Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 4694305, 10 pages

https://doi.org/10.1155/2017/4694305

## Bifurcations and Synchronization of the Fractional-Order Bloch System

^{1}School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, China^{2}State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Shaanxi, Xi’an 710049, China

Correspondence should be addressed to Xiaojun Liu

Received 13 February 2017; Accepted 6 March 2017; Published 22 March 2017

Academic Editor: Mohamed A. Hajji

Copyright © 2017 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, bifurcations and synchronization of a fractional-order Bloch system are studied. Firstly, the bifurcations with the variation of every order and the system parameter for the system are discussed. The rich dynamics in the fractional-order Bloch system including chaos, period, limit cycles, period-doubling, and tangent bifurcations are found. Furthermore, based on the stability theory of fractional-order systems, the adaptive synchronization for the system with unknown parameters is realized by designing appropriate controllers. Numerical simulations are carried out to demonstrate the effectiveness and flexibility of the controllers.

#### 1. Introduction

Nowadays, fractional calculus is a hot topic in the research field. It is well known that fractional calculus has an equally long history with classical calculus. It did not attract enough attention for the absence of geometrical interpretation and applications at the initial stage of development. As the development of technology and science continues, fractional calculus has been applied in many fields, such as control theory, dynamics, mathematics, mechanics, and physics [1–5].

As the research of fractional calculus moves along, many nonlinear systems with fractional orders are proposed and investigated. The chaos and bifurcations which are observed in integer-order systems are also found in fractional-order ones, such as fractional versions of Duffing system, Lorenz system, and Chen system [6–11]. It is well known that the Bloch system is very important for interpretation of the underlying physical process of nuclear magnetic resonance. Recently, the fractional-order Bloch equations with and without delay were studied [12–14]. Meanwhile, physical interest in the fractional-order Bloch equation has been growing [15, 16] with the goal of improving the modeling of relaxation, diffusion, and perfusion in biological tissues. In [17], for the fractional-order Bloch system, the chaotic dynamics including the chaotic attractors in different system parameters sets, bifurcations with the derivative order in commensurate-order case, were analyzed. Rich dynamics such as period-doubling and subharmonic cascade routes to chaos were found for the system in the commensurate-order case. Based on these results, we want to know the bifurcations of the fractional-order Bloch system with the variation of every order in incommensurate-order case as well as every system parameter.

Motivated by the above discussed, in this paper, the bifurcations with the derivative order in incommensurate-order case and system parameters are studied firstly. A series of period-doubling bifurcations and tangent bifurcations are obtained by numerical simulations. Meanwhile, different chaotic and periodic attractors are also observed. Furthermore, based on the stability theory of fractional-order systems, the adaptive synchronization of the fraction-order systems with uncertain parameters is realized by designing appropriate controllers. Numerical simulations are carried out to demonstrate the effectiveness and flexibility of the controllers.

The paper is organized as follows. In Section 2, the definitions for the fractional calculus and numerical algorithms for fractional differential equations are given. The bifurcations of the fractional-order Bloch system are investigated in detail in Section 3. In Section 4, the adaptive synchronization of the system is investigated. Numerical simulations are used to demonstrate the effectiveness of the controllers. Finally, we summarize the results in Section 5.

#### 2. Fractional Derivatives

There are many definitions for the general fractional derivative. The three most frequently used ones are the Grunwald-Letnikov definition and the Riemann-Liouville and the Caputo definitions. It is well known that the initial conditions for the fractional differential equations with Caputo derivatives take on the same form as those for the integer-order ones, which is very suitable for practical problems. Therefore, we will use the Caputo definition for the fractional derivatives in this paper.

The Caputo fractional derivative is defined as follows:

As the initial conditions for the fractional differential equations with Caputo derivatives take on the same form as those for the integer-order ones, we will use the Caputo definition for the fractional derivatives in this paper.

In the following, we will give the definitions of commensurate-order and incommensurate-order fractional-order systems [18].

*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.

Compared with the numerical algorithm for solving an ordinary differential equation, the numerical solution of a fractional differential equation is not easy to obtain. There are two approximation methods which can frequently be used for numerical computation on chaos and bifurcations with fractional differential equations. One is an improved version of Adams-Bashforth-Moulton algorithm based on the predictor-correctors scheme [19–21], which is a time domain approach. The other is a method, known as frequency domain approximation [22], based on numerical analysis of fractional-order systems in the frequency domain.

Simulation of fractional-order systems using the time domain methods is complicated and, due to long memory characteristics of these systems, requires a very long simulation time but on the other hand, it is more accurate [23]. Therefore, we employ the improved predictor-corrector algorithm for fractional-order differential equations in this paper.

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

The Bloch system is usually used to describe an ensemble of spins. The integer-order and fractional-order Bloch systems were studied in [17]. In this section, the bifurcations of the fractional-order Bloch system with the variation of different system parameters and derivative orders will be investigated.

The fractional-order Bloch system can be described as follows:where are the state variables, the derivative orders, and the system parameters. When the orders , system (2) has a chaotic attractor with system parameters , , , , , which is plotted in Figure 1. The initial conditions for the numerical simulation are , , and also used in the rest of the paper.