Full State Hybrid Projective Synchronization Research Articles

The Co-existence of Different Synchronization Types in Fractional-order Discrete-time Chaotic Systems with Non-identical Dimensions and Orders.

This paper is concerned with the co-existence of different synchronization types for fractional-order discrete-time chaotic systems with different dimensions. In particular, we show that through appropriate nonlinear control, projective synchronization (PS), full state hybrid projective synchronization (FSHPS), and generalized synchronization (GS) can be achieved simultaneously. A second nonlinear control scheme is developed whereby inverse full state hybrid projective synchronization (IFSHPS) and inverse generalized synchronization (IGS) are shown to co-exist. Numerical examples are presented to confirm the findings.

Coexistence of identical synchronization, antiphase synchronization and inverse full state hybrid projective synchronization in different dimensional fractional-order chaotic systems

The topic related to the coexistence of different synchronization types between fractional-order chaotic systems is almost unexplored in the literature. Referring to commensurate and incommensurate fractional systems, this paper presents a new approach to rigorously study the coexistence of some synchronization types between nonidentical systems characterized by different dimensions and different orders. In particular, the paper shows that identical synchronization (IS), antiphase synchronization (AS), and inverse full state hybrid projective synchronization (IFSHPS) coexist when synchronizing a three-dimensional master system with a fourth-dimensional slave system. The approach, which can be applied to a wide class of chaotic/hyperchaotic fractional-order systems in the master-slave configuration, is based on two new theorems involving the fractional Lyapunov method and stability theory of linear fractional systems. Two examples are provided to highlight the capability of the conceived method. In particular, referring to commensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Rössler system of order 2.7 and the hyperchaotic four-dimensional Chen system of order 3.84. Finally, referring to incommensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Lü system of order 2.955 and the hyperchaotic four-dimensional Lorenz system of order 3.86.

A fractional-order form of a system with stable equilibria and its synchronization

There has been an increasing interest in studying fractional-order chaotic systems and their synchronization. In this paper, the fractional-order form of a system with stable equilibrium is introduced. It is interesting that such a three-dimensional fractional system can exhibit chaotic attractors. Full-state hybrid projective synchronization scheme and inverse full-state hybrid projective synchronization scheme have been designed to synchronize the three-dimensional fractional system with different four-dimensional fractional systems. Numerical examples have verified the proposed synchronization schemes.

Inverse full state hybrid projective synchronization for chaotic maps with different dimensions

A new synchronization scheme for chaotic (hyperchaotic) maps with different dimensions is presented. Specifically, given a drive system map with dimension n and a response system with dimension m, the proposed approach enables each drive system state to be synchronized with a linear response combination of the response system states. The method, based on the Lyapunov stability theory and the pole placement technique, presents some useful features: (i) it enables synchronization to be achieved for both cases of n < m and n > m; (ii) it is rigorous, being based on theorems; (iii) it can be readily applied to any chaotic (hyperchaotic) maps defined to date. Finally, the capability of the approach is illustrated by synchronization examples between the two-dimensional Hénon map (as the drive system) and the three-dimensional hyperchaotic Wang map (as the response system), and the three-dimensional Hénon-like map (as the drive system) and the two-dimensional Lorenz discrete-time system (as the response system).

A new approach to study the coexistence of some synchronization types between chaotic maps with different dimensions

When synchronizing chaotic systems, an interesting phenomenon that may occur is the coexistence of several synchronization types. Referring to discrete-time chaotic (hyperchaotic) systems, this paper presents a new approach to rigorously study the coexistence of some synchronization types between maps with different dimensions. By exploiting a Lyapunov-based approach, the paper first analyzes the coexistence of full state hybrid projective synchronization and generalized synchronization when the drive system is a three-dimensional map and the response system is a two-dimensional map. Successively, the coexistence of three different synchronization types is illustrated, i.e., inverse projective synchronization, inverse generalized synchronization and Q–S synchronization are proved to coexist between two-dimensional drive systems and three-dimensional response systems. Finally, by exploiting a pole placement technique, the coexistence of antiphase synchronization and dislocated full state hybrid projective synchronization is proved to be achieved between three-dimensional drive system maps and two-dimensional response system maps. Since synchronization is achieved in finite time, this last case represents an example of dead beat coexistence of some synchronization types in maps with different dimensions. Several numerical examples of coexistence of synchronization types are illustrated, with the aim to show the effectiveness of the novel approach developed herein.

On inverse full state hybrid projective synchronization of chaotic dynamical systems in discrete-time

In this paper, the problem of inverse full-state hybrid projective synchronization (IFSHPS) for general discrete chaotic systems is introduced and investigated. Based on a new controllers and Lyapunov stability theory, a new different nonlinear control schemes are designed to study (IFSHPS) for 2D, 3D and n-D chaotic systems in discrete-time. Theoretical analysis and numerical simulations are shown to verify the results.

On Full-State Hybrid Projective Synchronization of General Discrete Chaotic Systems

The problems of full-state hybrid projective synchronization (FSHPS) and inverse full-state hybrid projective synchronization (IFSHPS) for general discrete chaotic systems are investigated in 2D. Based on nonlinear control method and Lyapunov stability theory, new controllers are designed to study FSHPS and IFSHPS, respectively, for 2D arbitrary chaotic systems in discrete-time. Numerical example and simulations are used to validate the main results of this paper.

ADAPTIVE FULL STATE HYBRID PROJECTIVE SYNCHRONIZATION IN THE IDENTICAL AND DIFFERENT CYQY HYPER-CHAOTIC SYSTEMS

This paper further investigates the adaptive full state hybrid projective synchronization (FSHPS) of hyper-chaotic systems — CYQY system with fully unknown parameters and perturbations. Based on the Lyapunov stability theory, adaptive controllers and updating laws of parameters can be designed for achieving the FSHPS of the CYQY hyper-chaotic systems with the same and different structures. Two groups numerical simulations are provided to verify the effectiveness of the proposed scheme.

Full State Hybrid Projective Synchronization and Parameters Identification for Uncertain Chaotic (Hyperchaotic) Complex Systems

We introduce the definition of full state hybrid projective synchronization (FSHPS) with complex scaling factors for chaotic and hyperchaotic complex systems and design adaptive FSHPS schemes for uncertain chaotic complex systems under bounded disturbances with all possible situations of unknown parameters. The proposed schemes guarantee adaptive FSHPS between two chaotic complex systems with a small error bound and the convergence factors and dynamical control strength are added to regulate the convergence speed and increase robustness. Then we draw on the sufficient condition and necessary condition that the unknown parameters converge to their true values based on the persistency of excitation (PE) and linear independence (LI). At last, we realize adaptive FSHPS between uncertain complex Chen and Lü systems, which verify the feasibility and effectiveness of the presented schemes.

Chaos Synchronization of the Modified Sprott E System

In this paper, chaos synchronization of the modified Sprott E system is investigated. Firstly, the chaotic attractors on different phase planes of the system are got by means of numerical simulation. When , the abundance dynamical behavior of the system is presented by the global bifurcation graph. Then, the chaos synchronization of the system by the full state hybrid projective synchronization (FSHPS) method is realized.

A New Method for Full State Hybrid Projective Synchronization of Different Fractional Order Chaotic Systems

In this paper, the full state hybrid projective synchronization (FSHPS) between two different fractional order chaotic systems is investigated. i.e., the fractional order Chen system and the fractional order Lorenz system. Based on the synchronization error system feedback linearization theory, a new method combining feedback control is proposed for theFSHPSin fractional order chaotic systems. Numerical simulations are presented to verify the effectiveness and the feasibility of the synchronization scheme.

Continuous-time chaotic systems: Arbitrary full-state hybrid projective synchronization via a scalar signal

Referring to continuous-time chaotic systems, this paper presents a new projective synchronization scheme, which enables each drive system state to be synchronized with a linear combination of response system states for any arbitrary scaling matrix. The proposed method, based on a structural condition related to the uncontrollable eigenvalues of the error system, can be applied to a wide class of continuous-time chaotic (hyperchaotic) systems and represents a general framework that includes any type of synchronization defined to date. An example involving a hyperchaotic oscillator is reported, with the aim of showing how a response system attractor is arbitrarily shaped using a scalar synchronizing signal only. Finally, it is shown that the recently introduced dislocated synchronization can be readily achieved using the conceived scheme.

Full state hybrid projective synchronization of a novel incommensurate fractional order hyperchaotic system using adaptive mechanism

In this paper, full state hybrid projective synchronization of two new incommensurate fractional hyperchaotic systems are presented. The synchronization is achieved under a master–slave configuration in which these systems have different fractional orders. The synchronization scheme and control technique are performed subject to parameter uncertainty in both master and slave systems. The main idea of such asymptotical synchronization is an adaptive mechanism which employs Lyapunov stability criterion. Numerical simulations support the proposed techniques.

Robust adaptive full state hybrid synchronization of chaotic complex systems with unknown parameters and external disturbances

This paper studies the robust adaptive full state hybrid projective synchronization (FSHPS) scheme for a class of chaotic complex systems with uncertain parameters and external disturbances. By introducing a compensator and using nonlinear control and adaptive control, the robust adaptive FSHPS scheme is derived, which can eliminate the influence of uncertainties effectively and achieve adaptive FSHPS of the chaotic (hyperchaotic) complex systems asymptotically with a small error bound. The adaptive laws of the unknown parameters are given, and the sufficient conditions of realizing FSHPS are derived as well. Moreover, we also discuss the case that parameters of chaotic complex system are complex. Finally, the complex Chen system and Lu system, and the hyperchaotic complex Lorenz system are taken as two examples and the numerical simulations are provided to verify the effectiveness and robustness of the proposed control scheme.

Arbitrary full-state hybrid projective synchronization for chaotic discrete-time systems via a scalar signal

In this paper we present a new projective synchronization scheme, where two chaotic (hyperchaotic) discrete-time systems synchronize for any arbitrary scaling matrix. Specifically, each drive system state synchronizes with a linear combination of response system states. The proposed observer-based approach presents some useful features: i) it enables exact synchronization to be achieved in finite time (i.e., dead-beat synchronization); ii) it exploits a scalar synchronizing signal; iii) it can be applied to a wide class of discrete-time chaotic (hyperchaotic) systems; iv) it includes, as a particular case, most of the synchronization types defined so far. An example is reported, which shows in detail that exact synchronization is effectively achieved in finite time, using a scalar synchronizing signal only, for any arbitrary scaling matrix.

ADAPTIVE FULL STATE HYBRID PROJECTIVE SYNCHRONIZATION OF UNIFIED CHAOTIC SYSTEM WITH UNKNOWN PARAMETERS

This paper studies full state hybrid projective synchronization of the unified chaotic system with unknown parameters. Based on the Lyapunov stability theory, an adaptive controller is designed. It is proved theoretically that the controller can make the states of the dynamical system and the response system with unknown parameters asymptotically full state hybrid projective synchronized. Numerical simulations show the effectiveness of the scheme.

Dead-beat full state hybrid projective synchronization for chaotic maps using a scalar synchronizing signal

This paper provides a contribution to the topic of full state hybrid projective synchronization (FSHPS) by introducing an observer-based approach that enables synchronization to be achieved via a scalar synchronizing signal. The method is based on a theorem that assures dead-beat synchronization (i.e., exact synchronization in finite time) to a wide class of discrete-time chaotic (hyperchaotic) systems. Two examples, involving the hyperchaotic Grassi–Miller map and the hyperchaotic double scroll map, show that FSHPS can be effectively achieved in finite time using a scalar synchronizing signal only.

Projective synchronization for unified hyperchaotic system

The projective synchronization in the unified hyperchaotic system using two control method is studied in this paper. Based on the Lyapunov stability theory, the full state hybrid projective synchronization is achieved via an adaptive controller. It is proved theoretically that the controller can make the states of the dynamical system and the response system with known parameters achieve the asymptotically full state hybrid projective synchronization. An active controller is designed to achieve the projective synchronization. Finally, Numerical simulations show the effectiveness of the two schemes.

LAG FULL STATE HYBRID PROJECTIVE SYNCHRONIZATION IN DIFFERENT FRACTIONAL-ORDER CHAOTIC SYSTEMS

In this paper, lag full state hybrid projective synchronization (LFSHPS) in fractional-order chaotic systems is first studied. We show that LFSHPS does exist in fractional-order chaotic systems. Based on active control theory, synchronization schemes for LFSHPS of the fractional-order chaotic systems are given. Numerical simulations are provided to illustrate and verify the effectiveness of the proposed methods.

Hybrid Projective Synchronization of a New Hyperchaotic System

The full state hybrid projective synchronization, which includes complete synchronization, anti-synchronization and projective synchronization as its special item, is a novel type of chaos synchronization, so the research in this area has high practical significance. This paper is involved with full state hybrid projective synchronization of a new hyperchaotic system. Based on the stability criterion of linear system, full state hybrid projective synchronization is achieved by appropriate linear separation. Numerical simulation is presented to demonstrate the effectiveness of the proposed method.