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

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## Abstract

**:**

## 1. Introduction

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

**A. Chua Oscillator Model**

**B. Modified Chua Oscillator Model**

**C. Dynamics of New Model**

- (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$${y}_{1}^{e}=\frac{{\sum}_{i=1}^{i}\left({m}_{k}-{m}_{k-1}\right){\xi}_{2k}^{b}}{{m}_{1}},\phantom{\rule{0ex}{0ex}}{y}_{2}^{e}=\frac{{\sum}_{i=1}^{i}\left({m}_{k}-{m}_{k-1}\right){\xi}_{2k}^{b}}{{m}_{2}},\phantom{\rule{0ex}{0ex}}\vdots \phantom{\rule{0ex}{0ex}}{y}_{i}^{e}=\frac{{\sum}_{k=1}^{i}\left({m}_{k}-{m}_{k-1}\right){\xi}_{2k}^{b}}{{m}_{{q}_{2-1}}},$$
- 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:$${x}_{j}^{e}=\left(\frac{a}{b}\right)\frac{{\sum}_{k=1}^{j}\left({m}_{k}-{m}_{k-1}\right){\xi}_{k}^{b}}{{m}_{{q}_{1-1}}},$$
- 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:$${z}_{r}^{e}={y}_{i}^{e}-{x}_{j}^{e},i\in \left[0,1,2\dots .{q}_{{\xi}_{2}}\right],j\in \left[0,1,2\dots .{q}_{{\xi}_{1}}\right]$$

- (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**

## 3. Design of CNN—Based New Chaotic System

## 4. CNN Circuit Implementation

**N**is the first state variable generator; ${a}_{1}=\frac{{R}_{5}}{{R}_{3}}$, ${a}_{12}=\frac{{R}_{5}}{{R}_{2}}$, ${s}_{11}=\frac{{R}_{5}}{{R}_{4}}$, ${s}_{12}=\frac{{R}_{5}}{{R}_{1}}$._{1}**N**is the second state variable generator; ${s}_{23}=\frac{{R}_{10}}{{R}_{8}}$, ${s}_{21}=\frac{{R}_{10}}{{R}_{7}}$._{2}**N**is the third state variable generator; ${a}_{32}=\frac{{R}_{14}}{{R}_{13}}$, ${s}_{32}=\frac{{R}_{14}}{{R}_{12}}$._{3}**N**is an inverting amplifier block with unity gain._{4}- Both ${Y}_{1}\mathrm{and}{Y}_{2}$ are the nonlinearities of the CNN.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**PWL function $h\left({\xi}_{j}\right)$: (

**a**) Odd number of scroll; (

**b**) Even number of scroll.

**Figure 5.**2D-grid of scroll chaotic attractor on $yz$-plane. (

**a**) $2\times 2$ grid (

**b**) $3\times 3$ grid.

**Figure 7.**The CNN circuit scheme of system (13). The components are chosen as follows: ${\mathrm{R}}_{1}=0.1428\mathrm{k}\mathsf{\Omega}$, ${\mathrm{R}}_{2}=0.066\mathrm{k}\mathsf{\Omega}$, ${\mathrm{R}}_{3}=0.00396\mathrm{k}\mathsf{\Omega}$, ${\mathrm{R}}_{4}=0.4\mathrm{k}\mathsf{\Omega}$, ${\mathrm{R}}_{5}={R}_{7}={R}_{8}={R}_{9}={R}_{18}={R}_{19}={R}_{20}={R}_{22}={R}_{23}={R}_{24}={R}_{25}={R}_{26}={R}_{27}={R}_{30}={R}_{31}={R}_{32}={R}_{34}={R}_{35}=1\mathrm{k}\mathsf{\Omega}$, ${\mathrm{R}}_{6}=10\mathrm{k}\mathsf{\Omega}$, ${\mathrm{R}}_{11}=8\mathrm{k}\mathsf{\Omega}$, ${\mathrm{R}}_{12}=0.1\mathrm{k}\mathsf{\Omega}$, ${\mathrm{R}}_{13}=0.027778\mathrm{k}\mathsf{\Omega}$, ${\mathrm{R}}_{14}=1\mathrm{k}\mathsf{\Omega}$, ${\mathrm{R}}_{15}=5\mathrm{k}\mathsf{\Omega}$, ${R}_{16}={R}_{28}=10$ k$\mathsf{\Omega}$, ${R}_{17}={R}_{29}=1000$ k$\mathsf{\Omega}$, ${R}_{21}={R}_{33}=199$ k$\mathsf{\Omega}$, ${\mathrm{C}}_{1}={\mathrm{C}}_{2}={\mathrm{C}}_{3}=100\mathrm{nF}$.

**Figure 8.**PSpice results. Phase portraits of $n\times m$ grid scroll attractors of CNN circuit as: (

**a**) $2\times 2$; (

**b**) $3\times 3$ on ${X}_{3}{X}_{2}$-projection, $x=0.5$ V/div, $y=2$ V/div.

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 |

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 |

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**MDPI and ACS Style**

Ali, A.M.; Ramadhan, S.M.; Tahir, F.R.
A Novel 2D—Grid of Scroll Chaotic Attractor Generated by CNN. *Symmetry* **2019**, *11*, 99.
https://doi.org/10.3390/sym11010099

**AMA Style**

Ali AM, Ramadhan SM, Tahir FR.
A Novel 2D—Grid of Scroll Chaotic Attractor Generated by CNN. *Symmetry*. 2019; 11(1):99.
https://doi.org/10.3390/sym11010099

**Chicago/Turabian Style**

Ali, Ahmed M., Saif M. Ramadhan, and Fadhil R. Tahir.
2019. "A Novel 2D—Grid of Scroll Chaotic Attractor Generated by CNN" *Symmetry* 11, no. 1: 99.
https://doi.org/10.3390/sym11010099