Lag projective synchronization of fractional-order delayed chaotic systems

doi:10.1016/j.jfranklin.2018.10.024

Journal of the Franklin Institute

Volume 356, Issue 3, February 2019, Pages 1522-1534

https://doi.org/10.1016/j.jfranklin.2018.10.024 Get rights and content

Abstract

This paper considers the lag projective synchronization of fractional-order delayed chaotic systems. The lag projective synchronization is achieved through the use of comparison principle of linear fractional equation at the presence of time delay. Some sufficient conditions are obtained via a suitable controller. The results show that the slave system can synchronize the past state of the driver up to a scaling factor. Finally, two different structural fractional order delayed chaotic systems are considered in order to examine the effectiveness of the lag projective synchronization. Feasibility of the proposed method is validated through numerical simulations.

Introduction

Chaos synchronization is one of the hot topics in nonlinear science and it has been widely studied in various fields, including physics, secure communication networks, control processing and information science [1], [2], [3], [4], [5], [6], [7], [8], [9]. In 1990, Pecora and Carroll attempted to synchronize two identical systems with different initial conditions [10]. Since then many kinds of synchronization have been considered in chaotic systems, including complete synchronization [11], impulsive synchronization [12], [13], lag synchronization [14], [15], projective synchronization [16], [17], exponential synchronization [18], H_∞ synchronization and filter [19,20]. Among all kinds of synchronization, projective synchronization, in which both master and slave systems can be synchronized up to a scaling factor, is an effective scheme in both theoretical and applicable manners. In recent years, projective synchronization of chaotic systems has attracted increasing attention.

Fractional calculus is the generalization of integer-order calculus to arbitrary order, and it owns the description of memory and hereditary properties of several processes. Compared to classical integer-order models, fractional-order models could exhibit more complex dynamical behaviors of systems. In fact, many fractional order systems act as chaotic systems, such as fractional order Liu system [21], fractional order L $\ddot{u}$ system [22], fractional order Chen system [23], fractional order R $\ddot{o}$ ssler system [24], fractional order Chua’s circuit [25]. Nonetheless, methods and results of synchronization of integer-order chaotic systems cannot be simply extended to the case of the fractional order systems. Recently, some interesting results of the synchronization control of fractional order chaotic systems have been reported in [26], [27].

It has to be noted that complete synchronization of chaotic systems is not practically possible due to the finite speed of signals. Therefore, it is potentially required for the system’s response to synchronize the drive system with a propagation delay σ. For example, in a telephony system a propagation delay is always induced when sending voice signals from the sender to the receiver. Therefore, it is important and necessary to study the lag synchronization. In [14], authors discussed the lag projective synchronization of fractional chaotic (hyperchaotic) systems. The lag projective synchronization of two chaotic systems, with different fractional orders, was studied in [15]. However, the effect of time-delay was not considered in most of the previous studies. In fact, time-delayed differential models unavoidable exist in real world, and time delay may cause oscillation and instability behaviors. Time delayed fractional order chaotic systems and their synchronization have been given great attention [28], as challenging and interesting topics. To the best of our knowledge, some few research works considered the lag projective synchronization of fractional order chaotic systems with time delay.

Motivated by the above discussion, in this paper, the lag projective synchronization of fractional order chaotic systems with time delay is considered. The main contributions of this paper lies in the following aspects. First, a suitable controller is designed to achieve the lag projective synchronization of fractional order chaotic systems with time delay, which is a key contribution since time delay was not taken into account in all previous studies. Second, based on the comparison principle of linear fractional order systems with time delay, synchronization conditions of fractional order delayed chaotic systems are proposed here. Moreover, the finding results can be used to realize the complete synchronization, anti-synchronization, generalized synchronization and lag synchronization. Finally, the results obtained in this paper are less conservative and more general.

Section snippets

Preliminaries and model description

Riemann Liouville and Caputo fractional derivatives are the mostly used definitions of fractional order derivatives. From the view of Laplace transform of fractional derivative, the Caputo fractional derivative only requires initial conditions in terms of integer-order derivatives, thus it is more convenient in the engineering applications. In this paper, we only adopt Caputo derivative.

Definition 1

[29] The Caputo fractional derivative of order α for a function x(t) is defined by:

$D^{α} x (t) = \frac{1}{Γ (n - α)} \int_{0}^{t} {(t - τ)}^{n - α - 1}$

Lag projective synchronization

This section discusses the design of a suitable controller that can achieve the lag projective synchronization between the slave system and the master system.

The error vector is defined as: $e (t) = y (t) - β x (t - σ),$ where $e (t) = {(e_{1} (t), \dots, e_{n} (t))}^{T} \in R^{n},$ β is the projective coefficient, σ is the propagation delay.

When combining (1), (2) and (7), the system’s error can be expressed as: $\begin{matrix} D^{α} e (t) & = & \bar{A} e (t) + \bar{B} e (t - τ) + β (\bar{A} - A) x (t - σ) \\ + β (\bar{B} - B) x (t - τ - σ) - β F (x (t - σ), x (t - τ - σ)) \\ + G (y (t), y (t - τ)) + U (t) . \end{matrix}$ When selecting the control input

Numerical simulations

The following fractional-order delayed chaotic system is considered as the master system: $\begin{matrix} {\begin{matrix} D^{α} x_{1} (t) = x_{3} (t) - a x_{1} (t) + x_{1} (t) x_{2} (t - τ), \\ D^{α} x_{2} (t) = 1 - b x_{2} (t) - x_{1}^{2} (t - τ), \\ D^{α} x_{3} (t) = - x_{1} (t - τ) - c x_{3} (t), \end{matrix} \end{matrix}$ where $x (t) = {(x_{1} (t), x_{2} (t), x_{3} (t))}^{T}$ is the state vector, $α = 0.92$ . $a = 3, b = 0.1, c = 1,$ $τ = 0.01$ . The initial values are selected as $x_{1} (0) = 0.1, x_{2} (0) = 4, x_{3} (0) = 0.5$ . Using these parameters can help system (23) to have a chaotic attractor, which is shown in Fig. 1.

The slave system is given as: $\begin{matrix} {\begin{matrix} D^{α} y_{1} (t) = l (y_{2} (t) - y_{1} (t)) + u_{1} (t), \\ D^{α} y_{2} (t) = m y_{1} (t - τ) - y_{1} (t \end{matrix} \end{matrix}$

Conclusions

Time delay can badly affect oscillation and instability behavior of dynamical systems, and therefore it is necessary to be taken into account. In this paper, the lag projective synchronization of delayed fractional order chaotic systems is investigated. For generality, an effective controller is designed, with the implementation of a fractional inequality and comparison principle of linear fractional equation with time delay. Sufficient conditions are achieved, ensuring the master–slave systems

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^☆: Project supported by the National Natural Science Foundation of China (Grant Nos. 11571016 and 61573096), the Natural Science Foundation of Anhui Province (Grant No. 1608085MA14) and the Natural Science Foundation of Anhui Education Department (Grant No. KJ2018A0365).

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Lag projective synchronization of fractional-order delayed chaotic systems☆

Abstract

Introduction

Section snippets

Preliminaries and model description

Lag projective synchronization

Numerical simulations

Conclusions

Chaos Solitons Fractals

Phys. Lett. A

Appl. Math. Comput.

Phys. Lett. A

Neural Netw.

Physica A

Chaos Solitons Fractals

Physica A

Physica A

Chaos Solitons Fractals

Nonlinear Anal. Hybrid Syst.

Commun. Nonlinear Sci. Numer. Simul.

Physica A

Discret. Dyn. Nat. Soc.

Neural Netw.

J. Frankl. Inst.

Automatica

Math. Comput. Simulat.

IET Control Theory & Appl.

Applications of Fractional Calculus in Physics