Dimensional Fractional Order Chaotic Systems Research Articles

Study of Generalized Synchronization and Anti-synchronization Between Different Dimensional Fractional-Order Chaotic Systems with Uncertainties

Study of Generalized Synchronization and Anti-synchronization Between Different Dimensional Fractional-Order Chaotic Systems with Uncertainties

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.

Fractional analysis of co-existence of some types of chaos synchronization

Chaotic dynamics and synchronization of fractional-order systems have attracted much attention recently. Based on stability theory of fractional-order systems and stability theory of integer-order systems, this paper deals with the problem of coexistence of various types of synchronization between different dimensional fractional-order chaotic systems. To illustrate the capabilities of the novel schemes proposed herein, numerical and simulation results are given.

Fractional chaos synchronization schemes for different dimensional systems with non-identical fractional-orders via two scaling matrices

By using two scaling matrices, the synchronization problem of different dimensional fractional order chaotic systems in different dimensions is developed in this article. The controller is designed to assure that the synchronization of two different dimensional fractional order chaotic systems is achieved using the fractional-order Lyapunov direct method. Numerical examples and computer simulations are used to validate numerically the proposed synchronization schemes.

Modified Function Projective Synchronization of Fractional Order Chaotic Systems with Different Dimensions

The modified function projective synchronization of different dimensional fractional-order chaotic systems with known or unknown parameters is investigated in this paper. Based on the stability theorem of linear fractional-order systems, the adaptive controllers with corresponding parameter update laws for achieving the synchronization are given. The fractional-order chaotic system and hyperchaotic system are applied to achieve synchronization in both reduced order and increased order. The corresponding numerical results coincide with theoretical analysis.

Generalized Projective Synchronization of Fractional Order Chaotic Systems with Different Dimensions

The generalized projective synchronization of different dimensional fractional order chaotic systems is investigated. According to the stability theory of linear fractional order systems, a sufficient condition to realize synchronization is obtained. The fractional order chaotic and hyperchaotic systems are applied to achieve synchronization in both reduced and increased dimensions. The corresponding numerical results coincide with theoretical analysis.

Hybrid projective synchronization of chaotic fractional order systems with different dimensions

The hybrid projective synchronization of different dimensional fractional order chaotic systems is investigated in this paper. It is shown that the slave system can be synchronized with the projection of the master system generated through state transformation. Based on the stability theorem of linear fractional order systems, a suitable controller for achieving the synchronization is given. The hybrid projective synchronization between the fractional order chaotic system and hyperchaotic system is successfully achieved in both reduced order and increased order. The corresponding numerical results verify the effectiveness of the proposed method.