a novel memristive hyperchaotic jerk system with multiple cubic nonlinearity

Memristor-based chaotic systems with infinite equilibria are interesting because they generate extreme multistability. Their initial state-dependent dynamics can be explained in a reduced-dimension model by converting the incremental integration of the state variables into system parameters. However, this approach cannot solve memristive systems in the presence of nonlinear terms other than the memristor term. In addition, the converted state variables may suffer from a degree of divergence. To ...