Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System
<p>I-V curves with different frequency inputs: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mrow> <mo>=</mo> <mn>2</mn> </mrow> <mo>,</mo> <mo> </mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mrow> <mo>=</mo> <mn>1</mn> </mrow> </mrow> </semantics></math><b>,</b> (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mrow> <mo>=</mo> <mn>10</mn> </mrow> <mo>,</mo> <mo> </mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mrow> <mo>=</mo> <mn>1</mn> </mrow> </mrow> </semantics></math>.</p> "> Figure 2
<p>Circuit implementation of the three-terminal memristor.</p> "> Figure 3
<p>I-V curves of Multisim simulation circuit (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>10</mn> <mo>,</mo> <mo> </mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 4
<p>Periodic orbit of the system of Equation (8).</p> "> Figure 5
<p>Chaotic orbits of system Equation (11), (<b>a</b>) time series of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, (<b>b</b>) time series of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>4</mn> </msub> </mrow> </semantics></math> and Hamiltonian.</p> "> Figure 6
<p>Numerical characteristics of the system of Equation (19): (<b>a</b>) 3D view of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>γ</mi> <mrow> <mo>=</mo> <mn>0</mn> </mrow> </mrow> </semantics></math>, (<b>b</b>) Lyapunov exponents with <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, (<b>c</b>) 3D view of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, (<b>d</b>) phase portrait of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, (<b>e</b>) Poincaré map of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, (<b>f</b>) Hamiltonian energy, (<b>g</b>) Lyapunov exponent (<span class="html-italic">LE</span>s) with large coefficients, and (<b>h</b>) Poincaré map with large coefficients.</p> "> Figure 7
<p>Dynamical analysis with different initial conditions: (<b>a</b>) Bifurcation within initial <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>20</mn> </mrow> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) bifurcation within initial <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>, (<b>c</b>) bifurcation within initial <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>1.2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>, (<b>d</b>) chaotic orbits with <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, (<b>e</b>) quasiperiodic orbits with <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, (<b>f</b>) chaotic orbits with <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mn>1.2</mn> </mrow> </semantics></math>.</p> "> Figure 7 Cont.
<p>Dynamical analysis with different initial conditions: (<b>a</b>) Bifurcation within initial <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>20</mn> </mrow> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) bifurcation within initial <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>, (<b>c</b>) bifurcation within initial <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>1.2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>, (<b>d</b>) chaotic orbits with <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, (<b>e</b>) quasiperiodic orbits with <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, (<b>f</b>) chaotic orbits with <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mn>1.2</mn> </mrow> </semantics></math>.</p> "> Figure 8
<p>Dynamical analysis with fixed <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mrow> <mi>c</mi> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math>, (<b>a</b>) Bifurcation of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>50</mn> </mrow> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mrow> <mi>c</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0.02</mn> </mrow> </semantics></math>, (<b>b</b>) <span class="html-italic">LE</span>s of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>50</mn> </mrow> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mrow> <mi>c</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0.02</mn> </mrow> </semantics></math>, (<b>c</b>) bifurcation of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>50</mn> </mrow> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mrow> <mi>c</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, (<b>d</b>) <span class="html-italic">LE</span>s of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mrow> <mn>50</mn> </mrow> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mrow> <mi>c</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> "> Figure 9
<p>Circuit implementation of system Equation (15), with electronic parameters: <math display="inline"><semantics> <mrow> <mrow> <mi mathvariant="normal">R</mi> <mn>1</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>2</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>3</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>5</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>6</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>7</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>8</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>10</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>11</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>12</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>13</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>14</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>15</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>17</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>19</mn> <mo>=</mo> <mn>10</mn> <mo> </mo> <mi mathvariant="normal">K</mi> <mi mathvariant="sans-serif">Ω</mi> <mo>;</mo> </mrow> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mi mathvariant="normal">R</mi> <mn>4</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>9</mn> <mo>=</mo> <mn>5</mn> <mo> </mo> <mi mathvariant="normal">K</mi> <mi mathvariant="sans-serif">Ω</mi> <mo>;</mo> </mrow> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mi mathvariant="normal">R</mi> <mn>16</mn> <mo>,</mo> <mi mathvariant="normal">R</mi> <mn>18</mn> <mo>=</mo> <mn>100</mn> <mo> </mo> <mi mathvariant="normal">K</mi> <mi mathvariant="sans-serif">Ω</mi> <mo>;</mo> </mrow> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mi mathvariant="normal">C</mi> <mn>1</mn> <mo>,</mo> <mi mathvariant="normal">C</mi> <mn>2</mn> <mo>,</mo> <mi mathvariant="normal">C</mi> <mn>3</mn> <mo>,</mo> <mi mathvariant="normal">C</mi> <mn>4</mn> <mo>,</mo> <mi mathvariant="normal">C</mi> <mn>5</mn> <mo>=</mo> <mn>10</mn> <mo> </mo> <mi>nF</mi> </mrow> <mo>.</mo> </mrow> </semantics></math></p> "> Figure 10
<p>Phase portrait of numerical simulation: (<b>a</b>) Phase portrait of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math>, (<b>b</b>) phase portrait of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>4</mn> </msub> </mrow> </semantics></math>.</p> "> Figure 11
<p>Implementation of circuit: (<b>a</b>) Phase portrait of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math>, (<b>b</b>) phase portrait of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>4</mn> </msub> </mrow> </semantics></math>, (<b>c</b>) implementation of the circuit, (<b>d</b>) implementation of the circuit.</p> ">
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
5.3. Dynamical Analysis with Fixed and Varied
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]
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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
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 StyleWang, 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