Unified Chaotic System Research Articles

Simple adaptive control strategy for a class of systems subjected to real parametric uncertainty with application to unified chaotic system

In this paper, a relatively simplified adaptive control strategy for set of chaotic systems is presented. The results utilise the basic linear state feedback control strategy blended with algebraic matrix Riccati equation (AMRE) to derive the control mechanism for chaotic systems operating in uncertain environment. The proposed approach utilises lesser number of control inputs in comparison to methods available in literature. Moreover, the controller structure turns out to be simple linear combination of states of system for the chaotic systems belonging to the proposed class, which are otherwise handled with nonlinear controller design strategies. The Lyapunov stability approach is exploited to analytically derive the control function and the parametric updation laws for uncertain parameters. Extensive simulation results are elaborated for set of chaotic systems to demonstrate effectiveness and validity of the presented approach.

Robust finite-time tracking for a square fully actuated class of nonlinear systems

In this paper, the robust finite-time tracking problem is addressed for a square fully actuated class of nonlinear systems subjected to disturbances and uncertainties. Firstly, two applicable lemmas are derived and novel nonlinear sliding surfaces (manifolds) are defined by applying these lemmas. Secondly, by developing the nonsingular terminal sliding mode control, two different types of robust nonlinear control inputs are designed to meet and accomplish the aforementioned finite-time tracking objective. The global finite-time stability of the closed-loop nonlinear system is evaluated analytically and mathematically. The proposed control inputs are utilized to tackle and solve two interesting issues containing (a): the finite-time tracking problem of the unified chaotic system and (b): the finite-time synchronization of two non-identical hyperchaotic systems. Finally, based on MATLAB software, two numerical simulations are carried out to illustrate and demonstrate the effectiveness and performance of the proposed robust finite-time nonlinear control schemes.

A novel secure communications scheme based on chaotic modulation, recursive encryption and chaotic masking

The aim of this paper is to propose a new secure communications approach, obtained by combining into one scheme chaotic modulation, recursive encryption and chaotic masking. The objective is achieved by designing control laws to synchronize two hyperchaotic Lorenz systems with encrypted states via a backstepping technique. The encryption process relies on a Unified chaotic system, driven by the hyperchaotic Lorenz system, as well as on a recursive encryption algorithm that exploits an n-fold composition of an invertible function. A novel theorem is proved, which assures that the transmitted message is recovered at the receiver side by virtue of the synchronization of the two Lorenz systems via backstepping design. Numerical simulations are carried out using image and text as transmitted signals, with the aim to show the effectiveness of the conceived chaos-based encryption scheme.

Passivity based sliding mode control and synchronization of a perturbed uncertain unified chaotic system

The problem of passivity-based sliding mode control is considered, for the unified chaotic system having uncertainties and perturbations. Using a combination between sliding mode method and the passivity method, an appropriate switching surface for the uncertain and perturbed system is designed, and then through passivity theory, the stability of the system is ensured. The proposed design is applied in two control problems, the first one being the stabilization of the uncertain perturbed unified chaotic system and the second one is the synchronization between two unified chaotic systems, where only the slave system contains uncertainties and perturbations. Finally, the effectiveness of the proposed methods are illustrated through numerical simulations, and also through a microcontroller implementation.

A New Six-Term 3D Unified Chaotic System

In this study, four different 3D five-term chaotic flows are unified and a novel six-term 3D unified chaotic system with three nonlinearities is introduced. Firstly, the theoretical system via an electronic circuit is realized, and then the basic dynamical properties of the proposed unified chaotic system are numerically and analytically analyzed, i.e., sensitivity to initial conditions, equilibrium points, eigenvalues, Kaplan–Yorke dimensions, dissipativity, Lyapunov exponents and bifurcation diagrams. Investigation results clearly present that this is a new unified chaotic system and earns further detailed disquisition.

The fractional-order unified chaotic system: A general cascade synchronization method and application

In this paper, we extend the cascade synchronization of integer-order chaotic systems to function cascade synchronization of fractional-order chaotic systems. The nice feature of our method is that a fractional-order chaotic system will synchronize to another system via a scaling function. The rich choices of the scaling function turn our method more general than some existing methods. As applications, we take a fractional-order unified chaotic system as an illustrative example to test the effectiveness.

Numerical Analysis, Circuit Simulation, and Control Synchronization of Fractional-Order Unified Chaotic System

The traditional method of solving fractional chaotic system has the problem of low precision and is computationally cumbersome. In this paper, different fractional-order calculus solutions, the Adams prediction–correction method, the Adomian decomposition method and the improved Adomian decomposition method, are applied to the numerical analysis of the fractional-order unified chaotic system. The result shows that different methods have higher precision, smaller computational complexity, and shorter running time, in which the improved Adomian decomposition method works best. Then, based on the fractional-order chaotic circuit design theory, the circuit diagram of fractional-order unified chaotic system is designed. The result shows that the circuit simulation diagram of fractional-order unified chaotic system is basically consistent with the phase space diagram obtained from the numerical solution of the system, which verifies the existence of the fractional-order unified chaotic system of 0.9-order. Finally, the active control method is used to control and synchronize in the fractional-order unified chaotic system, and the experiment result shows that the method can achieve synchronization in a shorter time and has a better control performance.

Parameter estimation for fractional-order chaotic systems by improved bird swarm optimization algorithm

The essence of parameter estimation for fractional-order chaotic systems is a multi-dimensional parameter optimization problem, which is of great significance for implementing fractional-order chaos control and synchronization. Aiming at the parameter estimation problem of fractional-order chaotic systems, an improved algorithm based on bird swarm algorithm is proposed. The proposed algorithm further studies the social behavior of the original bird swarm algorithm and optimizes the foraging behavior in the original bird swarm algorithm. This method is applied to parameter estimation of fractional-order chaotic systems. Fractional-order unified chaotic system and fractional-order Lorenz system are selected as two examples for parameter estimation systems. Numerical simulation shows that the algorithm has better convergence accuracy, convergence speed and universality than bird swarm algorithm, artificial bee colony algorithm, particle swarm optimization and genetic algorithm.

Adaptive projective synchronization for a class of switched chaotic systems

This paper addresses the problem of projective synchronization of chaotic systems and switched chaotic systems by adaptive control methods. First, a necessary and sufficient condition is proposed to show how many state variables can realize projective synchronization under a linear feedback controller for the chaotic systems. Then, accordingly, a new algorithm is given to select all state variables that can realize projective synchronization. Furthermore, according to the results of the projective synchronization of chaotic systems, the problem of projective synchronization of the switched chaotic systems comprised by the unified chaotic systems is investigated, and an adaptive global linear feedback controller with only one input channel is designed, which can realize the projective synchronization under the arbitrary switching law. It is worth mentioning that the proposed method can also realize complete synchronization of the switched chaotic systems. Finally, the numerical simulation results verify the correctness and effectiveness of the proposed method.

Control of Fractional-Order Unified Chaotic Systems Subject to External Disturbances Using Twisting Algorithm with Fractional Integral Sliding Surface

This paper focuses on output feedback control of fractional-order unified chaotic systems subject to external disturbances. The output feedback controller is a combination of a high-order sliding mode controller and an observer. The controller based on a twisting algorithm makes use of a fractional integral sliding surface. The observer is a nonlinear state observer. Stability conditions for the output feedback control system are formulated as a linear matrix inequality problem. Numerical studies are given, showing the effectiveness of the method.

Explosive Synchronization in Complex Dynamical Networks Coupled with Chaotic Systems

Explosive synchronization (ES), as one kind of abrupt dynamical transitions in nonlinearly coupled systems, has become a hot spot of modern complex networks. At present, many results of ES are based on the networked Kuramoto oscillators and little attention has been paid to the influence of chaotic dynamics on synchronization transitions. Here, the unified chaotic systems (Lorenz, Lü and Chen) and R?ssler systems are studied to report evidence of an explosive synchronization of chaotic systems with different topological network structures. The results show that ES is clearly observed in coupled Lorenz systems. However, the continuous transitions take place in the coupled Chen and Lü systems, even though a big shock exits during the synchronization process. In addition, the coupled R?ssler systems will keep synchronous once the entire network is completely synchronized, although the coupling strength is reduced. Finally, we give some explanations from the dynamical features of the unified chaotic systems and the periodic orbit of the R?ssler systems.

The Secure Transmission of Videos Using the Karhunen-Loéve (K-L) Decomposition and the Synchronization of the Unified Chaotic System with the Hyperchaotic Chen System

A unique secure communication scheme that can be used for the transmission of gray-scale and color videos is presented in this paper. The proposed scheme is developed by using the Karhunen-Loéve (K-L) decomposition and the synchronization of the unified chaotic system with the hyperchaotic Chen system. First, the gray-scale or color video is represented by a set of N frames. In order to reduce the data, the K-L decomposition is used to come up with data coefficients and eigenfunctions that optimally obtain the crux of the N frames. Using only the most energetic eigenfunctions to approximate the original frames results in computational savings. The data coefficients corresponding to the most energetic eigenfunctions are encrypted and transmitted using a master system composed of a combination of the unified chaotic system and the hyperchaotic Chen system. At the receiver end, these coefficients are recovered and a controller of the sliding mode type is utilized forcing the master and slave systems to synchronize. Simulation results illustrate how the proposed control law is able to synchronize the master and the slave systems. In addition, a demonstration of the recovery of the original frames using the decrypted data coefficients along with the eigenfunctions of the frame is provided. The presented simulations indicate that the proposed scheme results in an excellent performance.

Synchronization of unified chaotic systems via adaptive nonsingular fast terminal sliding mode control

This paper investigates the robust synchronization problem for a class of unified chaotic systems with input nonlinearity via an adaptive nonsingular fast terminal sliding mode control. The proposed scheme not only ensures that the synchronization errors are fully derived to zero in the presence of system uncertainties and external disturbances, but also copes with the singularity and chattering phenomena. Using the adaptation laws, the estimation of unknown uncertainties and disturbances can be realized. The efficiency of the proposed scheme is verified in simulation results.

Chaos synchronization of TSUCS unified chaotic system, a modified function projective control method

Chaos synchronization of TSUCS unified chaotic system, a modified function projective control method

Comparison of the influences of nodal dynamics on network synchronized regions

A new piecewise linear unified chaotic (PLUC) system is firstly presented, and then its fundamental dynamical behaviors are analyzed. This modified chaotic system, as well as the unified chaotic (UC) one, is taken as network nodal oscillators for investigating the difference of influences of nodal dynamics on the bifurcation of network synchronized regions. It is found that beyond the greatly similar bifurcation modes between PLUC and UC networks, the synchronized regions in PLUC networks are far narrower at almost each parameter a than those in UC networks for most of inner coupling matrices, indicating the PLUC node makes the network more difficult to synchronization. Our numerical investigations show that this phenomenon is closely related with nodal dynamical properties, such as the boundary of attractors, the largest Lyapunov exponent and Lyapunov dimension.

Chaotic and non-chaotic strange attractors of a class of non-autonomous systems.

In this paper, the dynamics of a class of non-autonomous systems, generated from a unified chaotic autonomous system, is studied. It is found, via parameter modulation, that they have chaotic and non-chaotic strange attractors (NCSA). Several representative systems are constructed to illustrate the complex strange dynamics. The first example exhibits Lorenz-like behavior and Chen-like behavior at different time intervals. The second illustrates the existence of NCSA, which is constructed by "joining" the chaotic Chen system and a system with regular dynamics. The third is constructed based on the topological structure of the original autonomous systems, which has complex transient dynamics at the beginning, with a periodic orbit as the omega-limit set. The last one has quasi-periodic coefficients, yielding strange dynamics. These examples demonstrate that non-autonomous systems can have extremely rich and interesting dynamics under certain conditions.

Comparative performances of synchronisation between different classes of chaotic systems using three control techniques

This paper puts forward the comparative performances for synchronisation between (i) systems from different chaotic system families, (ii) systems from the Unified Chaotic System (UCS) family, (iii) a hyperchaotic and a chaotic systems and (iv) identical chaotic systems. Three different well-known control techniques, i.e. Nonlinear Active Control (NAC), Sliding Mode Control (SMC) and Adaptive Control (AC) are used for synchronisation between various pairs of chaotic and/or hyperchaotic systems. Performances of NAC, SMC and AC techniques are investigated and compared with synchronisation of different pairs of chaotic systems based on the error dynamics and required control inputs. The integral square error and required control energy measures are considered for comparison. Finally, a generalised view on the use of the control techniques for synchronisation is finally proposed. Moreover, a new chaotic system is proposed and its qualitative analysis is done to illustrate the chaotic behaviour of the system. The new system is used as an example of synchronisation. MATLAB simulation results are presented which reflect the successful achievement of the objectives.

Adaptative synchronization in multi-output fractional-order complex dynamical networks and secure communications

This work deals with the adaptative synchronization of complex dynamical networks with fractional-order nodes and its application in secure communications employing chaotic parameter modulation. The complex network is composed of multiple fractional-order systems with mismatch parameters and the coupling functions are given to realize the network synchronization. We introduce a fractional algebraic synchronizability condition (FASC) and a fractional algebraic identifiability condition (FAIC) which are used to know if the synchronization and parameters estimation problems can be solved. To overcome these problems, an adaptative synchronization methodology is designed; the strategy consists in proposing multiple receiver systems which tend to follow asymptotically the uncertain transmitters systems. The coupling functions and parameters of the receiver systems are adjusted continually according to a convenient sigmoid-like adaptative controller (SLAC), until the measurable output errors converge to zero, hence, synchronization between transmitter and receivers is achieved and message signals are recovered. Indeed, the stability analysis of the synchronization error is based on the fractional Lyapunov direct method. Finally, numerical results corroborate the satisfactory performance of the proposed scheme by means of the synchronization of a complex network consisting of several fractional-order unified chaotic systems.

Dynamical Analysis of Memristive Unified Chaotic System and Its Application in Secure Communication

In this paper, a novel memristive unified chaotic system is proposed. The fundamental dynamical behaviors of such system are investigated by calculating phase portraits, Poincare map, Lyapunov exponents, and bifurcation diagram. Also, the parameters and initial conditions that affect the properties of the memristive system are studied, In addition, a new scheme employing the adaptive parameter modulation synchronization of two memristive unified chaotic systems is proposed. In this scheme, the original signal is modulated into the parameter of the memristive unified systems, and the corresponding receiver system can succeed in recovering the original signal by utilizing the estimated parameter. Furthermore, the proposed memristive unified chaotic system can be applied into secure communication. The simulation results prove the effectiveness of the proposed scheme.

Identical and Non-identical Synchronization of Three Scroll Unified Chaotic System (TSUCS) with Unknown Parameter Using a Modified Function Projective Control Method

This paper investigates the synchronization problem of the three-scroll unified chaotic system (TSUCS) using a modified function projective synchronization (MFPS) method. It is assumed that the parameters of both the drive and response unified chaotic systems are uncertain, while all state variables are available. First, identical synchronization of the TSUCS system is discussed with MFPS method. Then, the MFPS synchronization of TSUCS and Lu, as the two non-identical unified chaotic systems is studied. Parameter estimation strategies and feedback controllers for synchronization of the drive-response systems are designed based on the Lyapunov stability theorem and the MFPS method. The proposed synchronization scheme can be effectively utilized for synchronization of other unified chaotic systems.