Modified Projective Synchronization Research Articles

Modified projective synchronization of fractional-order hyperchaotic memristor-based Chua’s circuit

This paper investigates the modified projective synchronization (MPS) between two hyperchaotic memristor-based Chua circuits modeled by two nonlinear integer-order and fractional-order systems. First, a hyperchaotic memristor-based Chua circuit is suggested, and its dynamics are explored using different tools, including stability theory, phase portraits, Lyapunov exponents, and bifurcation diagrams. Another interesting property of this circuit was the coexistence of attractors and the appearance of mixed-mode oscillations. It has been shown that one can achieve MPS with integer-order and incommensurate fractional-order memristor-based Chua circuits. Finally, examples of numerical simulation are presented, showing that the theoretical results are in good agreement with the numerical ones.

Dynamic analysis and adaptive modified projective synchronization for systems with Atangana-Baleanu-Caputo derivative: A financial model with nonconstant demand elasticity

The inherent instability of the financial system itself may lead to chaos and unpredictable economic disorder. It is meaningful to study the stability and control theory of financial chaotic system. Based on the definition of Atangana-Baleanu-Caputo fractional derivative, this work extends the integer-order financial chaotic system with nonconstant demand elasticity to a fractional-order system, and analyzes its nonlinear dynamic properties. The dynamic behavior of the system is discussed by phase diagram, bifurcation diagram, the Largest Lyapunov Exponent (LLE) and 0–1 test method. An adaptive controller is designed to realize the modified projection synchronization of the system. The results can help to improve the understanding of the complex financial system, and provide theoretical support for the formulation of financial intervention strategies.

Hybrid Delayed Synchronizations of Complex Chaotic Systems in Modulus-Phase Spaces and Its Application

Compared with chaotic systems over the real number field, complex chaotic dynamics have some unique properties. In this paper, a kind of novel hybrid synchronizations of complex chaotic systems is discussed analytically and numerically. Between two nonidentical complex chaotic systems, modified projective synchronization (MPS) in the modulus space and complete synchronization in the phase space are simultaneously achieved by means of active control. Based on the Lyapunov stability theory, a controller is developed, in which time delay as an important consideration is involved. Furthermore, a switch-modulated digital secure communication system based on the proposed synchronization scheme is carried out. Different from the previous works, only one set of drive-response chaotic systems can implement switch-modulated secure communication, which could simplify the complexity of design. Furthermore, the latency of a signal transmitted between transmitter and receiver is simulated by channel delay. The corresponding numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.

A Memristor-Based Complex Lorenz System and Its Modified Projective Synchronization

The aim of this paper is to introduce and investigate a novel complex Lorenz system with a flux-controlled memristor, and to realize its synchronization. The system has an infinite number of stable and unstable equilibrium points, and can generate abundant dynamical behaviors with different parameters and initial conditions, such as limit cycle, torus, chaos, transient phenomena, etc., which are explored by means of time-domain waveforms, phase portraits, bifurcation diagrams, and Lyapunov exponents. Furthermore, an active controller is designed to achieve modified projective synchronization (MPS) of this system based on Lyapunov stability theory. The corresponding numerical simulations agree well with the theoretical analysis, and demonstrate that the response system is asymptotically synchronized with the drive system within a short time.

Complex dynamical behavior and modified projective synchronization in fractional-order hyper-chaotic complex Lü system

In this paper, a dynamic system, the fractional-order hyper-chaotic complex Lü system is proposed for the first time and its rich dynamics are investigated through numerical simulations. It is found that the variation of both the system parameters and the fractional order can induce bifurcations, and the parameter ranges of chaos are different. Furthermore, based on the Open-Plus-Closed-Loop (OPCL) technique and the stability theory of fractional-order dynamical system, a scheme of modified projective synchronization (MPS) for the fractional-order hyper-chaotic complex Lü system is presented, which can effectively achieve modified projective synchronization in a class of high dimensional fractional-order systems. Examples of numerical simulations are presented to show the validity of the proposed control strategy.

An approach to achieve modified projective synchronization between different types of fractional-order chaotic systems with time-varying delays

In this paper, modified projective synchronization (MPS) for different fractional-order chaotic systems with variable time delays is investigated. By using active control method, and analysis of the error dynamical systems, a suitable controller for achieving the modified projective synchronization is given. Finally the proposed method illustrated by two examples in two conditions and performance of proposed method is shown.

A new fractional-order hyperchaotic system and its modified projective synchronization

A new four-dimensional (4D) autonomous hyperchaotic system is investigated at first. It is found that the fractional-order version of the system goes through periodical status to chaos and hyperchaos while the order number q is increased. We verify the realizability of the new fractional-order hyperchaotic system via a practical electronic circuit. Furthermore, an active integral sliding mode control (ISMC) scheme is proposed to achieve the modified projective synchronization (MPS) of two different fractional-order hyperchaotic systems. The sliding mode controller of MPS is derived based on the stability theory of the fractional-order system and Lyapunov stability theorem. The presented method can guarantee the synchronization error asymptotically stable and enhance the robustness of MPS. Finally, the MPS between the fractional-order Chen hyperchaotic system and the new fractional-order hyperchaotic system is considered as an example, our simulation results demonstrate the effectiveness and robustness of the proposed method.

Adaptive Control for Modified Projective Synchronization-Based Approach for Estimating All Parameters of a Class of Uncertain Systems: Case of Modified Colpitts Oscillators

A method of estimation of all parameters of a class of nonlinear uncertain dynamical systems is considered, based on the modified projective synchronization (MPS). The case of modified Colpitts oscillators is investigated. Through a suitable transformation of the dynamical system, sufficient conditions for achieving synchronization are derived based on Lyapunov stability theory. Global stability and asymptotic robust synchronization of the considered system are investigated. The proposed approach offers a systematic design procedure for robust adaptive synchronization of a large class of chaotic systems. The combined effect of both an additive white Gaussian noise (AWGN) and an artificial perturbation is numerically investigated. Results of numerical simulations confirm the effectiveness of the proposed control strategy.

Modified Projective Synchronization between Different Fractional-Order Systems Based on Open-Plus-Closed-Loop Control and Its Application in Image Encryption

A new general and systematic coupling scheme is developed to achieve the modified projective synchronization (MPS) of different fractional-order systems under parameter mismatch via the Open-Plus-Closed-Loop (OPCL) control. Based on the stability theorem of linear fractional-order systems, some sufficient conditions for MPS are proposed. Two groups of numerical simulations on the incommensurate fraction-order system and commensurate fraction-order system are presented to justify the theoretical analysis. Due to the unpredictability of the scale factors and the use of fractional-order systems, the chaotic data from the MPS is selected to encrypt a plain image to obtain higher security. Simulation results show that our method is efficient with a large key space, high sensitivity to encryption keys, resistance to attack of differential attacks, and statistical analysis.

Synchronization of Underactuated Unknown Heavy Symmetric Chaotic Gyroscopes via Optimal Gaussian Radial Basis Adaptive Variable Structure Control

This brief proposes modified projective synchronization (MPS) methods for underactuated unknown heavy symmetric chaotic gyroscope systems via optimal Gaussian radial basis adaptive variable structure control. Chaotic gyroscope systems are considered as underactuated systems where a control input is designed to synchronize the two degree of freedoms interactions. Until now, no investigation of this subject with one control input has been presented. The importance of obtaining synchronization objectives is specified when the dynamics of gyroscope system are unknown. In this brief, using the neural variable structure control technique, a control law is established that guarantees the MPS of underactuated unknown chaotic gyros. In the neural variable structure control, Gaussian radial basis functions are utilized to estimate online the system dynamic functions. Adaptation laws of the online estimator are derived in the sense of the Lyapunov function. Moreover, online and offline optimizers are applied to optimize the energy of the control signal. The proposed solution is generalized to chaos control of the mentioned gyroscopes. Numerical simulations are presented to verify the proposed synchronization methods.

Generation and Modified Projective Synchronization for a Class of New Hyperchaotic Systems

A class of new hyperchaotic systems with different nonlinear terms is proposed, and the existence of hyperchaos is exhibited by calculating their Lyapunov exponent spectrums. Then the universal theories on modified projective synchronization (MPS) of the systems with general form which linearly depends on unknown parameters or time-varying parameters, are investigated by presenting an adaptive control strategy together with parameter update laws and a nonlinear control scheme based on Lyapunov stability theory. Subsequently, the presented control methods are applied to achieve MPS of the new hyperchaotic systems, and their effectiveness is illustrated by numerical simulations.

Modified projective synchronization of a fractional-order hyperchaotic system with a single driving variable

In this paper, the modified projective synchronization (MPS) of a fractional-order hyperchaotic system is investigated. We design the response system corresponding to the drive system on the basis of projective synchronization theory, and determine the sufficient condition for the synchronization of the drive system and the response system based on fractional-order stability theory. The MPS of a fractional-order hyperchaotic system is achieved by transmitting a single variable. This scheme reduces the information transmission in order to achieve the synchronization, and extends the applicable scope of MPS. Numerical simulations further demonstrate the feasibility and the effectiveness of the proposed scheme.

Designing Hopf limit circle to dynamical systems via modified projective synchronization

In order to affirmatively utilize the characteristics of Hopf limit circle, a control method to design Hopf circle with proper characteristics into dynamical system is established based on the modified projective synchronization (MPS). The proposed method may serve as a complete solution to design a stable Hopf limit circle, which can simultaneously achieve the following three properties: with the desired amplitudes and shape changes, with the pre-specified location center, and at a pre-specified system parameter location. In contrast to the methods based on Hopf bifurcation theory, the new method is independent of the verbose procedures for the bifurcation critical conditions and the stability analysis. Numerical examples demonstrate the effectiveness of the proposed method.

Synchronization and control of hyperchaotic complex Lorenz system

The aim of this paper is to investigate the phenomenon of projective synchronization (PS) and modified projective synchronization (MPS) of hyperchaotic attractors of hyperchaotic complex Lorenz system which has been introduced recently in our work. The control problem of these attractors is also studied. Our system is a 6-dimensional continuous real autonomous hyperchaotic system. The active control method based on Lyapunov function is used to study PS and MPS of this system. The problem of hyperchaos control is treated by adding the complex periodic forcing. The control performances are verified by calculating Lyapunov exponents. Numerical simulations are implemented to verify the results of these investigations.

Line spectra reduction and vibration isolation via modified projective synchronization for acoustic stealth of submarines

The potential of chaotifying vibration isolation systems to reduce line spectra of underwater vehicle and improve its capability of concealment have recently reported. Notice that for an isolation system, the amplitude of chaotifying vibration is, in general, larger than that of its specified periodic motion subject to the inherent characteristic of chaotic behavior. Therefore, there exists a conflict between the line spectra reduction based on chaotifying vibration responses and the capability of vibration isolation. In this paper, a method based on the modified projective synchronization (MPS) is developed to improve vibration isolation while the chaotifying vibration signals with broad band frequency are utilized to achieve line spectra reduction. The key to this solution is the scaling factors of MPS which enable us to proportionally diminish the vibration amplitudes of the isolated equipment. The feasibility of the methodology in practical engineering is illustrated with the application to acoustic stealth of submarines.