Master Chaotic Systems Research Articles

Secure message recovery in presence of the cyber-attack using discrete-time chaotic modulation approach

The secure message recovery problem in presence of cyber-attack is considered in this paper. The message is modulated by a master chaotic system which may be vulnerable to a cyber-attack and then transmitted to a receiver. By designing a novel observer at the receiver side, the message is recovered correctly. All the design process is performed in discrete-time domain which is suitable for digital communications. Besides, there are no restrictions on the message signal and nonlinear term signature matrix which makes the proposed approach more suitable in the real applications. Numerical simulations are introduced to show the feasibility and the superior performance of the proposed scheme in comparison with the conventional methods.

Secure Data Transmission and Image Encryption Based on a Digital-Redesign Sliding Mode Chaos Synchronization

In this paper, a novel image encryption algorithm based on chaotic synchronization is proposed. First, a digital-redesign sliding mode controller (SMC) is developed to guarantee the chaos synchronization. The digital redesign method makes it possible to transform a proposed continuous-time SMC to discrete-time SMC whilst maintaining the performance of the robust synchronization. Then, the secret keys are embedded in the state equations of the master chaotic system, such that the secret keys do not appear in the public channel, and utilize the chaotic synchronization to achieve secure communication for transmitting the secret keys from transmitter to receiver. Second, an image encryption algorithm integrating the S-box with chaotic synchronization is established, where the S-box is created by the secret key transmitted from the transmitter. Finally, a detailed analysis of the image encryption algorithm based on chaos synchronization is included to verify the feasibility and effectiveness of this proposed approach.

Secure Data Transmission Based on Adaptive Chattering-Free Sliding Mode Synchronization of Unified Chaotic Systems

This paper is concerned with a novel secure data transmission design based on adaptive synchronization of master and slave unified chaotic systems. First, by introducing an augmented error state, an adaptive continuous sliding mode control (SMC) is derived to guarantee the synchronization of unified chaotic systems. Then, the secret message embedded in the master chaotic system can be transmitted from transmitter to receiver. Different from previous works using discontinuous SMC, the undesired chattering phenomenon can be fully eliminated, and it becomes possible to precisely recover the embedded secret message at the receiver. Last, an example is given to illustrate the success of secure data transmission with the continuous SMC developed in this paper.

Uncertainty and disturbance estimator based robust synchronization for a class of uncertain fractional chaotic system via fractional order sliding mode control

This paper deals with a finite time robust synchronization problem of a class of uncertain fractional chaotic/hyper-chaotic systems with a novel fractional sliding mode control technique. Firstly, a fractional order sliding surface is proposed to mimic the behavior of master chaotic system. Then, a fractional order sliding mode control (FOSMC) methodology is derived analytically for convergence of all the synchronizing errors to zero in finite time. Finally, the derived control strategy is augmented with an auxiliary control based on uncertainty and disturbance estimator (UDE) for ensuring the robustness of the closed loop system dynamics in the presence of system uncertainties. Further, the uncertainties with unknown bounds are tackled for depicting the practical scenario and these results are also applicable to the N-dimensional uncertain chaotic as well as hyper-chaotic systems. Moreover, Mittag-Leffler and fractional order Lyapunov results are utilized to prove the stability and finite time convergence. Also, the proposed method delivers chatter-free control signal which is a major issue in sliding mode. MATLAB simulations are carried out to verify the efficacy and robustness of the derived results by considering two examples from literature.

Coexistence of generalized synchronization and inverse generalized synchronization between chaotic and hyperchaotic systems

In this paper, we present new schemes to synchronize different dimensional chaotic and hyperchaotic systems. Based on coexistence of generalized synchronization (GS) and inverse generalized synchronization (IGS), a new type of hybrid chaos synchronization is constructed. Using Lyapunov stability theory and stability theory of linear continuous-time systems, some sufficient conditions are derived to prove the coexistence of generalized synchronization and inverse generalized synchronization between 3D master chaotic system and 4D slave hyperchaotic system. Finally, two numerical examples are illustrated with the aim to show the effectiveness of the approaches developed herein.

Robust finite-time composite nonlinear feedback control for synchronization of uncertain chaotic systems with nonlinearity and time-delay

This paper proposes a combination of finite-time robust-tracking theory and composite nonlinear feedback approach for the finite-time and high performance synchronization of the chaotic systems in the presence of the external disturbances, parametric uncertainties, Lipschitz nonlinearities and time delays. The composite nonlinear feedback control technique develops an accurate and high-performance response for the following of the master chaotic system and the finite time concept provides the convergence of the error signals to zero in the finite time. Therefore, in this work, we will develop a new finite time robust tracking and model following control approach based on the composite nonlinear feedback scheme. Using the Lyapunov stability approach, we have proved that the tracking errors of the uncertain chaotic system converge to the origin in the finite time. Moreover, a sufficient criterion is derived to guarantee the robust asymptotic stability of the synchronization error dynamics. Simulation results on Chua's chaotic system are presented to prove the performance of the suggested controller compared to the other technique.

Chaos synchronization of uncertain chaotic systems using composite nonlinear feedback based integral sliding mode control

This paper proposes a combination of composite nonlinear feedback and integral sliding mode techniques for fast and accurate chaos synchronization of uncertain chaotic systems with Lipschitz nonlinear functions, time-varying delays and disturbances. The composite nonlinear feedback method allows accurate following of the master chaotic system and the integral sliding mode control provides invariance property which rejects the perturbations and preserves the stability of the closed-loop system. Based on the Lyapunov- Krasovskii stability theory and linear matrix inequalities, a novel sufficient condition is offered for the chaos synchronization of uncertain chaotic systems. This method not only guarantees the robustness against perturbations and time-delays, but also eliminates reaching phase and avoids chattering problem. Simulation results demonstrate that the suggested procedure leads to a great control performance.

Globally exponential multi switching-combination synchronization control of chaotic systems for secure communications

This article introduces the global exponential multi switching combination synchronization (GEMSCS) for three different chaotic systems with known parameters in the master-slave system configuration. The proposed GEMSCS scheme establishes the global exponential stability of the synchronization error at the origin with different combinations of state variables of the two master chaotic systems with the state variables of a slave chaotic system in diverse manners. Consequently, it increases the complexity level of the information signal in secure communications. To study the GEMSCS, an efficient nonlinear control algorithm is designed. The Lyapunov direct theorem is used to accomplish the global exponential stability of the synchronization error at the origin. The stability conditions are derived analytically. To show the effectiveness and advantages of the proposed GEMSCS control approach, two numerical examples are presented. The computer based simulation results are compared with the reported works in the relevant literature. This article also extends the idea of GEMSCS to the secure communication using the chaotic masking technique. Using the GEMSCS strategy, the information signal is recovered at the receiving system with good accuracy and high speed while the parameters of the transmitter and receiver systems mismatch. At the end, some future research problems related to this work are suggested.

On a simple approach for Q-S synchronisation of chaotic dynamical systems in continuous-time

In this paper, the problem of Q-S synchronisation for arbitrary dimensional chaotic dynamical systems in continuous-time is investigated. Based on nonlinear control method, we would like to present a constructive scheme to study the Q-S synchronisation between n-dimensional master chaotic system and m-dimensional slave chaotic system in arbitrary dimension. The new derived synchronisation result is proved using Lyapunov stability theory. In order to verify the effectiveness of the proposed method, our approach is applied to some typical chaotic systems and numerical simulations are given to validate the derived results.

Model reference fractional order control using type-2 fuzzy neural networks structure: Implementation on a 2-DOF helicopter

In this paper, an adaptive learning algorithm is proposed for an interval type-2 fuzzy fractional order controller. The use of fractional order controller adds more degrees of freedom which makes it possible to obtain superior performance in comparison with ordinary differential controllers. A fractional order reference model is used to define the desired trajectory of the nonlinear dynamic system. The structure of the system is based on the feedback error learning method. The stability of the adaptation laws is proved using Lyapunov theory. In order to test the efficiency and efficacy of the proposed learning and the control algorithm, the trajectory tracking problem of a magnetic rigid spacecraft is studied. The simulation results show that the proposed control algorithm outperforms the case when ordinary differential fuzzy controller is used. Furthermore, it is shown that it is possible to define a master chaotic system as a reference model and obtain synchronization between the two chaotic systems using the proposed approach. In the simulation part the synchronization between two Duffing–Holmes system is also achieved. In order to show the implementability of the proposed method, it is used to control a real time laboratory setup 2-DOF helicopter. It is shown that the proposed fractional order controller can be implemented in a low cost embedded system and can successfully control a highly nonlinear dynamic system.

Event-triggered heterogeneous master-slave synchronization with sampled-data control

As is well known, chaos synchronization has a good performance in secret communication. However, in most existing researches, chaos synchronization is realized between master chaotic system and slave chaotic system with real-time communication. A lot of network bandwidths are wasted in useless communication which may bring great economic costs and increase the likelihood of network congestion. In this paper, the synchronization for the heterogeneous master-slave system is studied. Compared with the homogeneous synchronization, the heterogeneous synchronization has a broad application prospect. Then the sampling controller based on event trigger is designed to achieve the heterogeneous master-slave synchronization which can save network bandwidth, with the control performance maintained. Because transmission time delay is universal in communication system, the delay model of the system is constructed and utilized firstly. Based on the proposed system model, the master-slave heterogeneous synchronization problem is equivalently converted into the asymptotical stability problem of a time-delay system by using the input delayed approach. Then, according to the Lyapunov stability theory and linear matrix inequality, we may rigouously prove that the synchronization of heteogeneous master-slave chaotic system can be achieved. Meanwhile, the heterogeneous master-slave system synchronization conditions and sampling controller can be given based on the constructing Lyapunov-Krasovskii functional with the Wirtinger inequality and free weight matrix method. The sampling controller actually is a stateback controller. But for the purpose of reducing the network usage rate, wheter the state can be transmitted can be determined by the event-triggered rules. The event-triggered rules are designed based on the error state between master chaotic system and slave chaotic system, and the synchronization performance index can be determined by choosing the system parameter. The designed sampling controller updates the control parameters when the event-triggered rules are satisfied. Finally, numerical simulation experiments are employed to verify the correctness and effectiveness of the proposed method. Results indicate that the hetergeneous master-slave chaotic system with transimission time delay can indeed achieve synchronization by event-triggerd sample control. Moreover, the number of communications between master chaotic system and slave chaotic system is less than before and the synchroization performance is also at an idea level.

Synchronization of Unknown Uncertain Chaotic Systems Via Adaptive Control Method

In this paper, an adaptive control scheme is offered to synchronize two different uncertain chaotic systems. It is assumed that the whole dynamics of both master and slave chaotic systems and their bounds are unknown and different. The error system stabilization is achieved in two cases: with input nonlinearities and without input nonlinearities. We design an adaptive control scheme based on the state boundedness property of the chaotic systems. The proposed method does not need any information about nonlinear/linear terms of the chaotic systems. It only uses an adaptive feedback control strategy. The stability of the proposed controllers is proved by using the Lyapunov stability theory. Finally, the designed adaptive controllers are applied to synchronize two different pairs of the chaotic systems (Lorenz–Chen and electromechanical device–electrostatic transducer).

Delay-Dependant Stability of Chaotic Systems with Exponential Synchronization

This study is concerned with the synchronization problem for a class of chaotic systems with multiple time-delays and transmitted noise. A dynamic compensator is proposed to achieve the synchronization between master and slave chaotic systems with multiple timedelays. First, the neural-network (NN) model is employed to approximate the chaotic systems with multiple time delays. Then, a linear differential inclusion (LDI) state-space representation is established for the dynamics of the NN model. Based on the LDI state-space representation, an exponential stability criterion of error dynamics derived in terms of Lyapunov's direct method is proposed to ensure that the trajectories of the slave system can approach those of master system. Subsequently, the stability condition of this criterion is reformulated into a linear matrix inequality (LMI). Based on the LMI, a fuzzy controller is synthesized to realize the exponential synchronization of the chaotic master-slave systems. Finally, this study provides a numerical example with simulations is given to illustrate the concepts discussed throughout this work.

Observer-Based Synchronization of Switched Chaotic Systems

This paper presents a control scheme for the synchronization of switched chaotic systems. The general case that the error state information between master and slave chaotic systems is not available for design the controller, an adaptive observer-based error system is constructed and the controller is designed. Utilizing common Lyapunov function method, a synchronization criterion is given. Simulation shows the effectiveness and efficiency of the proposed scheme.

Robust adaptive synchronization for a class of chaotic systems with actuator failures and nonlinear uncertainty

This paper is concerned with the robust adaptive synchronization problem for a class of chaotic systems with actuator failures and unknown nonlinear uncertainty. Combining adaptive method and linear matrix inequality (LMI) technique, a novel type of robust adaptive reliable synchronization controller is proposed, which can eliminate the effect of actuator fault and nonlinear uncertainty on systems. After solving a set of LMIs, synchronization error between the master chaotic and slave chaotic systems can converge asymptotically to zero. Finally, illustrate examples about chaotic Chua’s circuit system and Lorenz systems are provided to demonstrate the effectiveness and applicability of the proposed design method.

Chaos Synchronization of Uncertain Lorenz System via Single State Variable Feedback

This paper treats the problem of chaos synchronization for uncertain Lorenz system via single state variable information of the master system. By the Lyapunov stability theory and adaptive technique, the derived controller is featured as follows: 1) only single state variable information of the master system is needed; 2) chaos synchronization can also be achieved even if the perturbation is occurred in some parameters of the master chaotic system. Finally, the effectiveness of the proposed controllers is also illustrated by the simulations as well as rigorous mathematical proofs.

Adaptive Synchronization of Unknown Chaotic Systems Using Mamdani Fuzzy Approach

The synchronization problem for a general-class of unknown chaotic systems is addressed. Fuzzy systems in Mamdani type are employed to provide an approximate model of the master chaotic system using an adaptive approach. Within this, a number of fuzzy systems are utilized to approximate the unknown nonlinear functions of the master system. An approximate model similar to the master system is constructed which is the slave system. The error dynamics between the master and slave chaotic systems is used to build a suitable control input and fuzzy systems’ parameters adaptive laws to force the slave system to be synchronized with master system. The stability of the overall synchronization system is derived based on Lyapunov stability theory. Its shown that under appropriate assumptions, the proposed approach guarantees the boundness and asymptotic convergence of synchronization errors to a small neighborhood of origin. An extensive simulation study is performed on both Duffing and Rössler chaotic systems to show the effectiveness of the proposed scheme.

Exponential networked synchronization of master-slave chaotic systems with time-varying communication topologies

The networked synchronization problem of a class of master-slave chaotic systems with time-varying communication topologies is investigated in this paper. Based on algebraic graph theory and matrix theory, a simple linear state feedback controller is designed to synchronize the master chaotic system and the slave chaotic systems with a time-varying communication topology connection. The exponential stability of the closed-loop networked synchronization error system is guaranteed by applying Lyapunov stability theory. The derived novel criteria are in the form of linear matrix inequalities (LMIs), which are easy to examine and tremendously reduce the computation burden from the feedback matrices. This paper provides an alternative networked secure communication scheme which can be extended conveniently. An illustrative example is given to demonstrate the effectiveness of the proposed networked synchronization method.

Control and synchronization of chaotic systems using a novel indirect model reference fuzzy controller

This paper presents a robust indirect model reference fuzzy control scheme for control and synchronization of chaotic nonlinear systems subject to uncertainties and external disturbances. The chaotic system with disturbance is modeled as a Takagi–Sugeno fuzzy system. Using a Lyapunov function, stable adaptation laws for the estimation of the parameters of the Takagi–Sugeno fuzzy model are derived as well as what the control signal should be to compensate for the uncertainties. The synchronization of chaotic systems is also considered in the paper. It is shown that by the use of an appropriate reference signal, it is possible to make the reference model follow the master chaotic system. Then, using the proposed model reference fuzzy controller, it is possible to force the slave to act as the reference system. In this way, the chaotic master and the slave systems are synchronized. It is shown that not only can the initial values of the master and the slave be different, but also there can be parametric differences between them. The proposed control scheme is simulated on the control and the synchronization of Duffing oscillators and Genesio–Tesi systems.

SECURE COMMUNICATION BASED ON A HYBRID OF CHAOS AND ICA ENCRYPTIONS

Chaos and independent component analysis (ICA) encryptions are two novel schemes for secure communications. In this paper, a new scheme combining chaos and ICA techniques is proposed to enhance the security level during communication. In this scheme, a master chaotic system is embedded at the transmitter. The message signal is mixed with a chaotic signal and a Gaussian white noise into two mixed signals and then transmitted to the receiver through the public channels. A signal for synchronization is transmitted through another public channel to the receiver where a slave chaotic system is embedded to reproduce the chaotic signal. A modified ICA is used to recover the message signal at the receiver. Since only two of the three transmitted signals contain the information of message signal, a hacker would not be able to retrieve the message signal by using ICA even though all the transmitted signals are intercepted. Spectrum analyses are used to prove that the message signal can be securely hidden under this scheme.