Discrete Dynamics in Nature and Society

Volume 2019, Article ID 8341514, 15 pages

https://doi.org/10.1155/2019/8341514

## Bifurcations of a New Fractional-Order System with a One-Scroll Chaotic Attractor

^{1}School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, China^{2}School of Sciences, Xi’an University of Architecture and Technology, Xi’an 710055, China^{3}State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China^{4}School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an 710021, China

Correspondence should be addressed to Xiaojun Liu; moc.621@2593ttebylf

Received 29 August 2018; Revised 27 October 2018; Accepted 12 November 2018; Published 1 January 2019

Guest Editor: Tarek F. Ibrahim

Copyright © 2019 Xiaojun Liu et al. 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

In this paper, a new fractional-order system which has a chaotic attractor of the one-scroll structure is presented. Firstly, the stability of the equilibrium points of the system is investigated. And based on the stability analysis, the generation conditions of the one-scroll structure for the attractor are determined. In a commensurate-order case, bifurcations with the variation of a system parameter are investigated as derivative orders decrease from 0.99. In an incommensurate-order case, bifurcations with the variation of a derivative order are analyzed as other orders decrease from 1.

#### 1. Introduction

Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders. Although the seeds of fractional derivatives were planted over 300 years ago, the development of fractional calculus is very slow at an early stage for the absence of geometrical interpretation and applications. In the recent several decades, it has been applied to almost every field of science, engineering, economics, secures communication, and so on [1–5].

It is well known that fractional calculus is very suitable for the description of properties of various real materials. Meanwhile, fractional calculus provides an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected. The advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials [6].

Many new fractional-order systems were presented in recent years. Meanwhile, rich and complex dynamics, such as periodic solutions, chaos windows, all kinds of bifurcations, boundary, and interior crises were observed in these systems. For example, Chua system with a derivative order 2.7 makes chaos motion [7]. Chaotic dynamics of a damped van der Pol equation with fractional order is investigated in [8]. Chen studied the nonlinear dynamics and chaos in a fractional-order financial system [9]. A periodically forced complex Duffing’s oscillator was proposed, and chaos for the system was studied in detail [10]. Bifurcation, chaos control, and synchronization were investigated for a fractional-order Lorenz system with the complex variables [11]. In [12], boundary and interior crises were determined in a fractional-order Duffing system by a global numerical computation method. A proposed standard for the publication of new chaotic systems was studied in [13]. The authors investigated a simple chaotic flow with a plane of equilibria in [14].

It is well known that bifurcation theory concerns the changes in qualitative or topological structures of limiting motions such as equilibriums, periodic solutions, homoclinic orbits, heteroclinic orbits, and invariant tori for nonlinear evolution equations as some relevant parameters in the equations vary. Generally, the subject can be traced back to the very earlier work of Poincare around 1892 [15]. Nowadays, it is a fundamental tool to analyze nonlinear problems which enables us to understand how and when a system organizes new states and patterns near the original “trivial” one when a control parameter crosses a critical value. For fractional-order systems, a bifurcation implies a qualitative or topological change in dynamics with a variation of a system parameter or derivative order, and bifurcation analysis becomes harder due to the nonlocal property of the operator of fractional calculus. Many references have studied the bifurcations of fractional-order systems [16–20]. However, these investigations mainly focus on the bifurcation of a fractional-order system as a system parameter or a derivative order varies. To our knowledge, few works concerns bifurcations with the variation of both a system parameter and a derivative order or with the variation of both a derivative order and other orders.

Compared with integer-order chaotic systems, fractional-order chaotic systems with more complex dynamic characteristics and more system parameters can provide higher security for secure communication [21, 22]. In [23], the authors investigated the synchronization of a three-dimensional integer-order system. The differential equations of the system with simple structure were similar to those of the Lorenz system. It is well known that the Lorenz system has a chaotic attractor with double-scroll structure. The system in [23] has a chaotic attractor with only one-scroll and very abundant dynamic behaviors. Motivated by the above, in this paper, a corresponding fractional-order system is proposed and studied. Firstly, the stability of equilibrium points of the system is investigated. In a commensurate-order case, bifurcations with the variation of a system parameter are investigated as derivative orders decrease from 0.99. In an incommensurate-order case, bifurcations with the variation of a derivative order are analyzed as the other orders decrease from 1. Period-doubling and saddle-node bifurcations can be observed from the bifurcation diagrams by numerical simulations.

The remainder of the paper is organized as follows. In Section 2, the definitions of the fractional calculus and related preliminaries are given. A new fractional-order system with one-scroll attractor is presented in Section 3. In Section 4, bifurcations in the two cases of commensurate-order and incommensurate-order are analyzed, respectively. Conclusions of the paper are drawn in Section 5.

#### 2. Fractional Derivatives and Preliminaries

##### 2.1. Definitions

Fractional calculus can be considered as a generalization of integration and differentiation. The operator of fractional calculus can be defined bywhere denotes the derivative order and corresponds to the real part of . The numbers and represent the limits of the operator.

In general, three definitions of fractional derivative are used frequently, namely, the Grunwald-Letnikov definition, the Riemann-Liouville, and the Caputo definitions [6, 24].

The Grunwald-Letnikov definition (GL) derivative with fractional order can be described bywhere the symbol represents the integer part.

The Riemann-Liouville (RL) definition iswhere denotes the gamma function.

The Caputo (C) fractional derivative is defined as follows:

For a fractional differential equation which is defined by Caputo derivatives, the initial condition takes on the same form as those for the integer-order ones, which can be measured easily in applications. For this reason, the Caputo derivative will be adopted in the rest of the paper.

##### 2.2. Numerical Methods

Due to the nonlocal property of the operator of fractional calculus, it is not easy to obtain the numerical solutions for a fractional differential equation. Generally speaking, two approximation methods are frequently used, namely, an improved version of Adams-Bashforth-Moulton algorithm based on the predictor-correctors scheme and the frequency domain approximation [25–28]. For the accuracy [29], we will employ the improved predictor-corrector algorithm to solve a fractional differential equation in this paper.

In order to get the approximate solution of a fractional-order chaotic system by the improved predictor-corrector algorithm, the following equation is considered:which is equivalent to the Volterra integral equation Now, set , . The corrector formula for (6) can thus be discretized as follows:where predicted values are determined by the following formula:andThe error estimate of this approach is , where .

##### 2.3. The Stability of a Fractional-Order System

For fractional-order systems, the stability analysis of equilibrium points is complex and difficult due to the nonlocal property of fractional calculus. Here, the definitions of commensurate-order and incommensurate-order fractional-order systems will be given firstly.

*Definition 1. *For a fractional-order system, which can be described by , where is the state vector, is the fractional derivative orders vector, and . The fractional-order system is commensurate-order when all the derivative orders satisfy ; otherwise it is an incommensurate-order system [30].

In order to investigate the stability of equilibrium points for fractional-order systems, the following lemma is used frequently.

Lemma 2. *For a commensurate fractional-order system, the equilibrium points of the system are asymptotically stable if all the eigenvalues at the equilibrium satisfy the following condition:where is the Jacobian matrix of the system evaluated at the equilibria [31].*

#### 3. A New Fractional-Order System

In this section, a new fractional-order system which consist of three differential equations is proposed and can be denoted as follows:where , , and are state variables of the system, and the system parameters, and , , and derivative orders.

When the derivative orders are selected as , the system parameters and , and initial conditions , system (11) is chaotic. In Figures 1(a) and 1(b), a chaotic attractor with a one-scroll on three-dimensional space and projected onto plane are depicted. In this case, the corresponding Lyapunov exponents are , , and . When the derivative order and , the corresponding attractors are displayed in Figures 1(c)–1(f).