# Multistability Dynamics Analysis and Digital Circuit Implementation of Entanglement-Chaos Symmetrical Memristive System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of Entanglement-Chaos System

_{1}= 2.03, LE

_{2}= 0.00, LE

_{3}= −9.02. The LEs of System (3) accord with the principle of the 3D chaotic system LEs (+,0,−); System (3) is chaotic. It can be seen from the attractor trajectory diagram of the system in Figure 2 that the attractor of the system has symmetric characteristics.

## 3. Analysis of Memristive Entanglement System

#### 3.1. Memristive Entanglement System

#### 3.2. Dissipative Analysis

#### 3.3. Equilibrium Stability Analysis

_{1}, c

_{2}, c

_{3}, and c

_{4}are the coefficients of the characteristic equation.

_{4}> 0. However, it is calculated that c

_{4}= 0 in Equation (11). Therefore, the equilibrium point is always unstable.

#### 3.4. 0–1 Test

#### 3.5. Multistability Analysis

## 4. Complexity Analysis of the System

#### 4.1. Completeness Calculation

#### 4.2. Complexity Analysis

#### 4.3. C0 Complexity Diagram

## 5. Digital Implementation

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Attractor trajectory diagrams of System (3). (

**a**) x−y phase diagram; (

**b**) x−z phase diagram.

**Figure 3.**Bifurcation diagram and LEs of System (3) with $\mathrm{x}\left(0\right)\in [-50,50]$. (

**a**) Bifurcation diagram; (

**b**) LEs.

**Figure 5.**System phase diagram trajectory in different projections of chaotic System (5) from initial values (0.1, 3, 2, 5) and (0.1, 3, 2, 6). (

**a**) System phase diagram trajectory in the x−y plane; (

**b**) system phase diagram trajectory in the x−z plane.

**Figure 6.**p−s plane diagrams in different projections of memristive chaotic System (2). (

**a**) Initial values (0.1, 3, 2, 6); (

**b**) initial values (0.1, 3, 2, 5).

**Figure 7.**Bifurcation diagram and LEs of System (2) with initial condition [x

_{0}, y

_{0}, z

_{0}, w

_{0}] = [x(0), 3, 2, 5]. (

**a**) Bifurcation diagram; (

**b**) LEs.

**Figure 8.**Various coexisting attractors with different values. (

**a**) x(0) = −100 and x(0) = 100; (

**b**) x(0) = 8 and x(0) = 10.

**Figure 9.**Bifurcation diagram and LEs of System (2) with initial condition [x

_{0}, y

_{0}, z

_{0}, w

_{0}] = [0.1, 3, 2, w(0)$\in $[−100, 100]]. (

**a**) Bifurcation diagram; (

**b**) LEs.

**Figure 10.**Various coexisting attractors with different values: (

**a**) $w\left(0\right)=-100$ and $w\left(0\right)=100$; (

**b**) $w\left(0\right)=8$ and $w\left(0\right)=10$; (

**c**) $w\left(0\right)=2$ and $w\left(0\right)=75$.

**Figure 11.**Bifurcation diagram and LEs of System (5) with initial condition [x0, y0, z0, w0] = [0.1, 3, 2, w(0)$\in $[−10,10]]. (

**a**) Bifurcation diagram; (

**b**) Les.

**Figure 12.**Bifurcation and LEs of the system with a varying in [0, 10]. (

**a**) Bifurcation of the system with [x(0), y(0), z(0), w(0)] = [0.1, 3, 2, 5]; (

**b**) bifurcation of the system with [x(0), y(0), z(0), w(0)] = [0.1, 3, 2, 6]; (

**c**) LEs of the system with [x(0), y(0), z(0), w(0)] = [0.1, 3, 2, 5]; (

**d**) LEs of the system with [x(0), y(0), z(0), w(0)] = [0.1, 3, 2, 6].

**Figure 13.**Complexity of the system with the parameters varying. (

**a**) x

_{0}varying; (

**b**) w

_{0}varying; (

**c**) a varying.

**Figure 15.**Oscilloscope display phase diagram of System (13) from initial values (0.1, 3, 2, 5). (

**a**) Phase portraits in the x-y plane; (

**b**) phase portraits in the x-z plane.

**Figure 16.**Oscilloscope display phase diagram of systems (13) from initial values (0.1, 3, 2, 6). (

**a**) Phase portraits in the x-y plane; (

**b**) phase portraits in the x-z plane.

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**MDPI and ACS Style**

Lei, T.; Zhou, Y.; Fu, H.; Huang, L.; Zang, H. Multistability Dynamics Analysis and Digital Circuit Implementation of Entanglement-Chaos Symmetrical Memristive System. *Symmetry* **2022**, *14*, 2586.
https://doi.org/10.3390/sym14122586

**AMA Style**

Lei T, Zhou Y, Fu H, Huang L, Zang H. Multistability Dynamics Analysis and Digital Circuit Implementation of Entanglement-Chaos Symmetrical Memristive System. *Symmetry*. 2022; 14(12):2586.
https://doi.org/10.3390/sym14122586

**Chicago/Turabian Style**

Lei, Tengfei, You Zhou, Haiyan Fu, Lili Huang, and Hongyan Zang. 2022. "Multistability Dynamics Analysis and Digital Circuit Implementation of Entanglement-Chaos Symmetrical Memristive System" *Symmetry* 14, no. 12: 2586.
https://doi.org/10.3390/sym14122586