Multistability Dynamics Analysis and Digital Circuit Implementation of Entanglement-Chaos Symmetrical Memristive System
<p>Block diagram of constructing the entanglement-chaos system.</p> "> Figure 2
<p>Attractor trajectory diagrams of System (3). (<b>a</b>) <span class="html-italic">x</span>−<span class="html-italic">y</span> phase diagram; (<b>b</b>) <span class="html-italic">x</span>−<span class="html-italic">z</span> phase diagram.</p> "> Figure 3
<p>Bifurcation diagram and LEs of System (3) with <math display="inline"><semantics> <mrow> <mi mathvariant="normal">x</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>50</mn> <mo>,</mo> <mn>50</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math>. (<b>a</b>) Bifurcation diagram; (<b>b</b>) LEs.</p> "> Figure 4
<p>Pinned hysteresis loop of the memristor with different frequencies.</p> "> Figure 5
<p>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). (<b>a</b>) System phase diagram trajectory in the <span class="html-italic">x</span>−<span class="html-italic">y</span> plane; (<b>b</b>) system phase diagram trajectory in the <span class="html-italic">x</span>−<span class="html-italic">z</span> plane.</p> "> Figure 6
<p><span class="html-italic">p</span>−<span class="html-italic">s</span> plane diagrams in different projections of memristive chaotic System (2). (<b>a</b>) Initial values (0.1, 3, 2, 6); (<b>b</b>) initial values (0.1, 3, 2, 5).</p> "> Figure 7
<p>Bifurcation diagram and LEs of System (2) with initial condition [<span class="html-italic">x</span><sub>0</sub>, <span class="html-italic">y</span><sub>0</sub>, <span class="html-italic">z</span><sub>0</sub>, <span class="html-italic">w</span><sub>0</sub>] = [<span class="html-italic">x</span>(0), 3, 2, 5]. (<b>a</b>) Bifurcation diagram; (<b>b</b>) LEs.</p> "> Figure 8
<p>Various coexisting attractors with different values. (<b>a</b>) <span class="html-italic">x</span>(0) = −100 and <span class="html-italic">x</span>(0) = 100; (<b>b</b>) <span class="html-italic">x</span>(0) = 8 and <span class="html-italic">x</span>(0) = 10.</p> "> Figure 9
<p>Bifurcation diagram and LEs of System (2) with initial condition [<span class="html-italic">x</span><sub>0</sub>, <span class="html-italic">y</span><sub>0</sub>, <span class="html-italic">z</span><sub>0</sub>, <span class="html-italic">w</span><sub>0</sub>] = [0.1, 3, 2, <span class="html-italic">w</span>(0)<math display="inline"><semantics> <mo>∈</mo> </semantics></math>[−100, 100]]. (<b>a</b>) Bifurcation diagram; (<b>b</b>) LEs.</p> "> Figure 10
<p>Various coexisting attractors with different values: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mo>−</mo> <mn>100</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>75</mn> </mrow> </semantics></math>.</p> "> Figure 11
<p>Bifurcation diagram and LEs of System (5) with initial condition [<span class="html-italic">x</span>0, <span class="html-italic">y</span>0, <span class="html-italic">z</span>0, <span class="html-italic">w</span>0] = [0.1, 3, 2, <span class="html-italic">w</span>(0)<math display="inline"><semantics> <mo>∈</mo> </semantics></math>[−10,10]]. (<b>a</b>) Bifurcation diagram; (<b>b</b>) Les.</p> "> Figure 12
<p>Bifurcation and LEs of the system with a varying in [0, 10]. (<b>a</b>) Bifurcation of the system with [<span class="html-italic">x</span>(0), <span class="html-italic">y</span>(0), <span class="html-italic">z</span>(0), <span class="html-italic">w</span>(0)] = [0.1, 3, 2, 5]; (<b>b</b>) bifurcation of the system with [<span class="html-italic">x</span>(0), <span class="html-italic">y</span>(0), <span class="html-italic">z</span>(0), <span class="html-italic">w</span>(0)] = [0.1, 3, 2, 6]; (<b>c</b>) LEs of the system with [<span class="html-italic">x</span>(0), <span class="html-italic">y</span>(0), <span class="html-italic">z</span>(0), <span class="html-italic">w</span>(0)] = [0.1, 3, 2, 5]; (<b>d</b>) LEs of the system with [<span class="html-italic">x</span>(0), <span class="html-italic">y</span>(0), <span class="html-italic">z</span>(0), <span class="html-italic">w</span>(0)] = [0.1, 3, 2, 6].</p> "> Figure 12 Cont.
<p>Bifurcation and LEs of the system with a varying in [0, 10]. (<b>a</b>) Bifurcation of the system with [<span class="html-italic">x</span>(0), <span class="html-italic">y</span>(0), <span class="html-italic">z</span>(0), <span class="html-italic">w</span>(0)] = [0.1, 3, 2, 5]; (<b>b</b>) bifurcation of the system with [<span class="html-italic">x</span>(0), <span class="html-italic">y</span>(0), <span class="html-italic">z</span>(0), <span class="html-italic">w</span>(0)] = [0.1, 3, 2, 6]; (<b>c</b>) LEs of the system with [<span class="html-italic">x</span>(0), <span class="html-italic">y</span>(0), <span class="html-italic">z</span>(0), <span class="html-italic">w</span>(0)] = [0.1, 3, 2, 5]; (<b>d</b>) LEs of the system with [<span class="html-italic">x</span>(0), <span class="html-italic">y</span>(0), <span class="html-italic">z</span>(0), <span class="html-italic">w</span>(0)] = [0.1, 3, 2, 6].</p> "> Figure 13
<p>Complexity of the system with the parameters varying. (<b>a</b>) <span class="html-italic">x</span><sub>0</sub> varying; (<b>b</b>) <span class="html-italic">w</span><sub>0</sub> varying; (<b>c</b>) <span class="html-italic">a</span> varying.</p> "> Figure 14
<p>C0 complexity in the <span class="html-italic">w</span>(0)−<span class="html-italic">x</span>(0) plane.</p> "> Figure 15
<p>Oscilloscope display phase diagram of System (13) from initial values (0.1, 3, 2, 5). (<b>a</b>) Phase portraits in the <span class="html-italic">x</span>-<span class="html-italic">y</span> plane; (<b>b</b>) phase portraits in the <span class="html-italic">x</span>-<span class="html-italic">z</span> plane.</p> "> Figure 16
<p>Oscilloscope display phase diagram of systems (13) from initial values (0.1, 3, 2, 6). (<b>a</b>) Phase portraits in the <span class="html-italic">x</span>-<span class="html-italic">y</span> plane; (<b>b</b>) phase portraits in the <span class="html-italic">x</span>-<span class="html-italic">z</span> plane.</p> ">
Abstract
:1. Introduction
2. Description of Entanglement-Chaos System
3. Analysis of Memristive Entanglement System
3.1. Memristive Entanglement System
3.2. Dissipative Analysis
3.3. Equilibrium Stability Analysis
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
- Chua, L.O. Memristor—The missing circuit element. IEEE Trans. Circuit Theory 1971, 18, 507–519. [Google Scholar] [CrossRef]
- 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]
- Ebong, I.E.; Mazumder, P. CMOS and memristor-based neural network design for position detection. Proc. IEEE 2012, 100, 2050–2060. [Google Scholar] [CrossRef]
- Yoon, J.H.; Wang, Z.; Kim, K.M.; Wu, H.; Ravichandran, V.; Xia, Q.; Hwang, C.S.; Yang, J.J. An artificial nociceptor based on a diffusive memristor. Nat. Commun. 2018, 9, 417. [Google Scholar] [CrossRef] [Green Version]
- Itoh, M.; Chua, L.O. Memristor oscillators. Int. J. Bifurc. Chaos 2008, 18, 3183–3206. [Google Scholar] [CrossRef]
- Bao, B.; Zhu, Y.; Ma, J.; Bao, H.; Wu, H.; Chen, M. Memristive neuron model with an adapting synapse and its hardware experiments. Sci. China Technol. Sci. 2021, 64, 1107–1117. [Google Scholar] [CrossRef]
- Chen, M.; Li, M.; Yu, Q.; Bao, B.C.; Xu, Q.; Wang, J. Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chua’s circuit. Nonlinear Dyn. 2015, 81, 215–226. [Google Scholar] [CrossRef]
- Yao, W.; Wang, C.; Sun, Y.; Zhou, C. Exponential multistability of memristive Cohen-Grossberg neural networks with stochastic parameter perturbations. Appl. Math. Comput. 2020, 386, 125483. [Google Scholar] [CrossRef]
- He, S.; Sun, K.; Peng, Y.; Wang, L. Modeling of discrete fracmemristor and its application. AIP Adv. 2020, 10, 015332. [Google Scholar] [CrossRef]
- Chen, M.; Sun, M.; Bao, H.; Hu, Y.; Bao, B. Flux-Charge Analysis of Two-Memristor-Based Chua’s Circuit: Dimensionality Decreasing Model for Detecting Extreme Multistability. IEEE Trans. Ind. Electron. 2019, 67, 2197–2206. [Google Scholar] [CrossRef]
- Bao, B.C.; Hu, F.W.; Chen, M.; Xu, Q.; Yu, Y.J. Selfexcited and hidden attractors found simultaneously in a modified Chua’s circuit. Int. J. Bifurc. Chaos 2015, 25, 1550075. [Google Scholar] [CrossRef]
- Chen, M.; Yu, J.; Bao, B.C. Finding hidden attractors in improved memristor-based Chua’s circuit. Electron. Lett. 2015, 51, 462–464. [Google Scholar] [CrossRef]
- Leonov, G.A.; Kuznetsov, N.V.; Vagaitsev, V.I. Localization of hidden Chua’s attractors. Phys. Lett. A 2011, 375, 2230–2233. [Google Scholar] [CrossRef]
- Zhang, X.; Li, C.; Chen, Y.; Herbert, H.C.I.U. A memristive chaotic oscillator with controllable amplitude and frequency. Chaos Solitons Fractals 2020, 139, 110000. [Google Scholar] [CrossRef]
- Huang, L.; Zhang, X.; Zang, H.; Lei, T.; Fu, H. An Offset-Boostable Chaotic Oscillator with Broken Symmetry. Symmetry 2022, 14, 1903. [Google Scholar] [CrossRef]
- Deng, Q.; Wang, C.; Yang, L. Four-Wing Hidden Attractors with One Stable Equilibrium Point. Int. J. Bifurc. Chaos 2020, 30, 2050086. [Google Scholar] [CrossRef]
- Chen, M.; Wang, C.; Bao, H.; Ren, X.; Bao, B.C. Reconstitution for interpreting hidden dynamics with stable equilibrium point. Chaos Solitons Fractals 2020, 140, 110188. [Google Scholar] [CrossRef]
- Kuznetsov, N.V.; Leonov, G.A. Hidden attractors in dynamical systems: Systems with no equilibria, multistability and coexisting attractors. IFAC Proc. Vol. 2014, 47, 5445–5454. [Google Scholar] [CrossRef] [Green Version]
- Pham, V.-T.; Volos, C.; Jafari, S.; Kapitaniak, T. Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dyn. 2017, 87, 2001–2010. [Google Scholar] [CrossRef]
- Zhang, S.; Zeng, Y. A simple Jerk-like system without equilibrium: Asymmetric coexisting hidden attractors, bursting oscillation and double full Feigenbaum remerging trees. Chaos Solitons Fractals 2019, 120, 25–40. [Google Scholar] [CrossRef]
- Wei, Z.; Yousefpour, A.; Jahanshahi, H.; Kocamaz, U.E.; Moroz, I. Hopf bifurcation and synchronization of a five-dimensional self-exciting homopolar disc dynamo using a new fuzzy disturbance-observer-based terminal sliding mode control. J. Frankl. Inst. 2021, 358, 814–833. [Google Scholar] [CrossRef]
- Peng, Y.; He, S.; Sun, K. A higher dimensional chaotic map with discrete memristor. AEU-Int. J. Electron. Commun. 2020, 129, 153539. [Google Scholar] [CrossRef]
- Lai, Q.; Akgul, A.; Zhao, X.; Pei, H. Various types of coexisting attractors in a new 4D autonomous chaotic system. Int. J. Bifurc. Chaos 2017, 27, 1750142. [Google Scholar] [CrossRef]
- Zhang, X.; Li, Z. Hidden extreme multistability in a novel 4D fractional-order chaotic system. Int. J. Non-Linear Mech. 2019, 111, 14–27. [Google Scholar] [CrossRef]
- Zhang, S.; Zeng, Y.; Li, Z.; Wang, M.; Le, X. Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability. Chaos 2018, 28, 013113. [Google Scholar] [CrossRef] [PubMed]
- Li, Q.; Zeng, H.; Li, J. Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria. Nonlinear Dyn. 2015, 79, 2295–2308. [Google Scholar] [CrossRef]
- Chen, H.; Lei, T.; Lu, S.; Dai, W.; Qiu, L.; Zhong, L. Dynamics and Complexity Analysis of Fractional-Order Chaotic Systems with Line Equilibrium Based on Adomian Decomposition. Complexity 2020, 2020, 5710765. [Google Scholar] [CrossRef]
- Zhang, H.; Sun, K.; He, S. A fractional-order ship power system with extreme multistability. Nonlinear Dyn. 2021, 106, 1027–1040. [Google Scholar] [CrossRef]
- Lu, Y.M.; Min, F.H. Dynamic analysis of symmetric behavior in flux-controlled memristor circuit based on field programmable gate array. Acta Phys. Sin. 2019, 68, 130502. [Google Scholar] [CrossRef]
- Li, H.M.; Yang, Y.F.; Li, W.; He, S.B.; Li, C.L. Extremely rich dynamics in a memristor-based chaotic system. Eur. Phys. J. Plus 2020, 135, 579. [Google Scholar] [CrossRef]
- Yuan, F.; Wang, G.; Wang, X. Extreme multistability in a memristor-based multi-scroll hyper-chaotic system. Chaos 2016, 26, 073107. [Google Scholar] [CrossRef] [PubMed]
- Zhang, H.; Liu, X.; Shen, X.; Liu, J. Chaos entanglement: A new approach to generate chaos. Int. J. Bifurc. Chaos 2013, 23, 30014. [Google Scholar] [CrossRef]
- Lei, T.; Mao, B.; Zhou, X.; Fu, H. Dynamics Analysis and Synchronous Control of Fractional-Order Entanglement Symmetrical Chaotic Systems. Symmetry 2021, 13, 1996. [Google Scholar] [CrossRef]
- Ventra, M.D.; Pershin, Y.V.; Chua, L.O. Circuit elements with memory: Memristors, memcapacitors, and meminductors. Proc. IEEE 2009, 97, 1717–1724. [Google Scholar] [CrossRef] [Green Version]
- Muthuswamy, B. Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 2010, 20, 1335–1350. [Google Scholar] [CrossRef]
- Iu, H.H.C.; Yu, D.S.; Fitch, A.L.; Sreeram, V.; Chen, H. Controlling Chaos in a Memristor Based Circuit Using a Twin-T Notch Filter. IEEE Trans. Circuits Syst. I Regul. Pap. 2011, 58, 1337–1344. [Google Scholar] [CrossRef]
- Sun, K.H.; Liu, X.; Zhu, C.X. 0-1 test algorithm for chaos and its applications. Chin. Phys. B 2010, 19, 110510. [Google Scholar] [CrossRef]
- Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov exponents from a time series. Physica 1985, 16, 285–317. [Google Scholar] [CrossRef]
- Phillip, P.A.; Chiu, F.L.; Nick, S.J. Rapidly detecting disorder in rhythmic biological signals: A spectral entropy measure to identify cardiac arrhythmias. Phys. Rev. E 2009, 79, 011915. [Google Scholar]
- Shen, E.H.; Cai, Z.J.; Gu, F.J. Mathematical foundation of a new complexity measure. Appl. Math. Mech. 2005, 26, 1188–1196. [Google Scholar]
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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
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 StyleLei, 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
APA StyleLei, T., Zhou, Y., Fu, H., Huang, L., & Zang, H. (2022). Multistability Dynamics Analysis and Digital Circuit Implementation of Entanglement-Chaos Symmetrical Memristive System. Symmetry, 14(12), 2586. https://doi.org/10.3390/sym14122586