Chaos Control of a New Chaotic Dynamical System and Its Application
This paper was devoted to study the problem of controlling chaos in a new chaotic dynamical system. The conclusions were applied to detecting the frequency of a weak signal. Two different methods of control, proportional differential control (PD control)
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Chaos Control of a New Chaotic Dynamical System and Its Application Qiuhui Zhong and Chunrui Zhang
Abstract This paper was devoted to study the problem of controlling chaos in a new chaotic dynamical system. The conclusions were applied to detecting the frequency of a weak signal. Two different methods of control, proportional differential control (PD control) and time-delayed feedback, were used to suppress chaos. When k2 > 1, the system at the equilibrium was stable with PD control method. The conditions to control system from chaos to stable were given with time-delayed feedback method. Numerical simulations were presented to show these results. Keywords PD control • Time-delayed • Stability • Periodical status • Frequency
187.1
Introduction
In recent years, controlling chaos has become more interesting in academic research and practical applications. There are many practical reasons for controlling chaos. Firstly, chaotic system response with little meaningful information content is unlikely to be useful. Secondly, chaos causes irregular behavior in nonlinear dynamical systems, therefore, chaos should be eliminated as much as possible or totally sup-pressed. This paper gives two ways to control chaos. One is PD control [1, 2]. Character of system is not affected by this method. Another one is timedelayed feedback control [3–5]. After that, the results are applied to detecting the frequency of a weak signal.
Q. Zhong (*) • C. Zhang Department of Mathematics, Northeast Forestry University, Harbin, China e-mail: [email protected]; [email protected] S. Zhong (ed.), Proceedings of the 2012 International Conference on Cybernetics 1465 and Informatics, Lecture Notes in Electrical Engineering 163, DOI 10.1007/978-1-4614-3872-4_187, # Springer Science+Business Media New York 2013
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187.2
Q. Zhong and C. Zhang
A New Chaotic Attractor
A new chaotic system [6] is described by the following system: 8 x_ ¼ y > < y_ ¼ z > : z_ ¼ x z þ y2 0:2y3 :
(187.1)
where x, y and z are the state variables. The equilibrium point of system (187.1) is X0 ¼ ð0; 0; 0Þ. The eigenvalues of system (187.1) at X0 are λ1 ¼ 1:4656 and λ2;3 ¼ 0:2328 0:7926i. This implies that X0 is unstable.
187.3
Controlling Chaos via PD Control Method
In this paper, two methods are introduced. First method is PD control. The following n-dimensional dynamical system is considered: x_ ¼ Fðx; μ; tÞ
(187.2)
where x 2 Rn ; F ¼ ½f1 ; f2 ; ; fn is a n -dimensional smooth vector, and μ is a parameter of system (187.2). When μ is regulated, system (187.2) has a chaotic attractor. Let x0 i ¼ k1 xi þ k2 x_ i . System (187.2) becomes 8 x_ ¼ fj ðx1 ; x2 ; ; x0 i ; ; xn ; μÞ > < j (187.3) x_ i ¼ fi ðx1 ; x2 ; ; xi ; ; xn ; μÞ > : 0 x i ¼ k1 xi þ k2 x_i :; where j 6¼ i; j ¼ 1; 2; ; n. Without loss of generality, we assume that k1 ¼ 1. Let x0 ¼ x þ k2 x_ ¼ x þ k2 y. System (187.1) becomes 8 x_ ¼ y > < (187.4) y_ ¼ z > : z_ ¼ ðx þ k2 yÞ z þ y2 0:2y3 : The linearization of system (187.4) at X0 ð0; 0; 0Þ is 8 x_
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