A Novel 2D—Grid of Scroll Chaotic Attractor Generated by CNN

Abstract

The complex grid of scroll chaotic attractors that are generated through nonlinear electronic circuits have been raised considerably over the last decades. In this paper, it is shown that a subclass of Cellular Nonlinear Networks (CNNs) allows us to generate complex dynamics and chaos in symmetry pattern. A novel grid of scroll chaotic attractor, based on a new system, shows symmetry scrolls about the origin. Also, the equilibrium points are located in a manner such that the symmetry about the line $x=y$ has been achieved. The complex dynamics of system can be generated using CNNs, which in turn are derived from a CNN array ($1\times 3$) cells. The paper concerns on the design and implementation of $2\times 2$ and $3\times 3$ $2\mathrm{D}$-grid of scroll via the CNN model. Theoretical analysis and numerical simulations of the derived model are included. The simulation results reveal that the grid of scroll attractors can be successfully reproduced using PSpice.

1. Introduction

Since the treasure trove of Chua’s circuit [1], many scientists from different fields have been studying the double scroll attractors. Chua’s circuit is a paradigm for chaos, which in turn is deformed from $3\mathrm{D}$ system. Many bifurcation phenomena have been described in [2], such as Hopf bifurcation, Rossler’s spiral, double scroll, etc., all these special emphases lead to the double scroll attractor. Leon O. Chua and Lui-nian Lin in [3], presented a canonical circuit capable of realizing every member of Chua’s family using a three-region system. In [4], many strange attractors were found from Chua’s oscillator and illustrated the complexity of Chua’s oscillator. The implementation of a smooth nonlinearity with a cubic polynomial or higher order was presented in [5], to overcome some subtle features of a real circuit. In [6], Xiao Fan and Guanrong Chen designed a linear feedback controller composed with nonlinear modulo or sawtooth function to drive the original nonlinear autonomous system. Chen [7] introduced a canonical form of the generalized Lorenz system, which was deeply studied and found to be a new and useful tool for chaos synthesis. Moreover, Jinhu Lu and Guanrong Chen discovered a new chaotic system [8], that connects Lorenz attractor and Chen’s attractor. A chaotic system of $3\mathrm{D}$ quadratic smooth autonomous equations can generate two double-wing chaotic attractors [9]. A novel bounded $4\mathrm{D}$ chaotic system was presented in [10], in which hyperchaos, chaos, equasiperiodic and periodic behavior were studied. In [11], Serdar et al. worked on numerical, electronic circuit simulation and electronic circuit implementation of a Sprott chaotic system and its synchronization.

For powerful ability in engineering applications based-chaos, chaotic attractors with high complex dynamic structures have been interestingly investigated. M. E. Ylcin et al. presented configuration of 3- and 5-scroll attractors from a generalized Chua’s circuit that lead to n-scroll attractors [12]. In [13] $\left(1\mathrm{D}\right)$, ($2\mathrm{D})$, and ($3\mathrm{D})$ grid attractors the scroll generated due to equilibrium points. An approach proposed by Weihua Deng and Jinhu Lu in [14], generates multiscroll chaotic attractors including ($1\mathrm{D}$) $n$-scroll, ($2\mathrm{D}$) $n\times m$-grid of scroll, and ($3\mathrm{D}$) $n\times m\times l$-grid of scroll chaotic attractors. Jinhu Lu and Guanrong Chen survey the advantages of multiscroll chaotic attractors generation in potential applications and design theories [15]. Simin Yu et al. proposed a fourth-order double-tours circuit for creating $\left(1\mathrm{D}\right)$ n-tours, $\left(2\mathrm{D}\right)$ $n\times m$-tours, ($3\mathrm{D})$ $n\times m\times l$-tours chaotic attractors [16]. In [17], a systematic threshold control approach for generating multi-scroll chaotic attractors was based on both general Jerk’s and Chua’s circuits with sine nonlinearity. In [18], a method was proposed for generating n-, $m\times n$-, and $m\times n\times l$-grid-scroll chaotic attractor based series of hyperbolic tangent function. In [19], generation of multiscroll $n\times m$-dimensional scroll was obtained from a one-dimension scroll by changing and adapting the system equation by jerk system. Zuo et al. used a CCII or current conveyor circuit for generating $4\times 2\times 2\times 2$-scroll and $4\times 2\times 2\times 2\times 2$-scroll chaotic attractors which are basically fourth-order and fifth-order, respectively, via switching control [20]. Based on a general Jerk circuit, a new nonlinearity function for modulating and generating n-scroll chaotic attractors was proposed. This function is able to arbitrarily design the shapes, widths, swings, breakpoints, slopes, equilibrium points, and even the phase portrait of the n-scroll chaotic attractors and the chaos condition and dynamic mechanism of the jerk circuit are interestingly studied in [21]. A scheme is also designed for generating 3-scroll and 12-scroll chaotic attractors via switches and analog circuit realization. Simin Yu et al. suggested increasing the number of equilibrium points type index 2, by employing quadratic function, so they explored $n$- and $n\times m$-wing Lorenz attractors [22]. A sawtooth function can be used to generate multiscroll chaotic attractors by changing the number n of sawtooth function [23]. A simple circuit design was proposed in [24], capable of creating new grid multiscroll chaotic electronic oscillator. In [25], novel multiscroll attractors and multiwing hidden attractors in five dimensional memrestive system were proposed.

During the past two decades, different kinds of CNN-based chaotic circuits have facilitated the knowledge of the chaotic phenomenon by enhancing research on chaos through analog simulation. A $4\mathrm{D}$ autonomous chaotic system was observed by introducing a state feedback and new n-well potential function for creating a butterfly wing chaotic attractor [26]. As a nonlinear circuit, CNN can be considered as a paradigm generator for complex systems [27]. Reasonable feedback from the states of the cells made the CNN-based chaotic circuit able to produce complex dynamics [28]. After developing the procedures of the CNN, great complex dynamic systems and chaotic circuits are modelled, such as Chua’s circuit that was controlled by photoresistor nonlinear function [29]. Complex dynamics are proposed in [30] by fractional-order four-cell CNN, a system that exhibits high dynamical behavior such as periodic, chaotic and hyperchaotic. This dynamics behaviors are investigated numerically by simulations and theoretically by equations. A hyperbolic tangent function series is proposed in [31] to generate multi-scroll attractors such as $1\mathrm{D}$, $2\mathrm{D}$, and $3\mathrm{D}$ scroll attractors from an autonomous SC-CNN system. Different approaches for simulating the multivariable nonlinearities by switching states of CNN in [32] based chaotic systems, Lorenz kind system dynamics and others like this one are generated by switching CNNs. In [33] it is shown that this structure, SC-CNN, is able to introduce complex dynamics, many circuits that exhibit complex dynamic behaviors can be obtained from this structure by changing the values of the cloning templates. On the other hand, CNN is a novel class of information-processing systems; like a neural network, it is a large-scale nonlinear analog circuit which processes signals in real-time. It was proposed and investigated by Chua and Yang in theory and applications [34,35]. In distinct cases, the CNN consists of a homogenous array and its array has no thresholds, no inputs, and even no outputs, and a sphere of influence that extends only to the nearest neighbours so the CNN reduces to a nonlinear lattice [36]. The local activity means that when the system exhibits complexity, the CNN parameters must be chosen so that the cells or their neighbours are locally actively coupled.

In this paper, a new model of CNN is presented using a modified Chua system. It is based on Piece Wise Linear (PWL) functions and the equilibrium points are distributed in such a way that give a symmetrical grid of scroll chaotic attractors. Next, to build CNN model scheme, we consider the generalized cell model that was introduced by Arena et al. [28]. The designed CNNs with array can emulate such a system and generate a grid of scroll. The results showed that a grid of scrolls, 2 × 2 and 3 × 3, can be obtained using numerical simulations. The designed electronic circuit of a CNN model is also confirmed through the PSpice. The rest of paper is organized as follows. In Section 2, a new model for generation of a grid of scroll attractors is described. Its dynamical behavior is studied in Section 3. Section 4 covers a CNN based design to emulate the new system model and generation process of a grid of scroll using a CNN array. Section 5 describes the realization of a nonlinear electronics circuit and observation of a grid of scroll using PSpice. The conclusions are given finally.

2. The Generation of 2D-Grid of Scroll from a Modified Chua Model

The attention is to produce a complex chaotic attractor that has a grid of scrolls. The framework is the Chua chaotic oscillator model.

A. Chua Oscillator Model

The Chua oscillator model [37] is given by

$$\begin{gathered} \dot{x}=\alpha \left[y-f\left(x\right)\right], \\ \dot{y}=x-y+z, \\ \dot{z}=-\beta y-\gamma z, \end{gathered}$$

(1)

$$f\left(x\right)=dx+\frac{1}{2}\left(c-d\right)\left[\left|x+1\right|-\left|x-1\right|\right],$$

(2)

Selecting parameters $\alpha =9$, $\beta =14.28$, $c=1/7$, and $d=2/7$, a typical double scroll Chua’s attractor can be obtained, as shown in Figure 1. The equibilibrium points can be calculated as saddle index-2 $\left(\pm \frac{\left(\beta +\gamma \right)\left(d-c\right)}{\beta \left(d+1\right)+\gamma d},\pm \frac{\gamma \left(d-c\right)}{\beta \left(d+1\right)+\gamma d},\pm \frac{\beta \left(d-c\right)}{\beta \left(d+1\right)+\gamma d}\right)$ and saddle index-1 $\left(0,0,0\right)$. The double scrolls are evolving from index-2 equilibrium points as shown in Figure 1. Detailed studies of this model, its dynamics, and bifurcation can be found in [38].

B. Modified Chua Oscillator Model

To create the 2D-grid of scroll chaotic attractors, it is essential to design a grid of saddle index-2 equilibrium points. System model (1) through (2), therefore, is developed to be

$$\begin{gathered} \dot{x}=af\left(y\right)-bf\left(x\right), \\ \dot{y}=x-y+z, \\ \dot{z}=-\beta f\left(y\right), \end{gathered}$$

(3)

where $a$, $b$, and $\mathsf{\beta }$ are system parameters, suppose that $f\left(x\right)$ and $f\left(y\right)$ are PWL functions, which determine the number of scrolls in $x$ and $y$ directions, respectively. That is,

$$h\left({\xi }_j\right)=m_{{\xi }_{j q_{{\xi }_j}-1}}{\xi }_j+\frac{1}{2}{\sum }_{i=1}^{q_{{\xi }_j}-1}\left(m_{\xi j i-1}-m_{\xi j i}\right)\left(\left|{\xi }_j+{\xi }_{j_i}^b\right|-\left|{\xi }_j-{\xi }_{j_i}^b\right|\right),$$

(4)

where ${\xi }_jϵ\left(x,y\right)$, $jϵ\left[1,2\right]$, and $q_{{\xi }_j}\left(=2,3\dots \right)$ is the number of scrolls, $m_{\xi j i}$ and ${\xi }_{j_i}^b$ are the control parameters of PWL function. Figure 2 shows nonlinearity function.

C. Dynamics of New Model

Two essential features are focused on, the equilibrium points characteristics and bifurcation diagram to study the dynamics of the new model (3).

(1)

Equilibrium points: As mentioned earlier, the distribution of saddle index-2 equilibrium points over the $xy$-plane to get a $2\mathrm{D}$ grid of scroll chaotic attractor. By setting $\dot{x}=\dot{y}=\dot{z}=0$, the equilibrium points can be discussed as follows:

• The equilibrium points $\pm y_i^e\left(i=0,1,2,\dots ,q_{{\xi }_2}-1\right)$ are located on the $y$-direction in the state space.
  One can deduce all positive equilibrium points by using the recursive formula which is given by

  $$\begin{gathered} y_1^e=\frac{{\sum }_{i=1}^i\left(m_k-m_{k-1}\right){\xi }_{2k}^b}{m_1}, \\ {y}_2^e=\frac{{\sum }_{i=1}^i\left(m_k-m_{k-1}\right){\xi }_{2k}^b}{m_2}, \\ ⋮ \\ {y}_i^e=\frac{{\sum }_{k=1}^i\left(m_k-m_{k-1}\right){\xi }_{2k}^b}{m_{q_{2-1}}}, \end{gathered}$$

  (5)
• For the equilibrium points $\pm x_j^e\left(j=0,1,2,\dots ,q_{{\xi }_1}-1\right)$ which are located on the $x$-direction in the state space. The positive equilibrium points:

  $$x_j^e=\left(\frac{a}{b}\right)\frac{{\sum }_{k=1}^j\left(m_k-m_{k-1}\right){\xi }_k^b}{m_{q_{1-1}}},$$

  (6)
• For the equilibrium points $\pm z_r^e=\left(r=0,1,2,\dots ,\left(q_{{\xi }_1}-1\right)\left(q_{{\xi }_2}-1\right)\right)$ which are located on the z-direction in the state space. The positive equilibrium points:

  $$z_r^e=y_i^e-x_j^e,i\in \left[0,1,2\dots .q_{{\xi }_2}\right], j\in \left[0,1,2\dots .q_{{\xi }_1}\right]$$

  (7)

For example, when $q_{{\xi }_1}=q_{{\xi }_2}=3$, a grid of $3\times 3$ saddle index-2 equilibrium points can be obtained, as shown in Figure 3.

Obviously, system (3) with (4) has $(q_{{\xi }_1}-1)(q_{{\xi }_2}-1)$ equilibrium points. The corresponding jacobian matrices and their characteristic equations are, respectively,

$$J=\left[\begin{array}{ccc}-b{m}_{{\xi }_{1j}}& a{m}_{{\xi }_{2i}}& 0\\ 1& -1& 1\\ 0& -\beta m_{{\xi }_{2i}}& 0\end{array}\right]$$

(8)

and

$${\lambda }^3+{\lambda }^2\left(1+b{m}_{{\xi }_{1j}}\right)+\lambda \left(b{m}_{{\xi }_{1j}}+\beta m_{{\xi }_{2i}}-a{m}_{{\xi }_{2i}}\right)+\left(b{m}_{{\xi }_{1j}}\beta m_{{\xi }_{2i}}\right)=0$$

(9)

Theoretical analysis shows that all equilibrium points can be classified into saddle index-2. For example, when $\beta =10, \mathrm{a}=7$, $b=2.5, m_{{\xi }_{1o}}=1, m_{{\xi }_{11}}=-2$, $m_{{\xi }_{2o}}=2, m_{{\xi }_{21}}=-4$, and $q_{\xi j}=2$, system (3) with (4) has a 2x2-grid of equilibrium points. Similarly, for $\beta =10, \mathrm{a}=7$, $b=2.5, m_{{\xi }_{1o}}=1, m_{{\xi }_{11}}=-2$, $m_{{\xi }_{2o}}=2, m_{{\xi }_{21}}=-4$, and $q_{\xi j}=3$, system (3) with (4) has a 3 × 3-grid of equilibrium points. Obviously, the equilibrium points have eigenvalues ${\lambda }_1=-4.3268$, ${\lambda }_{2,3}=0.4134\pm j3.531$ and ${\lambda }_1=-5.755$, ${\lambda }_{2,3}=4.877\pm j3.772$, which are called saddle points of index-2 since the two complex conjugate eigenvalues have positive real parts.

(2)

Bifurcation diagram: To confirm the existence of chaos in the new system (3) with (4), assume that the case of $q_{\xi j}=3$. Then, $\beta \in \left[7,12\right]$. The bifurcation diagram of the parameter $\beta$ of system (3) with (4) can be obtained as illustrated in Figure 4.

D. Numerical Simulation Results

As mentioned in Section 2-C, there are $(q_{{\xi }_1}-1)(q_{{\xi }_2}-1)$ saddle index-2 equilibrium points. Considering the system parameters given in Section 2-C, a different 2D-grid of scroll chaotic attractors can be generated, as shown in Figure 5.

3. Design of CNN—Based New Chaotic System

This section introduces the design of chaotic system (3) with (4) based on CNN paradigm. A CNN can produce different forms of complex phenomena just like pattern formation, chaotic scroll, spiral, and autowaves. It works well as a merged model for complexity and a paradigm for simulating nonlinear partial differential equations (PDE’s). Arena et al. [28] introduced the generalized cell model which is characterised by the following state equation:

$${\dot{x}}_j=-x_j+a_j{y}_j+G_o+G_s+i_j,$$

(10)

with

$$y_j=\frac{1}{2}\left(\left|x_j+1\right|-\left|x_j-1\right|\right),$$

(11)

where $x_j$ is the single state variable of cell, $y_j$ is the single output state of cell, $a_j$, $G_o$, $G_s$, and $i_j$ are parameters defining the CNN template. It is worth noting that the dynamics of state variables of the single cell depend also on the state variables of neighbouring cells. According to model (10) with (11), the dynamic model of three fully connected CNN cells is

$$\begin{gathered} {\dot{x}}_1=-x_1+a_1{y}_1+a_{12}{y}_2+a_{13}{y}_3+\sum _{k=1}^3{s}_{1k}{x}_k+i_1, \\ {\dot{x}}_2=-x_2+a_{21}{y}_1+a_{22}{y}_2+a_{23}{y}_3+\sum _{k=1}^3{s}_{2k}{x}_k+i_2, \\ {\dot{x}}_3=-x_3+a_{31}{y}_1+a_{32}{y}_2+a_3{y}_3+\sum _{k=1}^3{s}_{3k}{x}_k+i_3, \end{gathered}$$

(12)

where $x_1$, $x_2$, and $x_3$ are state variables, $y_1$, $y_2$, and $y_3$ are the corresponding output. By choosing

$$a_{13}=a_{21}=a_{22}=a_{23}=a_{31}=a_{31}=a_{33}=0; s_{13}=s_{22}=s_{33}=0; i_1=i_2=i_3=0$$

(13)

model (12) becomes:

$$\begin{gathered} {\dot{x}}_1=-x_1+a_1{y}_1+a_{12}{y}_2+s_{11}{x}_1+s_{12}{x}_2, \\ {\dot{x}}_2=-x_2+s_{21}{x}_1+s_{23}{x}_3, \\ {\dot{x}}_3=-x_3+a_{32}{y}_2+s_{32}{x}_2+s_{33}{x}_3, \end{gathered}$$

(14)

The new chaotic system (3) with (4) can be designed by CNN model to produce a $2\mathrm{D}$-grid of scroll chaotic attractor. So, two different paradigm layouts for generation of $n\times m$ $2\mathrm{D}$-grid of scroll chaotic attractor by CNN cells are adopted with different CNN cell parameters values, as shown in

Table 1. Parameters of the designed Cellular Nonlinear Networks (CNN) model.

	Parameters	${\mathit{a}}_1$	${\mathit{a}}_{12}$	${\mathit{a}}_{32}$	${\mathit{s}}_{11}$	${\mathit{s}}_{12}$	${\mathit{s}}_{21}$	${\mathit{s}}_{23}$	${\mathit{s}}_{32}$	${\mathit{s}}_{33}$
Grid
2 × 2-scroll		15.0	−25.2	36.0	−1.5	7.0	1.0	1.0	−10.0	1.0
3 × 3-scroll		−7.5	46.2	−66.0	−1.625	15.4	1.0	1.0	−22.0	1.0

.

The design scheme of CNN cells connection is proposed for realizing model (14) as shown in Figure 6. Assuming that ${\mathrm{V}}_1=-{\mathrm{Y}}_1$ and ${\mathrm{V}}_2={\mathrm{Y}}_2$ and ${\mathrm{V}}_3={\mathrm{X}}_2$ for the first cell, ${\mathrm{V}}_1={\mathrm{X}}_1$ and ${\mathrm{V}}_2={\mathrm{X}}_3$ for the second cell, ${\mathrm{V}}_1=-{\mathrm{X}}_2$ and ${\mathrm{V}}_2=-{\mathrm{Y}}_2$ for the third cell.

4. CNN Circuit Implementation

In CNN-based implementations, a state variable of the system to be realized is the voltage across capacitor of the Op-Amp based circuit design which is used to implement the dynamics of a CNN cell. A circuit scheme for CNN model is proposed as shown in Figure 7 and it is govering by the following state equations:

$$\begin{gathered} \frac{d{X}_1}{dt}=\frac{1}{R_6{C}_1}\left(\frac{R_5}{R_2}{Y}_1+\frac{R_5}{R_3}{Y}_2+\frac{R_5}{R_4}{X}_1+\frac{R_5}{R_1}{X}_2\right), \\ \frac{d{X}_2}{dt}=\frac{1}{R_{11}{C}_2}\left(\frac{R_9}{R_7}{X}_1-X_2+\frac{R_9}{R_8}{X}_3\right), \\ \frac{d{X}_3}{dt}=\frac{1}{R_{15}{C}_3}\left(\frac{R_{14}}{R_{12}}{Y}_2+\frac{R_{14}}{R_{11}}{X}_2\right), \end{gathered}$$

(15)

It is mainly constructed by six blocks, which are as follows:

• N₁ is the first state variable generator; $a_1=\frac{R_5}{R_3}$, $a_{12}=\frac{R_5}{R_2}$, $s_{11}=\frac{R_5}{R_4}$, $s_{12}=\frac{R_5}{R_1}$.
• N₂ is the second state variable generator; $s_{23}=\frac{R_{10}}{R_8}$, $s_{21}=\frac{R_{10}}{R_7}$.
• N₃ is the third state variable generator; $a_{32}=\frac{R_{14}}{R_{13}}$, $s_{32}=\frac{R_{14}}{R_{12}}$.
• N₄ is an inverting amplifier block with unity gain.
• Both $Y_1 \mathrm{and} Y_2$ are the nonlinearities of the CNN.

The design of electronic circuit is based on capacitors, resistors, Op-Amps (TL084) of voltage supply equal to $\pm 12 V$. The nonlinearities functions, $Y_1 \mathrm{and} Y_2$, are implemented by using the circuitry block in such a different way so that it can be adapted by state of linkage switches. In accordance with the switches states that shown in

Table 2. The $ON-OFF$ switch linkages $K_0-K_3$.

	Switch	${\mathit{K}}_0$	${\mathit{K}}_1$	${\mathit{K}}_2$	${\mathit{K}}_3$
Grids
2 × 2-scroll		ON	OFF	ON	OFF
3 × 3-scroll		ON	ON	ON	ON

, one can get different grids of scroll chaotic attractors. The PSpice simulations show that the new system (3) mutates to a CNN-based chaotic system and for selected values of the parameters. It shows chaotic dynamical behavior. The phase portraits are shown in Figure 8, on $X_3{X}_2$-plane projection.

5. Conclusions

We have presented a novel third-order chaotic system and its complex dynamics. The new system is constructed by introducing two PWL functions that enable us to create a 2D-grid of scroll chaotic attractors. Such PWL function can be implemented by CNN model because its main block corresponds to saturation function. Dynamical analysis has been investigated, including equilibrium points and bifurcation. The equilibrium points distribution confirms the principle of symmetry. The locus y = x is the line of symmetry.

It can be noticed that this is the first CNN model reported in the literature that generates a 2D-grid of scroll chaotic attractor. Two different layouts generate a 2D-grid of scroll chaotic attractors by CNN cells. According to the selected parameters, a $2\times 2$-scroll and a $3\times 3$-scroll are confirmed. The system was implemented by using CNN cells. The PSpice environment results of the CNN model circuit design have demonstrated a good agreement with simulation results of the systems. The new system can be used in engineering applications based on chaos due to its capability of producing a grid of scroll attractors with a CNN circuitry.