A Four-Dimensional Chaotic System with Hidden Attractor and Its New Proposed Electronic Circuit

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1. Introduction

Electronic systems rule our modern world, bridging the gap between software and physical reality. From communication to industrial process control to transportation to entertainment, the field of electronics continues to grow in reach and complexity. A working knowledge of basic electrical engineering concepts is now a powerful tool in many fields, and this value is likely to grow in the coming decades [1].

In recent years, the chaotic circuit has received a lot of research interest due to the fact that it has been applied in abundant areas such as economic model simulation, secure communication, electronic circuits design, robotics, image processing, and neural networks [2]. [8].

In Section 2, we describe a 4-dimensional chaotic system with a hidden attractor; it consists of eight simple terms involving two quadratic nonlinearities. In Section 3, we proposed an electronic circuit to implement system (1.1). In Section 4, simulate the designed circuit by MultiSIM12. Finally Section 5 presents the main conclusions of this paper.

2. Description of Chaotic System with Hidden Attractor

First of all, let us review the autonomous four dimensional dynamical system [9].hich consists of eight simple terms involving two nonlinear terms.

The equations are shown as follow:

$\begin{array}{l}\stackrel{\dot{}}{x}=\rho \left(y-x\right)\\ \stackrel{\dot{}}{y}=ax-\delta xz+w\\ \stackrel{\dot{}}{z}=\phi xy-z\\ \stackrel{\dot{}}{w}=-kx\end{array}$ (1.1)

Given initial values $\left[{x}_{0},{y}_{0},{z}_{0},{w}_{0}\right]=\left[4,1,4,2\right]$ and parameters with the following values $\rho =10,\delta =40,a=296.5,\phi =10,k=8$ .

After solving the equations of the system:

$\begin{array}{l}10\left(y-x\right)=0\\ 396.5x-40xz+w=0\\ 10xy-z=0\\ -8x=0\end{array}$ (1.2)

We note that the chaotic system (1.1) only has a zero equilibrium point. Therefore, system (1.1) has a hidden attractor.

Now, By applying Wolf’s algorithm [10]. The Lypanuov exponents are determined as: ( ${L}_{1}=1.660748$ , ${L}_{2}=0.149599$ , ${L}_{3}=-0.068474$ and ${L}_{4}=-12.144118$ ). Also, the Lypanuov dimension Kaplan-Yorke dimension of this system is calculated as

${D}_{L}=3+\frac{{L}_{1}+{L}_{2}+{L}_{3}}{\left|{L}_{4}\right|}=3.143433471$ (1.3)

So the chaotic behavior of system (1.1) in ${R}^{3}$ is shown in Figures 1(a)-(c), Figure 1(a): $\left(y,x,w\right)$ space, Figure 1(b): $\left(z,w,x\right)$ space, Figure 1(c): $\left(y,z,w\right)$ space. Figures 2(a)-(c) shows the Phase portraits of system (1.1), Figure 2(a): $\left(x,w\right)$ plane, Figure 2(b): $\left(y,w\right)$ plane, Figure 2(c): $\left(w,z\right)$ plane.

In additional, one of the basic behaviors of chaotic dynamical systems is shown in Figure 3 which is the wave-form $x\left(t\right),y\left(t\right),z\left(t\right),w\left(t\right)$ for system (1.1).

3. Proposed Electronic Circuit

In this section we design an electronic circuit to implement the 4-D chaotic

(a)(b)(c)

Figure 1. (a)-(c) The chaotic behavior in ${R}^{3}$ .

(a)(b)(c)

Figure 2. (a)-(c) The phase portraits of system (1.1).

Figure 3. The wave form $x\left(t\right),y\left(t\right),z\left(t\right),w\left(t\right)$ for system (1.1).

system (1.1). It consists of the following electronic elements: resistors, capacitors, multipliers and operational amplifiers TL034CN.

Applying Kirchhoff’s laws [7]. we can obtain the equations of the corresponding circuit as follows:

$\begin{array}{l}\frac{\text{d}{V}_{x}}{\text{d}t}=\frac{1}{{R}_{1}{C}_{1}}\left({V}_{y}-{V}_{x}\right)\\ \frac{\text{d}{V}_{y}}{\text{d}t}=\frac{1}{{R}_{2}{C}_{2}}{V}_{x}-\frac{1}{{R}_{3}{C}_{2}}{V}_{x}{V}_{z}+\frac{1}{{R}_{4}{C}_{2}}{V}_{w}\\ \frac{\text{d}{V}_{z}}{\text{d}t}=\frac{1}{{R}_{5}{C}_{3}}{V}_{x}{V}_{y}-\frac{1}{{R}_{6}{C}_{3}}{V}_{z}\\ \frac{\text{d}{V}_{w}}{\text{d}t}=-\frac{1}{{R}_{7}{C}_{4}}{V}_{x}\end{array}$ (1.4)

where ${V}_{x},{V}_{y},{V}_{z},{V}_{w}$ are the output voltages, the fixed multipliers constant is ${k}_{m}=10\text{\hspace{0.17em}}\text{V}$ , consequently the outputs are ${V}_{xz}={V}_{x}{V}_{z}/{k}_{m}$ and ${V}_{xy}={V}_{x}{V}_{y}/{k}_{m}$ .

${V}_{x}=1\text{\hspace{0.17em}}\text{V}\cdot x,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{y}=1\text{\hspace{0.17em}}\text{V}\cdot y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{z}=1\text{\hspace{0.17em}}\text{V}\cdot z,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{t}^{\prime}=\tau \cdot t=100\text{\hspace{0.17em}}\mu \text{s}\cdot t$ (1.5)

Substitute (1.5) in equations of system (1.4) we get:

$\begin{array}{l}\frac{\text{d}x}{\text{d}{t}^{\prime}}=\frac{\tau}{{R}_{1}{C}_{1}}\left(y-x\right)\\ \frac{\text{d}y}{\text{d}{t}^{\prime}}=\frac{\tau}{{R}_{2}{C}_{2}}x-\frac{\tau}{{R}_{3}{C}_{2}}xz+\frac{\tau}{{R}_{4}{C}_{2}}w\\ \frac{\text{d}z}{\text{d}{t}^{\prime}}=\frac{\tau}{{R}_{5}{C}_{3}}xy-\frac{\tau}{{R}_{6}{C}_{3}}z\\ \frac{\text{d}w}{\text{d}t}=-\frac{\tau}{{R}_{7}{C}_{4}}x\end{array}$ (1.6)

Set side by side system (1.1) with system (1.6) gives posterior conditions:

$\begin{array}{l}\frac{\tau}{{R}_{1}{C}_{1}}=\rho ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\tau}{{R}_{2}{C}_{2}}=a,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\tau}{{R}_{3}{C}_{2}}=\delta ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\tau}{{R}_{4}{C}_{2}}=1,\\ \frac{\tau}{{R}_{5}{C}_{3}}=\phi ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\tau}{{R}_{6}{C}_{3}}=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\tau}{{R}_{7}{C}_{4}}=k\end{array}$ (1.7)

Use convenient values for resistances and capacitances as

$\begin{array}{l}{R}_{1}=10\text{\hspace{0.17em}}\Omega ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{R}_{2}=337.268\text{\hspace{0.17em}}\text{m}\Omega ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{R}_{3}=2.5\text{\hspace{0.17em}}\Omega ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{R}_{4}=100\text{\hspace{0.17em}}\Omega ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{R}_{5}=10\text{\hspace{0.17em}}\Omega ,\\ {R}_{6}=100\text{\hspace{0.17em}}\Omega ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{R}_{7}=12.5\text{\hspace{0.17em}}\Omega ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{1}={C}_{2}={C}_{3}={C}_{4}=1\text{\hspace{0.17em}}\text{mF}\end{array}$ (1.8)

We got the empirical electronic circuit (1.6) for system (1.1) with parameters $\rho =10,a=296.5,\delta =40,\phi =10,k=8$ .

4. The Simulation Results

In this section, we simulate the circuit designed to implement the chaotic system (1.1) electronically by MultiSIM12; where we show a circuit diagram of the chaotic system (1.1) in Figure 4.

The phase portraits of electronic circuit and the outputs voltages signals ${V}_{x},{V}_{y},{V}_{z},{V}_{w}$ versus time are presented in Figure 5.

From comparing Figure 5 which was obtained from MultiSIM 12 with Figure 2 and Figure 3 which was obtained from MATLAB, we note that between experimental achievements and numerical simulation there is a good qualitative agreement.

Figure 4. Diagram of the electronic circuit designed for chaotic system (1.1).

(a)(b)(c)(d)(e)(f)(g)

Figure 5. The right-hand side shows phase portraits of the proposed electronic circuit in (e) $\left({V}_{x},{V}_{w}\right)$ plane, (f) $\left({V}_{y},{V}_{w}\right)$ plane, (g) $\left({V}_{w},{V}_{z}\right)$ plane; While The left-hand side shows the output voltage signals as seen in (a) ( ${V}_{x}$ versus time), (b) ( ${V}_{y}$ versus time), (c) ( ${V}_{z}$ versus time), (d) ( ${V}_{w}$ versus time).

5. Conclusion

In this paper, the variety chaotic motions of a four-dimensional dynamical system with hidden attractor are investigated through Lyapunov exponents, wave-form and phase portraits. The Lyapunov exponents are determined as ( ${L}_{1}=1.660748$ , ${L}_{2}=0.149599$ , ${L}_{3}=-0.068474$ and ${L}_{4}=-12.144118$ ). Also, Kaplan-Yorke dimension is calculated as ${D}_{L}=3.143433471$ . Then, an electronic circuit is proposed to implement a chaotic system (1.1). The electronic circuit is designed by applying Kirchhoff’s laws, Voltages and time normalized well by dimensionless states variables. The designed circuit is simulated by the MultiSIM12 program; after observing the results, it becomes clear that the numerical results obtained from MATLAB are well in agreement with the experimental results obtained from the Multisim12 program, meaning that the simulation was done well.

Acknowledgements

The authors are very grateful to Mosul University/College of Computer Sciences and Mathematics for their supported, which helped to improve the quality of this work.

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