# Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System

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## Abstract

**:**

## 1. Introduction

## 2. Modeling of Three-Terminal Memristor

## 3. Modeling of Conservative Chaotic System Based on Three-Terminal Memristor

## 4. Equilibria and Their Stability of Three-Terminal Memristor Conservative System

## 5. Dynamical Analysis of Three-Terminal Memristor Conservative Chaotic System

#### 5.1. Memristor Effect in Chaos Generation

#### 5.2. Dynamical Analysis with Different Initial Conditions of ${H}_{c}$

#### 5.3. Dynamical Analysis with Fixed ${H}_{c}$ and Varied ${H}_{n}$

## 6. Circuit Implementation

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Strukov, D.B.; Snider, G.S.; Stewart, D.R.; Williams, R.S. The missing memristor found. Nature
**2008**, 453, 80–83. [Google Scholar] [CrossRef] [PubMed] - Francesco, C.; Carbajal, J.P. Memristors for the Curious Outsiders. Technologies
**2018**, 6, 118. [Google Scholar] [CrossRef] [Green Version] - Chua, L.O. Resistance switching memories are memristors. Appl. Phys.
**2011**, 102, 765–783. [Google Scholar] [CrossRef] [Green Version] - Borghetti, J.; Snider, G.S.; Kuekes, P.J.; Yang, J.J.; Stewart, D.R.; Stanley, R. ‘Memristive’ switches enable ‘stateful’ logic operations. Nature
**2010**, 464, 873–876. [Google Scholar] [CrossRef] [PubMed] - Wang, W.; Jia, X.; Luo, X.; Kurths, J.; Yuan, M. Fixed-time synchronization control of memristive MAM neural networks with mixed delays and application in chaotic secure communication. Chaos Solitons Fractals
**2019**, 126, 85–96. [Google Scholar] [CrossRef] - Miranda, E.; Sune, J. Memristors for Neuromorphic Circuits and Artificial Intelligence Applications. Materials
**2020**, 13, 938. [Google Scholar] [CrossRef] [Green Version] - Bao, H.; Hu, A.; Liu, W.; Bao, B. Hidden Bursting Firings and Bifurcation Mechanisms in Memristive Neuron Model With Threshold Electromagnetic Induction. IEEE Trans. Neural Netw. Learn. Syst.
**2020**, 31, 502–511. [Google Scholar] [CrossRef] - Widrow, B. An Adaptive Adaline Neuron Using Chemical Memristors. Technical Report. 1960. (Stanford Electronics Laboratories). Available online: www-isl.stanford.edu/~widrow/papers/t1960anadaptive.pdf (accessed on 30 December 2020).
- Diorio, C.; Hasler, P.; Minch, A.; Mead, C.A. A single-transistor silicon synapse. IEEE Trans. Electron Dev.
**1996**, 43, 1972–1980. [Google Scholar] [CrossRef] [Green Version] - Lai, Q.; Zhang, L.; Li, Z.; Stickle, W.F.; Williams, R.S.; Chen, Y. Ionic/electronic hybrid materials integrated in a synaptic transistor with signal processing and learning functions. Adv. Mater.
**2010**, 22, 2448–2453. [Google Scholar] [CrossRef] - Mouttet, B. Memristive systems analysis of 3-terminal devices. In Proceedings of the 2010 17th IEEE International Conference on Electronics, Circuits and Systems, Athens, Greece, 12–15 December 2010; pp. 930–933. [Google Scholar]
- Chua, L.O.; Kang, S.M. Memristive devices and systems. Proc. IEEE
**1976**, 64, 209–223. [Google Scholar] [CrossRef] - Sangwan, V.K.; Lee, H.S.; Bergeron, H.; Balla, I.; Beck, M.E.; Chen, K.S.; Hersam, M.C. Multi-terminal memtransistors from polycrystalline monolayer molybdenum disulfide. Nature
**2018**, 554, 500–504. [Google Scholar] [CrossRef] [PubMed] - Kapitaniak, T.; Mohammadi, S.; Mekhilef, S.; Alsaadi, F.; Hayat, T.; Pham, V. A new chaotic system with stable equilibrium: Entropy analysis, parameter estimation, and circuit design. Entropy
**2018**, 20, 670. [Google Scholar] [CrossRef] [PubMed] [Green Version] - David, S.; Fischer, C.; Machado, J. Fractional electronic circuit simulation of a nonlinear macroeconomic model. AEU Int. J. Electron. Commun.
**2018**, 84, 210–220. [Google Scholar] [CrossRef] - Ovchinnikov, I.V.; Ventra, M.D. Chaos as a symmetry-breaking phenomenon. Mod. Phys. Lett. B
**2019**, 33, 1950287. [Google Scholar] [CrossRef] - Qi, G. Modelings and mechanism analysis underlying both the 4D Euler equations and Hamiltonian conservative chaotic systems. Nonlinear Dyn.
**2019**, 95, 2063–2077. [Google Scholar] [CrossRef] - Qi, G.; Hu, J.; Wang, Z. Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos. Appl. Math. Model.
**2020**, 78, 350–365. [Google Scholar] [CrossRef] - Qi, G.; Hu, J. Modelling of both energy and volume conservative chaotic systems and their mechanism analyses. Commun. Nonlinear Sci. Numer. Simulat.
**2020**, 84, 105171. [Google Scholar] [CrossRef] - Frederickson, P.; Kaplan, J.L.; Yorke, E.D.; Yorke, J.A. The liapunov dimension of strange attractors. J. Differ. Equ.
**1983**, 49, 185–207. [Google Scholar] [CrossRef] [Green Version] - Muthuswamy, B. Implementing Memristor Based Chaotic Circuits. Int. J. Bifurc. Chaos
**2010**, 20, 1335–1350. [Google Scholar] [CrossRef] - Itoh, M.; Chua, L.O. Memristor Oscillators. Int. J. Bifurc. Chaos
**2008**, 18, 3183–3206. [Google Scholar] [CrossRef] - Feng, Y.; Rajagopal, K.; Khalaf, A.; Alsaadi, F.; Alsaadi, F.; Pham, V. A new hidden attractor hyperchaotic memristor oscillator with a line of equilibria. Eur. Phys. J. Spec. Top.
**2020**, 229, 1279–1288. [Google Scholar] [CrossRef] - Lu, H.; Petrzela, J.; Gotthans, T.; Rajagopal, K.; Jafari, S.; Hussain, I. Fracmemristor chaotic oscillator with multistable and antimonotonicity properties. J. Adv. Res.
**2020**, 25, 137–145. [Google Scholar] [CrossRef] [PubMed] - Biolek, Z.; Biolek, D.; Biolkova, V.; Kolka, Z. All Pinched Hysteresis Loops Generated by (alpha, beta) Elements: In What Coordinates They May be Observable. IEEE Access
**2020**, 8, 199179–199186. [Google Scholar] [CrossRef] - Biolek, Z.; Biolek, D.; Biolkova, V.; Kolka, Z. Higher-Order Hamiltonian for Circuits with (alpha,beta) Elements. Entropy
**2020**, 22, 412. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Machado, J.; Lopes, A. Multidimensional scaling locus of memristor and fractional order elements. J. Adv. Res.
**2020**, 25, 147–157. [Google Scholar] [CrossRef] - Deng, Y.; Li, Y. A memristive conservative chaotic circuit consisting of a memristor and a capacitor. Chaos
**2020**, 30, 013120. [Google Scholar] [CrossRef] - Vaidyanathan, S. A Conservative Hyperchaotic Hyperjerk System Based on Memristive Device, Advances in Memristors. Memristive Devices and Systems; Springer: Berlin/Heidelberg, Germany, 2017; Volume 701, pp. 393–423. [Google Scholar]
- Yuan, F.; Jin, Y.; Li, Y. Self-reproducing chaos and bursting oscillation analysis in a meminductor-based conservative system. Chaos
**2020**, 30, 053127. [Google Scholar] [CrossRef] - Chua, L.O. Memristor-the missing circuit element. IEEE Trans. Circuit Theory
**1971**, 18, 507–519. [Google Scholar] [CrossRef] - Bao, B.; Liu, Z.; Xi, J. Transient chaos in smooth memristor oscillator. Chin. Phys. B
**2010**, 19, 030510. [Google Scholar] - Faradja, P.; Qi, G. Hamiltonian-Based Energy Analysis for Brushless DC Motor Chaotic System. Int. J. Bifurc. Chaos
**2020**, 30, 2050112. [Google Scholar] [CrossRef] - Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov Exponents from a Time Series. Phys. D
**1985**, 16, 285–317. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**I-V curves with different frequency inputs: (

**a**) ${\omega}_{1}=2,{\omega}_{2}=1$

**,**(

**b**) ${\omega}_{1}=10,{\omega}_{2}=1$.

**Figure 3.**I-V curves of Multisim simulation circuit (

**a**) ${\omega}_{1}=2,{\omega}_{2}=1$ (

**b**) ${\omega}_{1}=10,{\omega}_{2}=1$.

**Figure 5.**Chaotic orbits of system Equation (11), (

**a**) time series of ${x}_{1}$, (

**b**) time series of ${x}_{1}{x}_{4}$ and Hamiltonian.

**Figure 6.**Numerical characteristics of the system of Equation (19): (

**a**) 3D view of ${x}_{1}-{x}_{2}-{x}_{3}$ with $\gamma =0$, (

**b**) Lyapunov exponents with $\gamma =1$, (

**c**) 3D view of ${x}_{1}-{x}_{2}-{x}_{3}$ with $\gamma =1$, (

**d**) phase portrait of ${x}_{1}-{x}_{2}$ with $\gamma =1$, (

**e**) Poincaré map of ${x}_{2}-{x}_{3}$ with ${x}_{1}=0$, (

**f**) Hamiltonian energy, (

**g**) Lyapunov exponent (LEs) with large coefficients, and (

**h**) Poincaré map with large coefficients.

**Figure 7.**Dynamical analysis with different initial conditions: (

**a**) Bifurcation within initial ${x}_{20}\in \left[-2,2\right]$, (

**b**) bifurcation within initial ${x}_{10}=\left[-2,2\right]$, (

**c**) bifurcation within initial ${x}_{10}\in \left[1,1.2\right]$, (

**d**) chaotic orbits with ${x}_{10}=1$, (

**e**) quasiperiodic orbits with ${x}_{10}=1.1$, (

**f**) chaotic orbits with ${x}_{10}=1.2$.

**Figure 8.**Dynamical analysis with fixed ${H}_{c0}$, (

**a**) Bifurcation of ${x}_{50}$ with ${H}_{c0}=0.02$, (

**b**) LEs of ${x}_{50}$ with ${H}_{c0}=0.02$, (

**c**) bifurcation of ${x}_{50}$ with ${H}_{c0}=2$, (

**d**) LEs of ${x}_{50}$ with ${H}_{c0}=2$.

**Figure 9.**Circuit implementation of system Equation (15), with electronic parameters: $\mathrm{R}1,\mathrm{R}2,\mathrm{R}3,\mathrm{R}5,\mathrm{R}6,\mathrm{R}7,\mathrm{R}8,\mathrm{R}10,\mathrm{R}11,\mathrm{R}12,\mathrm{R}13,\mathrm{R}14,\mathrm{R}15,\mathrm{R}17,\mathrm{R}19=10\mathrm{K}\mathsf{\Omega};$ $\mathrm{R}4,\mathrm{R}9=5\mathrm{K}\mathsf{\Omega};$ $\mathrm{R}16,\mathrm{R}18=100\mathrm{K}\mathsf{\Omega};$ $\mathrm{C}1,\mathrm{C}2,\mathrm{C}3,\mathrm{C}4,\mathrm{C}5=10\mathrm{nF}.$

**Figure 10.**Phase portrait of numerical simulation: (

**a**) Phase portrait of ${x}_{2}-{x}_{3}$, (

**b**) phase portrait of ${x}_{1}-{x}_{4}$.

**Figure 11.**Implementation of circuit: (

**a**) Phase portrait of ${x}_{2}-{x}_{3}$, (

**b**) phase portrait of ${x}_{1}-{x}_{4}$, (

**c**) implementation of the circuit, (

**d**) implementation of the circuit.

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**MDPI and ACS Style**

Wang, Z.; Qi, G.
Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System. *Entropy* **2021**, *23*, 71.
https://doi.org/10.3390/e23010071

**AMA Style**

Wang Z, Qi G.
Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System. *Entropy*. 2021; 23(1):71.
https://doi.org/10.3390/e23010071

**Chicago/Turabian Style**

Wang, Ze, and Guoyuan Qi.
2021. "Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System" *Entropy* 23, no. 1: 71.
https://doi.org/10.3390/e23010071