Synchronization of a new fractional-order hyperchaotic system

doi:10.1016/j.physleta.2009.04.063

Physics Letters A

Volume 373, Issues 27–28, 22 June 2009, Pages 2329-2337

https://doi.org/10.1016/j.physleta.2009.04.063 Get rights and content

Abstract

In this letter, a new fractional-order hyperchaotic system is proposed. By utilizing the fractional calculus theory and computer simulations, it is found that hyperchaos exists in the new fractional-order four-dimensional system with order less than 4. The lowest order to have hyperchaos in this system is 2.88. The results are validated by the existence of two positive Lyapunov exponents. Using the pole placement technique, a nonlinear state observer is designed to synchronize a class of nonlinear fractional-order systems. The observer method is used to synchronize two identical fractional-order hyperchaotic systems. In addition, the active control technique is applied to synchronize the new fractional-order hyperchaotic system and the fractional-order Chen hyperchaotic system. The two schemes, based on the stability theory of the fractional-order system, are rather simple, theoretically rigorous and convenient to realize synchronization. They do not require the computation of the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the proposed synchronization schemes.

Introduction

It is well known that the fractional calculus is a classical mathematical notion, with a history as long as calculus itself. But its applications to physics and engineering are just a recent subject of interest [1], [2]. It was found that many systems in interdisciplinary fields can be elegantly described with the help of fractional derivatives. For instance: fractional derivatives have been widely applied in mathematical modeling of viscoelastic materials [3]. Some electromagnetic problems can be depicted using fractional differ-integration [4]. In the fractional capacitor theory, if one of the capacitor electrodes has a rough surface, the current passing through it is proportional to the non-integer derivative of its voltage [5]. Also, the existing memory in dielectrics used in capacitors is justified by fractional derivative based models [6]. The anomalous diffusion phenomena in inhomogeneous media can be explained by non-integer derivative based equations of diffusion [7]. The electrode–electrotype interface is a sample of fractional-order processes because at metal–electrolyte interfaces the impedance is proportional to the non-integer order of frequency for small angular frequencies [8]. The resistance–capacitance–inductance (RLC) interconnect model of a transmission line is a fractional-order model [9]. Heat conduction as a dynamical process can be more adequately modeled by fractional-order models than integer-order models [10]. In the field of mechanics, it has been found that the water flow on a dyke with porous internal structure is proportional to the fractional derivative of the dynamic pressure at the water/dyke interface [11]. In economy, it has been known that some finance systems can display fractional order dynamics [12]. More examples for fractional-order dynamics can be found in Ref. [2]. Furthermore, applications of fractional calculus have been reported in many areas such as image processing [13], signal processing [14] and automatic control [15]. These examples and many other similar samples perfectly clarify the importance of consideration and analysis of dynamical systems with fractional-order models.

By now, many fractional-order differential systems such as the fractional-order Chua's circuit [16], the fractional-order Chen system [17], [18], the fractional-order Rössler system [19], the fractional-order Lü system [20], the fractional-order van der Pol system [21] and so forth, behave chaotically. However, there are many material differences between the ordinary differential equation systems (integer-order) and the corresponding fractional-order differential equation systems. Most of the properties or conclusions of the integer-order system cannot be simply extended to the case of the fractional-order one. So we will study the system reported in Ref. [22], and demonstrate that its hyperchaotic nature is preserved under fractional order setting.

Recently, synchronization of chaotic fractional-order systems starts to attract increasing attention due to its potential applications in secure communication and control processing [23], [24], [25], [26], [27], [28], [29]. For example, Gao et al. studied the master–slave synchronization of fractional order chaotic systems [25]. In Ref. [26], the linear and nonlinear drive-response synchronization methods were applied for synchronizing the fractional-order chaotic Arneodo systems only using a scalar drive signal. Also, generalized projective synchronization of two chaotic or hyperchaotic fractional-order systems have been presented in Refs. [27], [28]. To our best knowledge, in the aforementioned literatures, the authors are all concerned with the identical synchronization of fractional-order chaotic systems. However, in the practical applications, complete synchronization only occurs at a certain point in the parameter space, and it is difficult to achieve complete synchronization except under ideal conditions.

A hyperchaotic attractor is characterized as a chaotic attractor with more than one positive Lyapunov exponents which can increase the randomness and higher unpredictability of the corresponding system. It is believed that chaotic systems with higher-dimensional attractors have much wider applications. In fact, adopting higher-dimensional chaotic systems has been proposed for secure communication, and the presence of more than one Lyapunov exponent clearly improves the security of communication schemes by generating more complex dynamics. Interest in synchronization of the fractional-order hyperchaotic systems is again motivated by secure communication applications.

In our work, we found that hyperchaotic behavior does exist in the new fractional-order four-dimensional system with order as low as 2.88. The hyperchaotic dynamic behaviors of the system were demonstrated by computer simulations. Numerical results revealed that the new hyperchaotic system possesses two positive exponents. On the other hand, our aim is to synchronize two hyperchaotic fractional-order systems, which can be identical or nonidentical. To achieve this goal, the nonlinear state observer method was constructed to synchronize the new fractional-order hyperchaotic systems and the active control technique was used to synchronize the new fractional-order system and the fractional-order Chen hyperchaotic system. The two approaches, based on stability theory of fractional-order systems, are simple, theoretically rigorous and convenient to realize synchronization. They do not require the computation of the conditional Lyapunov exponents. Numerical simulations have performed the effectiveness and feasibility of the presented synchronization techniques.

This letter is organized as follows. In Section 2, the fractional derivative and its approximation are introduced. In Section 3, a new four-dimensional system is described both in its integer and fractional forms. The new fractional-order hyperchaotic system is numerically studied. In Section 4, based on the stability theory of the fractional-order system and the pole placement technique, a nonlinear state observer is constructed for synchronization in a class of nonlinear fractional-order systems. The observer method is used to synchronize two identical fractional-order hyperchaotic systems. Numerical simulations are used to show this process. In Section 5, the active control technique is applied to synchronize the new fractional-order hyperchaotic system and the fractional-order Chen hyperchaotic system. Numerical simulations are used to show this process. Finally, conclusions are drawn in Section 6.

Section snippets

Fractional derivative and its approximation

Fractional calculus is a generalization of integration and differentiation to a non-integer-order integro-differential operator ${}_{a}{D_{t}^{α}}$ is defined by

${}_{a}{D_{t}^{α}} = {\begin{matrix} \frac{d^{α}}{d t^{α}}, & R (α) > 0, \\ 1, & R (α) = 0, \\ \int_{a}^{t} {(d τ)}^{- α}, & R (α) < 0 . \end{matrix}$

There are some definitions for fractional derivatives [1]. The commonly used definition is Riemann–Liouville definition, defined by $D^{α} x (t) = \frac{d^{n}}{d t^{n}} J^{n - α} x (t), α > 0,$ where $n = ⌈ α ⌉$ , i.e. n is the first integer which is not less than α, $J^{β}$ is the β-order Riemann–Liouville integral operator which is described as follows $J^{β}$

A new fractional-order hyperchaotic system

Recently, Gao et al. proposed a new hyperchaotic system [22] from Chen system [40], which is described by ${\begin{matrix} \dot{x} = a (y - x), \\ \dot{y} = d x - x z + c y - w, \\ \dot{z} = x y - b z, \\ \dot{w} = x + k \end{matrix}$ where a, b, c, d and k are the real constants. When $a = 36$ , $b = 3$ , $c = 28$ , $d = - 16$ and $k = 0.5$ , the new system (8) is hyperchaotic. Fig. 1 displays the hyperchaotic attractors of the new hyperchaotic system.

Based on the above descriptions, we consider the fractional version of the new hyperchaotic system which is given by ${\begin{matrix} \frac{d^{α} x}{d t^{α}} = a (y - x), \\ \frac{d^{α} y}{d t^{α}} = d x - x z + c y - w, \\ \frac{d^{α} z}{d t^{α}} = x \end{matrix}$

Nonlinear state observer design

Suppose that the fractional-order chaotic drive system can be written as $\frac{d^{α} x}{d t^{α}} = A x + B f (x) + C,$ where $x \in R^{n}$ is an n-dimensional state vector of the system, $A \in R^{n \times n}$ , $B \in R^{n \times n}$ and $C \in R^{n}$ are constant matrices, $f (x) : R^{n} \to R^{n}$ is a nonlinear vector function in terms of x, and $0 < α ⩽ 1$ is the fractional order. Ax is the linear part of the system (10) and $B f (x)$ is the nonlinear part.

It is worth noting that the drive signal is an artificial output of system (10) which can be properly designed to drive the response

Synchronization between the different fractional-order hyperchaotic systems using the active control method

In this section, we focus on investigating synchronization between two different fractional-order hyperchaotic systems using the active control technique [44]. Let the new fractional-order hyperchaotic system be the drive system: ${\begin{matrix} \frac{d^{α} x_{1}}{d t^{α}} = a (y_{1} - x_{1}), \\ \frac{d^{α} y_{1}}{d t^{α}} = d x_{1} - x_{1} z_{1} + c y_{1} - w_{1}, \\ \frac{d^{α} z_{1}}{d t^{α}} = x_{1} y_{1} - b z_{1}, \\ \frac{d^{α} w_{1}}{d t^{α}} = x_{1} + k \end{matrix}$ which drives the fractional-order Chen hyperchaotic system [18] given as ${\begin{matrix} \frac{d^{α} x_{2}}{d t^{α}} = θ (y_{2} - x_{2}) + w_{2} + u_{1}, \\ \frac{d^{α} y_{2}}{d t^{α}} = β x_{2} - x_{2} z_{2} + γ y_{2} + u_{2}, \\ \frac{d^{α} z_{2}}{d t^{α}} = x_{2} y_{2} - δ z_{2} + u_{3}, \\ \frac{d^{α} w_{2}}{d t^{α}} = y_{2} z_{2} + λ w_{2} + u_{4} \end{matrix}$ where $U = {(u_{1}, u_{2}, u_{3}, u_{4})}^{T}$ is the active

Conclusion

In this letter, we present a new fractional-order hyperchaotic system with its order as low as 2.88 applying the fractional calculus techniques. Its dynamical behaviors are studied. Our results have been validated by the existence of two positive Lyapunov exponents and some phase diagrams. Moreover, the nonlinear state observer method and the active control technique are used to synchronize two identical or different fractional-order hyperchaotic systems based on the stability theory of the

Acknowledgements

The research is supported by the National Natural Science Foundation of China (Grant No. 60873133), National High Technology Research and Development Program of China (Grant No. 2007AA01Z478), and Natural Science Foundation of Educational Committee of He'nan Province of China (Grant No. 2008B520003).

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Physics Letters A

Synchronization of a new fractional-order hyperchaotic system

Abstract

Introduction

Section snippets

Fractional derivative and its approximation

A new fractional-order hyperchaotic system

Nonlinear state observer design

Synchronization between the different fractional-order hyperchaotic systems using the active control method

Conclusion

Acknowledgements

J. Electroanal. Chem.

Physica A

Chaos Solitons Fractals

Phys. A

Physica A

Chaos Solitons Fractals

Chaos Solitons Fractals

Phys. D

Chaos Solitons Fractals

Chaos Solitons Fractals

Phys. A

Phys. Lett. A

Chaos Solitons Fractals

Chaos Solitons Fractals

J. Math. Anal. Appl.

Phys. D

Chaos Solitons Fractals

Fractional Differential Equations

Applications of Fractional Calculus in Physics

J. Appl. Mech.

Electromagnetic Theory

J. Electroanal. Chem.