Complexity

Volume 2019, Article ID 1612752, 9 pages

https://doi.org/10.1155/2019/1612752

## Finite Time Synchronization for Fractional Order Sprott C Systems with Hidden Attractors

Shanghai University of Engineering Science, Shanghai, China

Correspondence should be addressed to Cui Yan; moc.621@2130nayiuc

Received 14 December 2018; Accepted 13 February 2019; Published 28 February 2019

Academic Editor: Xiaopeng Zhao

Copyright © 2019 Cui Yan 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

Fractional order systems have a wider range of applications. Hidden attractors are a peculiar phenomenon in nonlinear systems. In this paper, we construct a fractional-order chaotic system with hidden attractors based on the Sprott C system. According to the Adomain decomposition method, we numerically simulate from several algorithms and study the dynamic characteristics of the system through bifurcation diagram, phase diagram, spectral entropy, and C_{0} complexity. The results of spectral entropy and C_{0} complexity simulations show that the system is highly complex. In order to apply such research results to engineering practice, for such fractional-order chaotic systems with hidden attractors, we design a controller to synchronize according to the finite-time stability theory. The simulation results show that the synchronization time is short and the robustness is stable. This paper lays the foundation for the study of fractional order systems with hidden attractors.

#### 1. Introduction

Since Lorenz proposed the first chaotic system [1] in 1963, many chaotic systems [2–6] have been proposed successively. Nonlinear systems have also been used in many fields such as image encryption, secure communication, and UAV navigation.

Chaotic attractors include self-excited attractors and hidden attractors. The self-excited attractor is mainly caused by the unstable equilibrium point, while the hidden attractor is mainly caused by the existence of infinite equilibrium points and disjoint with the unstable equilibrium point. In the last half century people used Routh-Hurwitz criterion and Shilnikov theorem to determine the stability of equilibrium point and then determine whether the system has attractors or chaos. For the hidden attractor, the stable equilibrium point does not mean that the system is stable, which means that the previous judgment method cannot complete the judgment work. Since Leonov and kuznetsov [7] discovered the first Chua hidden attractor, many achievements [8–11] have been made. At present, the researches on hidden attractors are mostly of integer order and few of fractional order.

With the in-depth research, people found the applicable range of the fractional order system is bigger than integer order more [12], especially secure communication. Due to the fact that difficulty of the fractional order synchronization is higher than the integer order system, for the fractional order synchronous research started later than integer order. In 2003, Li Chunguang [13] realized the synchronization of fractional chaos system for the first time. After that, many synchronization methods [14, 15] of fractional-order chaotic systems have been proposed.

In this paper we combine the two hotspots of the fractional system and the hidden attractor. We construct a new chaotic system with hidden attractors based on Sprott C system by adding tiny perturbations. The spectral entropy and C_{0} complexity are the newest chaotic system characterization indicators, and we use these indicators to analyze the complex characteristics of new system. Then we realize chaotic synchronization on the basis of fractional finite time stability theory. These properties have significant application value to the area of secure communication and image encryption.

#### 2. Dynamic Analysis

##### 2.1. Fractional Differential Equation

Since fractional differential equations were proposed, many differential definitions have been proposed [16]. There are Grunwald-Letnikov (G-L) definition, Riemann-Liouville (R-L) definition, Caputo definition, etc., the most commonly used being G-L and R-L definition. Caputo definition is suitable for describing initial value problems of differential equations. In this paper, the definition of Caputo fractional-order differential equation is used to solve the chaos analysis of Sprott C system.

The expression of derivative of Caputo type is as follows:

C indicates that this is defined as the Caputo-type fractional order definition and ,* q* is the order of differential operator, and is Gamma function. Caputo differentiation involves the following properties.

Theorem 1 (see [16]). *Common differential equations can be described as follows:**The general solution of the above differential equation is**In the above formula, Mittag-Leffter function is*

##### 2.2. System Model

The Sprott C system was discovered by J. C. Sprott [17] through computer exhaustion. It consists of five elements, two of which are one of the simplest nonlinear systems and are easier to implement in an application. The Sprott C system has two equilibrium points and is symmetrical, with a pair of conjugate virtual roots at the equilibrium point. By adding a small perturbation term, the pure virtual root is transformed into a pair of conjugate eigenvalues with negative real parts and does not affect its chaotic characteristics. The Sprott C system appears chaotic and the corresponding feature data is stable after the above transformation, which is the hidden attractor. Liu [18] analyzed the stability and coupling synchronization problems of integer-order Sprott b and Sprott C systems and analyzed them by phase space trajectory and circuit simulation. Based on this, this paper extends it to fractional order, constructs hidden attractors, and studies its stability and synchronization problems. Sprott C original system is

The modified fractional system is

where is the order of nonlinear system and and is the disturbance parameter; let* a*=0.001 and the right side of (7) is equal to zero, so

By calculating system (8) we can found that system (7) only has two equilibria . The above two equilibrium points are symmetric, so we only discuss the properties of one of them.

##### 2.3. Hidden Attractor

The Jacobean matrix for the equilibrium point A is

As we all know that the characteristic polynomial , there are

The Jacobean for the equilibrium point B is

The eigenvalues are the same as the equilibrium point A. They are

It can be inferred from (10) and (12) that the real parts of all eigenvalues are negative. So the equilibrium points A and B are both stable when* a*= 0.001. However, the results of numerical simulation contradict the above theoretical analysis.

#### 3. Solution and Simulation

##### 3.1. Nonlinear Multiplier Sub Decomposition

Based on the improved Adomian algorithm [19], we decompose the nonlinear term of (7) into the following form:

Now, let

Then

Let all variables be equal to the corresponding coefficient values:

We can find out all the coefficients by using the above method. They are

Based on the above decomposed coefficients, the equations of the nonlinear system can be combined as follows:

##### 3.2. Numerical Simulation

Bifurcation diagrams are often used in dynamic analysis to observe system characteristics. However, the bifurcation diagram can only show the case where an independent variable changes with one parameter. At present, most studies only use the bifurcation diagram to judge the type of system classification. We use the order parameter* q* in system (7) as the variable of bifurcation graph and take the initial value as to get the numerical change of state variable y. As shown in Figure 1, the bifurcation diagram is drawn with the maximum value method.