﻿ 一个新四维混沌系统的分析、控制与电路实现 Analysis, Control and Circuit Implementation of a Novel 4D Chaotic System

Dynamical Systems and Control
Vol. 08  No. 02 ( 2019 ), Article ID: 29574 , 11 pages
10.12677/DSC.2019.82015

Analysis, Control and Circuit Implementation of a Novel 4D Chaotic System

Zhili Wang1,2, Hong Niu1, Dechu Tan1,3

1College of Electronic Information and Automation, Tianjin University of Science & Technology (TUST), Tianjin

2School of Electrical and Information Engineering, Tianjin University (TJU), Tianjin

3School of Automation and Electrical Engineering, University of Science and Technology Beijing (USTB), Beijing

Received: Mar. 11th, 2019; accepted: Mar. 22nd, 2019; published: Apr. 3rd, 2019

ABSTRACT

The dynamics of the novel four-dimensional (4D) chaotic system are presented. The analog circuit is derived from the modified module-based approach to chaotic circuit design and implemented by means of the A + D Lab platform to illustrate that the novel 4D chaotic system could generate chaos on hardware. For chaos control of the novel 4D chaotic system, the mathematical model of the controlled system is formulated, and a linear feedback controller only with one variable is designed based on the Lyapunov stability theory and implemented by circuit. The output signals of the circuit implementation show that the state variables of the controlled novel 4D chaotic system are no longer chaotic or periodic but asymptotically converge to zero, which indicates that the controlled system is asymptotically stable at the origin and the linear feedback controller is feasible and effective.

Keywords:Chaos, Novel Four-Dimensional Chaotic System, Chaos Control, Circuit Implementation

1天津科技大学，电子信息与自动化学院，天津

2天津大学，电气自动化与信息工程学院，天津

3北京科技大学，自动化学院，北京

1. 引言

2. 新四维混沌系统的动力学特性分析

2.1. 数学模型

$\begin{array}{l}\stackrel{˙}{x}=a\left(y-x\right),\\ \stackrel{˙}{y}=c\left(x+y\right)+z-xw,\\ \stackrel{˙}{z}=mx-y-hz,\\ \stackrel{˙}{w}=xy-bw,\end{array}$ (1)

2.2. 对称性

2.3. 耗散性

$\nabla V=\frac{\partial \stackrel{˙}{x}}{\partial x}+\frac{\partial \stackrel{˙}{y}}{\partial y}+\frac{\partial \stackrel{˙}{z}}{\partial z}+\frac{\partial \stackrel{˙}{w}}{\partial w}=-a+c-h-b=-\left(a+b+h-c\right)=-24<0,$

$\frac{\text{d}V}{\text{d}t}={\text{e}}^{-\left(a+b+h-c\right)},$

2.4. 平衡点分析

$\begin{array}{l}a\left(y-x\right)=0,\\ c\left(x+y\right)+z-xw=0,\\ mx-y-hz=0,\\ xy-bw=0,\end{array}$ (2)

$\begin{array}{l}{x}_{\text{o}}={y}_{\text{o}}=\sqrt{\frac{b\left(2ch+m-1\right)}{h}},\\ {z}_{\text{o}}=\frac{\left(m-1\right)}{h}\sqrt{\frac{b\left(2ch+m-1\right)}{h}},\\ {w}_{\text{o}}=\frac{2ch+m-1}{h},\end{array}$

$\begin{array}{ccc}\begin{array}{l}{S}_{1}=\left(0,0,0,0\right),\\ {S}_{2}=\left({x}_{\text{o}},{y}_{\text{o}},{z}_{\text{o}},{w}_{\text{o}}\right),\\ {S}_{3}=\left(-{x}_{\text{o}},-{y}_{\text{o}},-{z}_{\text{o}},{w}_{\text{o}}\right),\end{array}& \begin{array}{l}\\ \to \\ \end{array}& \begin{array}{l}{S}_{1}=\left(0,0,0,0\right),\\ {S}_{2}=\left(10.5762,10.5762,13.5980,37.2857\right),\\ {S}_{3}=\left(-10.5762,-10.5762,-13.5980,37.2857\right),\end{array}\end{array}$

$J=\left[\begin{array}{cccc}-a& a& 0& 0\\ c& c& 1& 0\\ m& -1& -h& 0\\ 0& 0& 0& -b\end{array}\right],$

$f\left(s\right)=\left(s+b\right)\left[{s}^{3}+\left(h-c+a\right){s}^{2}+\left(1-hc+ah-2ac\right)s+\left(a-2ach-am\right)\right]=0,$

$a=25$$b=3$$c=18$$m=19$$h=14$ 代入上式，求得其特征根分别为

${s}_{1}=26.8744\text{,}{s}_{2}=-33.2857,{s}_{3}=-14.5886,{s}_{4}=-3.0000\text{,}$

Table 1. Properties of equilibrium points of the novel 4D chaotic system

2.5. 初值敏感性

Figure 1. Curves of state variables of the novel 4D chaotic system (1) under different initial values: (a) t-x; (b) t-y; (c) t-z; (d) t-w

2.6. 新四维混沌系统的混沌吸引子

${d}_{L}=j+\frac{\underset{i=1}{\overset{j}{\sum }}{\lambda }_{i}}{|{\lambda }_{j+1}|}=2+\frac{2.6630+0.0016}{11.7829}=2.2261,$ (3)

Figure 2. Phase portraits of the novel 4D chaotic system (1): (a) x-y; (b) x-z; (c) x-w; (d) y-z; (e) y-w; (f) z-w

3. 新四维混沌系统的电路设计与硬件实现

3.1. 电路设计

$\begin{array}{l}\frac{\text{d}x}{\text{d}t}=-a{\tau }_{0}x-a{\tau }_{0}\left(-y\right),\\ \frac{\text{d}y}{\text{d}t}=-c{\tau }_{0}\left(-x\right)-c{\tau }_{0}\left(-y\right)-{\tau }_{0}\left(-z\right)-10{\tau }_{0}xw,\\ \frac{\text{d}z}{\text{d}t}=-m{\tau }_{0}\left(-x\right)-{\tau }_{0}y-h{\tau }_{0}z,\\ \frac{\text{d}w}{\text{d}t}=-10{\tau }_{0}\left(-x\right)y-b{\tau }_{0}w,\end{array}$ (4)

Figure 3. Circuit model of the system (4)

$\begin{array}{l}\frac{\text{d}x}{\text{d}t}=-\frac{1}{{\text{R}}_{1}{\text{C}}_{1}}x-\frac{1}{{\text{R}}_{2}{\text{C}}_{1}}\left(-y\right),\\ \frac{\text{d}y}{\text{d}t}=-\frac{1}{{\text{R}}_{3}{\text{C}}_{2}}\left(-x\right)-\frac{1}{{\text{R}}_{4}{\text{C}}_{2}}\left(-y\right)-\frac{1}{{\text{R}}_{5}{\text{C}}_{2}}\left(-z\right)-\frac{1}{{\text{10R}}_{6}{\text{C}}_{2}}xw,\\ \frac{\text{d}z}{\text{d}t}=-\frac{1}{{\text{R}}_{7}{\text{C}}_{3}}\left(-x\right)-\frac{1}{{\text{R}}_{8}{\text{C}}_{3}}y-\frac{1}{{\text{R}}_{9}{\text{C}}_{3}}z,\\ \frac{\text{d}w}{\text{d}t}=-\frac{1}{{\text{10R}}_{10}{\text{C}}_{4}}\left(-x\right)y-\frac{1}{{\text{R}}_{11}{\text{C}}_{4}}w.\end{array}$ (5)

3.2. 硬件实现

Figure 4. Circuit implementation of the novel 4D chaotic system

Figure 5. Phase portraits of the circuit implementation: (a) x-y; (b) x-z; (c) x-w; (d) y-z; (e) y-w; (f) z-w

4. 新四维混沌系统的稳定性控制

4.1. 控制器设计

$\begin{array}{l}\stackrel{˙}{\stackrel{˜}{x}}=a\left(\stackrel{˜}{y}-\stackrel{˜}{x}\right)+{u}_{\text{c}1},\\ \stackrel{˙}{\stackrel{˜}{y}}=c\left(\stackrel{˜}{x}+\stackrel{˜}{y}\right)+\stackrel{˜}{z}-\stackrel{˜}{x}\stackrel{˜}{w}+{u}_{\text{c}2},\\ \stackrel{˙}{\stackrel{˜}{z}}=m\stackrel{˜}{x}-\stackrel{˜}{y}-h\stackrel{˜}{z}+{u}_{\text{c}3},\\ \stackrel{˙}{\stackrel{˜}{w}}=\stackrel{˜}{x}\stackrel{˜}{y}-b\stackrel{˜}{w}+{u}_{\text{c}4},\end{array}$ (6)

${u}_{\text{c}}={\left[\begin{array}{cccc}{u}_{\text{c}1}& {u}_{\text{c}2}& {u}_{\text{c}3}& {u}_{\text{c4}}\end{array}\right]}^{T}={\left[\begin{array}{cccc}-{k}_{1}\stackrel{˜}{x}& -{k}_{2}\stackrel{˜}{y}& -{k}_{3}\stackrel{˜}{z}& -{k}_{4}\stackrel{˜}{w}\end{array}\right]}^{\text{T}},{k}_{1},\cdots ,{k}_{4}\ge 0$ (7)

$V\left(\stackrel{˜}{X}\right)=\frac{1}{2}\left({\stackrel{˜}{x}}^{2}+{\stackrel{˜}{y}}^{2}+{\stackrel{˜}{z}}^{2}+{\stackrel{˜}{w}}^{2}\right),$

Figure 6. Curves of state variables of circuit implementation of the controlled novel 4D chaotic system: (a) t-x; (b) t-y; (c) t-z; (d) t-w

$\begin{array}{c}\stackrel{˙}{V}\left(\stackrel{˜}{X}\right)=\stackrel{˜}{x}\stackrel{˙}{\stackrel{˜}{x}}+\stackrel{˜}{y}\stackrel{˙}{\stackrel{˜}{y}}+\stackrel{˜}{z}\stackrel{˙}{\stackrel{˜}{z}}+\stackrel{˜}{w}\stackrel{˙}{\stackrel{˜}{w}}\\ =-\left({k}_{1}+a\right){\stackrel{˜}{x}}^{2}+\left(a+c\right)xy-\left({k}_{2}-c\right){\stackrel{˜}{y}}^{2}+mxz-\left({k}_{3}+h\right){\stackrel{˜}{z}}^{2}-\left({k}_{4}+b\right){\stackrel{˜}{w}}^{2}\\ =\frac{1}{2}{\stackrel{˜}{X}}^{T}\left[\begin{array}{cccc}-2\left({k}_{1}+a\right)& a+c& m& 0\\ a+c& -2\left({k}_{2}-c\right)& 0& 0\\ m& 0& -2\left({k}_{3}+h\right)& 0\\ 0& 0& 0& -2\left({k}_{4}+b\right)\end{array}\right]\stackrel{˜}{X},\end{array}$

${k}_{2}>\frac{h{\left(a+c\right)}^{2}}{4ah-{m}^{2}}+c>\frac{{\left(a+c\right)}^{2}}{4a}+c,$

$\stackrel{˙}{V}\left(\stackrel{˜}{X}\right)$ 是负定的。又由于 $V\left(\stackrel{˜}{X}\right)$ 是径向无界的，所以被控系统(6)在原点处是全局渐近稳定的。现取 ${k}_{2}=45$ ，则控制器 ${u}_{\text{c}}$ 可表示为

$\begin{array}{c}{u}_{\text{c}}={\left[\begin{array}{cccc}{u}_{\text{c}1}& {u}_{\text{c}2}& {u}_{\text{c}3}& {u}_{\mathrm{c}4}\end{array}\right]}^{\text{T}}={\left[\begin{array}{cccc}0& -45\stackrel{˜}{y}& 0& 0\end{array}\right]}^{\text{T}}.\end{array}$ (8)

4.2. 被控系统的电路实现

${u}_{\text{c}2}=-\frac{1}{{\text{R}}_{12}{\text{C}}_{2}}\stackrel{˜}{y}=-{k}_{2}{\tau }_{0}\stackrel{˜}{y}\to \frac{1}{{\text{R}}_{12}{\text{C}}_{2}}={k}_{2}{\tau }_{0}\to {\text{R}}_{12}=\frac{1}{{k}_{2}{\tau }_{0}{\text{C}}_{2}}=22.2\text{\hspace{0.17em}}\text{kΩ,}$ (9)

5. 结论

Analysis, Control and Circuit Implementation of a Novel 4D Chaotic System[J]. 动力系统与控制, 2019, 08(02): 129-139. https://doi.org/10.12677/DSC.2019.82015

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