Fractional-order Discrete-time Chaotic Systems Research Articles

A new synchronization result for fractional-order discrete-time chaotic systems via bandlimited channels

Recently, digital chaotic communication with limited bandwidth has seized the attention of several researchers. This interest stems from the fact that all physical communication channels are limited in bandwidth for data transmission. In furtherance of this, researchers and investigators in this area have put forward various new communication strategies based on synchronizing and controlling coupled chaotic systems evolving in discrete time. In this article, we present a new robust communication scheme based on the synchronization of two fractional-order discrete-time chaotic systems connected by bandlimited channels. The major contribution is to give a sufficient condition in term of the existing and updated parameters of the incorporated fractional-order system on one hand and the inserted filter coefficients, on the other hand. Once applied for data communications, the currently drawn up synchronization scheme pledges to enhance the transmission robustness. Indeed, the new considered parameters act like novel secret keys which widen the cipher key space. Primary and pertinent simulation curves and results performed on Matlab software portray and illustrate the effectiveness of our approach.

An Unprecedented 2-Dimensional Discrete-Time Fractional-Order System and Its Hidden Chaotic Attractors

Some endeavors have been recently dedicated to explore the dynamic properties of the fractional-order discrete-time chaotic systems. To date, attention has been mainly focused on fractional-order discrete-time systems with “self-excited attractors.” This paper makes a contribution to the topic of fractional-order discrete-time systems with “hidden attractors” by presenting a new 2-dimensional discrete-time system without equilibrium points. The conceived system possesses an interesting property not explored in the literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of chaotic attractors. Bifurcation diagrams, computation of the largest Lyapunov exponents, phase plots, and the 0-1 test method are reported, with the aim to analyze the dynamics of the system, as well as to highlight the coexistence of chaotic attractors. Finally, an entropy algorithm is used to measure the complexity of the proposed system.

Synchronisation of integer-order and fractional-order discrete-time chaotic systems

This paper studies the synchronisation of integer- and fractional-order discrete-time chaotic systems with different dimensions. Control laws are proposed for the full-state hybrid projective synchronisation (FSHPS) of a master–slave pair, where the difference equations of the master have an integer order while those of the slave have a fractional order. Moreover, inverse FSHPS laws are proposed for a fractional-order master and an integer-order slave. The Lyapunov stability theory of integer-order maps and the stability theory of linear fractional-order maps are utilised to establish the asymptotic stability of the zero equilibrium corresponding to the synchronisation error system. Numerical results are presented to verify the findings of the study.

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.

On the Q–S Chaos Synchronization of Fractional-Order Discrete-Time Systems: General Method and Examples

In this paper, we propose two control strategies for the Q–S synchronization of fractional-order discrete-time chaotic systems. Assuming that the dimension of the response system m is higher than that of the drive system n, the first control scheme achieves n-dimensional synchronization whereas the second deals with the m-dimensional case. The stability of the proposed schemes is established by means of the linearization method. Numerical results are presented to confirm the findings of the study.

Generalized and inverse generalized synchronization of fractional-order discrete-time chaotic systems with non-identical dimensions

In this paper, we introduce two approaches to the generalized synchronized synchronization and the inverse generalized synchronization of fractional discrete-time chaotic systems with non-identical dimensions. The convergence of the proposed approaches is established by means of recently developed stability theory. Numerical results are presented based on well-known maps in the literature. Two examples are considered: a 3D generalized synchronization and a 2D inverse generalized synchronization.

A new contribution for the impulsive synchronization of fractional-order discrete-time chaotic systems

In this paper, we discuss and investigate the impulsive synchronization of fractional-order discrete-time chaotic systems. The proposed method is based on the impulsive synchronization theory used in the integer-order case on the one hand and the mathematical analysis of the fractional-order discrete-time systems on the other hand. Sufficient conditions for the stability of synchronization error system are given, and application example with numerical simulations is illustrated in order to verify that the proposed method is applicable and effective. Furthermore, in order to validate the proposed synchronization approach, we have also provided the experimental implementation results using Arduino Mega boards.