Discrete Dynamics in Nature and Society

Volume 2018, Article ID 8284121, 10 pages

https://doi.org/10.1155/2018/8284121

## Dynamics Analysis and Control of a Five-Term Fractional-Order System

^{1}School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an 710021, China^{2}School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, China

Correspondence should be addressed to Li-xin Yang; moc.361@nixilgnayadoaij

Received 31 May 2018; Revised 22 July 2018; Accepted 10 September 2018; Published 9 October 2018

Guest Editor: Abdul Qadeer Khan

Copyright © 2018 Li-xin Yang and Xiao-jun Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes a new fractional-order chaotic system with five terms. Firstly, basic dynamical properties of the fractional-order system are investigated in terms of the stability of equilibrium points, Jacobian matrices theoretically. Furthermore, rich dynamics with interesting characteristics are demonstrated by phase portraits, bifurcation diagrams numerically. Besides, the control problem of the new fractional-order system is discussed via numerical simulations. Our results demonstrate that the new fractional-order system has compound structure.

#### 1. Introduction

Recently, the study of fractional calculus has attracted great attention due to its potential applications in various fields [1–3]. As a branch of mathematical analysis, fractional calculus can be considered as the generalization of the conventional calculus. Although the fractional-order derivative theory has a more than 300-year-old history, its application of the theory is just gaining attention [4–6]. In fact, most of the systems in interdisciplinary fields can be described via fractional calculus [7–9]. Moreover, fractional-order model can provide an explicit description and give a further insight into physical process. That is, fractional-order systems can serve as a valuable tool in the modeling of many phenomena. In view of the fact that fractional calculus provides another good way to describe, predict, and control physical systems accurately, it has been applied to control system, physics, and system modeling. Moreover, with the development of interdisciplinary applications, people found that various research fields can be elegantly described with the help of fractional derivatives, such as viscoelastic bodies, quantitative finance, dielectric polarization, electromagnetic waves, and polymer physics.

On the other hand, chaos and its applications have been intensively investigated and developed in many fields of science. In [9], the authors have presented analytical proofs of fold Hopf bifurcation in hyperchaotic Chen system and given sufficient conditions for stability and instability of the bifurcation periodic orbits. Researchers have investigated chaos synchronization of fractional-order systems via linear control [10–12]. Fractional-order systems possess long-range memory behavior and display complexity dynamics characteristic compared to its integral-order counterpart. On the other hand, there exist many significant differences between fractional-order system and the corresponding integer-order differential systems. Generally speaking, fractional-order nonlinear system can display more rich dynamical behaviors such as various bifurcations under certain conditions which are different from the corresponding integer-order system [13, 14]. Several studies have explored the complex dynamical properties in many fractional-order systems, such as fractional-order Chen system [15] and fractional-order Duffing system [16]. Until now, many researchers have investigated the dynamics of several fractional-order chaotic systems and obtained many excellent results [17–19]. In addition, searching for new chaotic systems with fewer terms in autonomous differential equations has been developed with much interest by scientists. In [20], the author has investigated the dynamics of a five-term chaotic attractor. However, this system with fractional order has not been actively and deeply explored, and it is very interesting in a number of different fields. To the best of our knowledge, chaotic attractors with fewer than five terms in three fractional-order differential equations have never been investigated.

Motivated by the above, within this body of work, we focus on the dynamical behaviors of this fractional-order simplified system. This would be of mathematical and practical interests. Rich dynamical behaviors are studied via bifurcation diagrams with varying the system parameters and the fractional derivative orders. Moreover, the control problem of the new fractional system is also investigated.

The remainder of this paper is organized as follows. In Section 2, the definition for the fractional calculus is given. Section 3 is devoted to the investigation of the fractional-order new system. In Section 4, the control of the new fractional-order system is investigated. Conclusions of this paper will be drawn in Section 5.

#### 2. Fractional Calculus

Fractional calculus can be considered as a generalization of integration and differentiation to a noninteger-order integrodifferential operator which is described aswhere denotes derivative order and corresponds to the real part of . The numbers and represent the limits of the operator. At present, there are several definitions of the fractional-order differential system. Riemann–Liouville and Caputo definitions are considered the most common and efficient fractional derivatives [21, 22].

Firstly, the Riemann–Liouville (RL) definition of fractional derivatives can be written as follows.The Caputo fractional derivative is given by the following. In the above formulations, represents the Gamma function. Note that the Caputo derivative guarantees a straightforward connection between the types of the initial condition and the fractional derivative. Hence, the Caputo derivative is adopted in this paper.

#### 3. An Unusual Five-Term Fractional-Order System

At first, a simple chaotic integer-order system with five terms is described as follows [20]:when the parameters are selected as , , system consists of two quadratic nonlinearities and displays abundantly complex behaviors of chaotic dynamics.

In what follows, we suppose that the derivative orders are fractional; the equations of the fractional-order system are readily derived from the above integer-order counterpart where are the fractional derivatives orders.

In the next step, the dynamical behaviors of this new fractional-order system are investigated.

##### 3.1. Some Properties of the New Fractional-Order System

It should be noted that most of the theory for the integer-order dynamic system cannot be simply extended to the fractional-order system. Therefore, the sufficient conditions of the stability of the fractional-order systems are given [23].

Lemma 1. *An autonomous fractional-order system is asymptotically steady at the equilibrium, if all the eigenvalues of the Jacobian matrix of some equilibrium satisfywhere denotes the eigenvalues of matrix .*

Based on the above theorem, the equilibria of system (6) can be calculated by solving the following equations The system contains two equilibrium points, i.e.,and, for the equilibrium , the Jacobian matrix of (6) at points is obtained asThen we can obtain the eigenvalues of the Jacobian matrix as follows.For the second equilibrium , the Jacobian matrix of the fractional-order new system (6) is defined asBy computing , it is found that the eigenvalues of the Jacobian matrix are the same as those of the Jacobian matrix . That is, is a negative real number and ; then according to Lemma 1, the eigenvalues cannot satisfy (7) as , and two equilibrium points of the nonlinear fractional-order system (6) are all unstable.

##### 3.2. Chaos and Bifurcations with Fractional-Order Parameters

To investigate the existence of new attractors in the fractional-order system, this part is devoted to dynamics of system (6) by considering several values of the fractional derivatives orders.

First, system parameters are selected as , , and the order parameter is varied. Figure 1 shows several typical attractors for ; in this case, the fractional-order system is a commensurate-order system. The initial states of the new fractional-order system are taken as , , . From these figures, one can observe that the new fractional-order system exhibits rich dynamical behaviors.