Discrete Dynamics in Nature and Society

Volume 2015, Article ID 740721, 11 pages

http://dx.doi.org/10.1155/2015/740721

## An Alternate Iterative Differential Evolution Algorithm for Parameter Identification of Chaotic Systems

^{1}Institute of Systems Engineering, Tianjin University, Tianjin 300072, China^{2}School of Traffic & Transportation, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China

Received 16 April 2015; Accepted 4 August 2015

Academic Editor: Daniele Fournier-Prunaret

Copyright © 2015 Wanli Xiang 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

Parameter estimation of chaotic systems plays a key role for control and synchronization of chaotic systems. At first, the parameter estimation of chaotic systems is mathematically formulated as a global continuous optimization problem. Then through integrating two differential mutation strategies, an improved greedy selection mechanism and a population diversity balance scheme, an alternate iterative differential algorithm, called AIDE, is presented to solve the problem in this paper. Subsequently, experiments are tested on a set of cases of parameter estimation of chaotic systems and the results show that AIDE is better than or at least equal to DE/rand/1/bin, DE/best/1/bin, and other four well-known algorithms in all cases.

#### 1. Introduction

In 1963, Lorenz first found the classical chaotic attractor during the process of simulating the change of atmosphere through a three-dimensional autonomous system [1]. After that, the chaos theory is studied in detail by many researchers. In particular, control and synchronization of chaotic systems have a promising prospect in various fields like information science, medicine, biology, engineering, and so on. However, the chaotic systems to be controlled usually have some unknown parameters. Thus, the parameter estimation problem becomes the first key issue for solving the control and synchronization of chaotic systems.

Recently, numerous researchers had given much attention to the parameter estimation of chaotic systems [2–18]. In particular, great achievements over the parameter estimation of chaotic systems have been obtained by intelligent algorithms recently. For example, Dai et al. [19] transformed the problem of parameter estimation of chaotic systems into a global optimization problem through designing a suitable objective function and solved the optimization problem using genetic algorithm. Likewise, Chang [3] employed the differential evolution (DE) algorithm to estimate the unknown parameters of Rossler’s chaotic system. Next, Chang [4] proposed an improved differential evolution algorithm to estimate the unknown parameters of Chen and Lü systems. He et al. [16] employed particle swarm optimization (PSO) algorithm for solving the problem of parameter estimation of Lorenz system and found that PSO is better than genetic algorithm (GA). Gao and Tong [20] also proposed an improved particle swarm optimization algorithm to effectively estimate the unknown parameters of Lorenz system and Lorenz system with noise. Meanwhile, a novel chaotic ant swarm (CAS for short) algorithm was developed to estimate the unknown parameters of Logistic and Lorenz systems by Li et al. [5]. Later, Chang et al. [6] introduced an evolutionary programming (EP) algorithm to solve the problem of parameter estimation of the unified chaotic systems including Lorenz, Lü, and Chen systems. Furthermore, some problems of parameter estimation of chaotic systems with time delay or other characteristics were also solved by some intelligent optimization algorithms [7–9]. In order to increasingly improve the accuracy of parameter estimation of chaotic systems, there are still some researchers who have proposed some improved PSO [10, 11], improved DE [2], and hybrid evolutionary algorithms [12–14, 21] to identify the unknown parameters of chaotic systems. In particular, Wang et al. [14] skillfully hybridize Nelder-Mead Simplex Search (NM for short) and differential evolution (DE) to propose a hybrid algorithm, called NMDE. NMDE was successfully used to identify the unknown parameters of chaotic systems and obtained a better convergence performance.

In order to further improve the accuracy of the parameter estimation of chaotic systems, inspired by the existence of a few big evolution eras and small evolution eras in nature, we propose an alternate iterative differential evolution algorithm, in which two mutation strategies with different search abilities (exploration and exploitation) are employed to imitate the evolutionary behaviour of big evolution era and small evolution era, respectively.

The rest of the paper is organized as follows. In Section 2, the problem description over parameter estimation is described. In Section 3, the traditional differential evolution algorithm is briefly described. Subsequently, an alternate iterative differential evolution, called AIDE, is presented in detail in Section 4. Nextly, comprehensive experiments are conducted in Section 5 to validate the performance of AIDE. Finally, a conclusion is drawn in Section 6.

#### 2. Problem Description

In this paper, the following -dimensional chaotic system is considered:where represents the state vector of chaotic systems, is an initial state, namely, the start point of evolution of chaotic systems, is the real parameters values of chaotic systems, and is the number of the parameters [14, 16, 21, 22].

When estimating the unknown parameters of chaotic systems, often suppose that the structure of the systems is known in advance [12] and further suppose that all states of the systems can be measured. Thus, the estimated system can be described as follows:where denotes a state vector of the estimated system and is a parameter vector of the estimated system.

In this way, the problem of parameter estimation of chaotic systems is to find a set of suitable parameters of and make them very close to real values of . The corresponding objective function is to minimize the error between the state vector of the estimated system and the state vector of the original system. Accordingly, the problem can be converted into an optimization problem. Generally, its objective can be described by (3). The specific principle of parameter estimation of chaotic systems is shown in Figure 1:where represents the length of the sampled data for parameters estimation, and denote the state vector of the original system and the estimated system at time , respectively, and represents the 2-norm or the Euclidean norm.