PaperNo Access

A New Memristive Chaotic System with a Plane and Two Lines of Equilibria

Shaanxi Engineering Research Center of Controllable Neutron Source, School of Science, Xijing University, Xi’an 710123, P. R. China

E-mail Address: williamchristian@163.com

Search for more papers by this author

Abdul Jalil M. Khalaf

Ministry of Higher Education and Scientific Research, Baghdad, Iraq

E-mail Address: abduljaleel.khalaf@uokufa.edu.iq

Search for more papers by this author

Sajad Jafari

https://orcid.org/0000-0002-6845-7539

Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413, Iran

E-mail Address: sajadjafari83@gmail.com

Search for more papers by this author

Shirin Panahi

Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413, Iran

E-mail Address: sh.panahi89@gmail.com

Search for more papers by this author

Chunbiao Li

School of Electronic and Information Engineering, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, P. R. China

E-mail Address: goontry@126.com

Search for more papers by this author

, and

Iqtadar Hussain

Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar

E-mail Address: iqtadarqau@gmail.com

Search for more papers by this author

https://doi.org/10.1142/S0218127421500668Cited by:4 (Source: Crossref)

Abstract

A new 4D memristive chaotic system with an infinite number of equilibria is proposed via exhaustive computer search. Interestingly, such a new memristive system has a plane of equilibria and two other lines of equilibria. Lyapunov exponent and bifurcation analysis show that this system has chaotic solutions with coexisting attractors. The basins of attraction of the coexisting attractors show chaos, stable fixed-points, and unbounded solutions. Furthermore, the 2D parameter space of the system is explored to find the optimum values of the parameters using the ALO (Ant Lion Optimizer) optimization algorithm.

Keywords:

References

Arecchi, F., Meucci, R., Puccioni, G. & Tredicce, J. [1982] “ Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a $q$ $q$ -switched gas laser,” Phys. Rev. Lett. 49, 1217. Crossref, Web of Science, Google Scholar
Bao, B., Bao, H., Wang, N., Chen, M. & Xu, Q. [2017a] “ Hidden extreme multistability in memristive hyperchaotic system,” Chaos Solit. Fract. 94, 102–111. Crossref, Web of Science, Google Scholar
Bao, B., Jiang, T., Wang, G., Jin, P., Bao, H. & Chen, M. [2017b] “ Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability,” Nonlin. Dyn. 89, 1157–1171. Crossref, Web of Science, Google Scholar
Bao, H., Chen, M., Wu, H. & Bao, B. [2019a] “ Memristor initial-boosted coexisting plane bifurcations and its extreme multi-stability reconstitution in two-memristor-based dynamical system,” Sci. China Tech. Sci. 63, 603–613. Crossref, Google Scholar
Bao, H., Hu, A., Liu, W. & Bao, B. [2019b] “ Hidden bursting firings and bifurcation mechanisms in memristive neuron model with threshold electromagnetic induction,” IEEE Trans. Neural Netw. Learn. Syst. 31, 502–511. Crossref, Web of Science, Google Scholar
Bao, H., Zhang, Y., Liu, W. & Bao, B. [2020] “ Memristor synapse-coupled memristive neuron network: Synchronization transition and occurrence of chimera,” Nonlin. Dyn. 100, 937–950. Crossref, Web of Science, Google Scholar
Barati, K., Jafari, S., Sprott, J. C. & Pham, V. -T. [2016] “ Simple chaotic flows with a curve of equilibria,” Int. J. Bifurcation and Chaos 26, 1630034-1–6. Link, Web of Science, Google Scholar
Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L. V. & Sciuto, G. [2012] “ Memristive chaotic circuits based on cellular nonlinear networks,” Int. J. Bifurcation and Chaos 22, 1250070-1–13. Link, Web of Science, Google Scholar
Chen, G. & Dong, X. [1998] From Chaos to Order: Methodologies, Perspectives and Applications, Vol. 24 (World Scientific). Link, Google Scholar
Chen, G. & Ueta, T. [1999] “ Yet another chaotic attractor,” Int. J. Bifurcation and chaos 9, 1465–1466. Link, Web of Science, Google Scholar
Chen, G., Moiola, J. L. & Wang, H. O. [2000] “ Bifurcation control: Theories, methods, and applications,” Int. J. Bifurcation and Chaos 10, 511–548. Link, Web of Science, Google Scholar
Chen, B., Zhu, Y., Hu, J. & Principe, J. C. [2013] System Parameter Identification: Information Criteria and Algorithms (Newnes). Google Scholar
Chen, M., Li, M., Yu, Q., Bao, B., Xu, Q. & Wang, J. [2015] “ Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chua’s circuit,” Nonlin. Dyn. 81, 215–226. Crossref, Web of Science, Google Scholar
Chen, M., Sun, M., Bao, H., Hu, Y. & Bao, B. [2019] “ Flux-charge analysis of two-memristor-based chua’s circuit: Dimensionality decreasing model for detecting extreme multistability,” IEEE Trans. Indust. Electron. 67, 2197–2206. Crossref, Web of Science, Google Scholar
Chua, L. [1971] “ Memristor-the missing circuit element,” IEEE Trans. Circuit Th. 18, 507–519. Crossref, Web of Science, Google Scholar
Chua, L. O. & Kang, S. M. [1976] “ Memristive devices and systems,” Proc. IEEE 64, 209–223. Crossref, Web of Science, Google Scholar
Danca, M.-F., Kuznetsov, N. & Chen, G. [2017] “ Unusual dynamics and hidden attractors of the Rabinovich–Fabrikant system,” Nonlin. Dyn. 88, 791–805. Crossref, Web of Science, Google Scholar
Feudel, U. [2008] “ Complex dynamics in multistable systems,” Int. J. Bifurcation and Chaos 18, 1607–1626. Link, Web of Science, Google Scholar
Geltrude, A., Al-Naimee, K., Euzzor, S., Meucci, R., Arecchi, F. T. & Goswami, B. [2010] “ Controlling multistability in chaotic systems,” 2010 Complexity in Engineering (IEEE), pp. 112–114. Crossref, Google Scholar
Gotthans, T., Sprott, J. C. & Petrzela, J. [2016] “ Simple chaotic flow with circle and square equilibrium,” Int. J. Bifurcation and Chaos 26, 1650137-1–8. Link, Web of Science, Google Scholar
Hilborn, R. C. et al. [2000] Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (Oxford University Press on Demand). Crossref, Google Scholar
Itoh, M. & Chua, L. O. [2008] “ Memristor oscillators,” Int. J. Bifurcation and Chaos 18, 3183–3206. Link, Web of Science, Google Scholar
Jafari, S., Golpayegani, S. M. R. H., Jafari, A. H. & Gharibzadeh, S. [2012] “ Some remarks on chaotic systems,” Int. J. Gen. Syst. 41, 329–330. Crossref, Web of Science, Google Scholar
Jafari, S., Golpayegani, S. M. R. H. & Daliri, A. [2013a] “ Comment on ‘parameters identification of chaotic systems by quantum-behaved particle swarm optimization’ [Int. J. Comput. Math. 86(12) (2009), pp. 2225–2235],” Int. J. Comput. Math. 90, 903–905. Crossref, Web of Science, Google Scholar
Jafari, S., Golpayegani, S. M. R. H. & Darabad, M. R. [2013b] “ Comment on ‘parameter identification and synchronization of fractional-order chaotic systems’ [Commun. Nonlin. Sci. Numer. Simulat. 2012; 17:305–16],” Commun. Nonlin. Sci. Numer. Simulat. 18, 811–814. Crossref, Web of Science, Google Scholar
Jafari, S., Sprott, J. C., Pham, V.-T., Golpayegani, S. M. R. H. & Jafari, A. H. [2014] “ A new cost function for parameter estimation of chaotic systems using return maps as fingerprints,” Int. J. Bifurcation and Chaos 24, 1450134-1–18. Link, Web of Science, Google Scholar
Jo, S. H., Chang, T., Ebong, I., Bhadviya, B. B., Mazumder, P. & Lu, W. [2010] “ Nanoscale memristor device as synapse in neuromorphic systems,” Nano Lett. 10, 1297–1301. Crossref, Web of Science, Google Scholar
Kundu, S., Majhi, S., Muruganandam, P. & Ghosh, D. [2018] “ Diffusion induced spiral wave chimeras in ecological system,” Eur. Phys. J. Special Topics 227, 983–993. Crossref, Web of Science, Google Scholar
Kuznetsov, N. V., Kuznetsova, O. A., Leonov, G. A. & Vagaytsev, V. [2011] “ Hidden attractor in Chua’s circuits,” Int. Conf. Inform. Contr. 1, 279–283. Google Scholar
Kuznetsov, N., Leonov, G., Mokaev, T., Prasad, A. & Shrimali, M. [2018] “ Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system,” Nonlin. Dyn. 92, 267–285. Crossref, Web of Science, Google Scholar
Li, Q., Hu, S., Tang, S. & Zeng, G. [2014] “ Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation,” Int. J. Circuit Th. Appl. 42, 1172–1188. Crossref, Web of Science, Google Scholar
Li, C. & Sprott, J. C. [2014] “ Multistability in the Lorenz system: A broken butterfly,” Int. J. Bifurcation and Chaos 24, 1450131-1–7. Link, Web of Science, Google Scholar
Li, Q., Zeng, H. & Li, J. [2015a] “ Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria,” Nonlin. Dyn. 79, 2295–2308. Crossref, Web of Science, Google Scholar
Li, X., Rakkiyappan, R. & Velmurugan, G. [2015b] “ Dissipativity analysis of memristor-based complex-valued neural networks with time-varying delays,” Inform. Sci. 294, 645–665. Crossref, Web of Science, Google Scholar
Lorenz, E. N. [1963] “ Deterministic nonperiodic flow,” J. Atmosph. Sci. 20, 130–141. Crossref, Web of Science, Google Scholar
Lorenz, E. [2000] “ The butterfly effect,” The Chaos Avant-Garde Memories of the Early Days of Chaos Theory, World Scientific Series in Nonlinear Science Series A, Vol. 39 (World Scientific), pp. 91–94. Google Scholar
Lü, J., Chen, G. & Cheng, D. [2004] “ A new chaotic system and beyond: The generalized Lorenz-like system,” Int. J. Bifurcation and Chaos 14, 1507–1537. Link, Web of Science, Google Scholar
Ma, J., Zhou, P., Ahmad, B., Ren, G. & Wang, C. [2018] “ Chaos and multi-scroll attractors in RCL-shunted junction coupled jerk circuit connected by memristor,” PLoS ONE 13, e0191120. Crossref, Web of Science, Google Scholar
Majhi, S., Perc, M. & Ghosh, D. [2017] “ Chimera states in a multilayer network of coupled and uncoupled neurons,” Chaos 27, 073109. Crossref, Web of Science, Google Scholar
Martínez, P. J., Euzzor, S., Meucci, R. & Chacón, R. [2020] “ Suppression of chaos by incommensurate excitations: Theory and experimental confirmations,” Commun. Nonlin. Sci. Numer. Simulat. 83, 105137. Crossref, Web of Science, Google Scholar
Mazumder, P., Kang, S.-M. & Waser, R. [2012] “ Memristors: Devices, models, and applications,” Proc. IEEE 100, 1911–1919. Crossref, Web of Science, Google Scholar
Mirjalili, S. [2015] “ The ant lion optimizer,” Adv. Engin. Softw. 83, 80–98. Crossref, Web of Science, Google Scholar
Panahi, S., Shirzadian, T., Jalili, M. & Jafari, S. [2019] “ A new chaotic network model for epilepsy,” Appl. Math. Comput. 346, 395–407. Crossref, Web of Science, Google Scholar
Peng, Y., Sun, K., He, S. & Yang, X. [2018] “ Parameter estimation of a complex chaotic system with unknown initial values,” Eur. Phys. J. Plus 133, 1–13. Crossref, Web of Science, Google Scholar
Peng, Y., Sun, K., He, S. & Alamodi, O. [2019a] “ The influence of samples on meta-heuristic algorithm for parameter estimation of chaotic system,” Mod. Phys. Lett. B 33, 1950041. Link, Web of Science, Google Scholar
Peng, Y., Sun, K., He, S. & Peng, D. [2019b] “ Parameter identification of fractional-order discrete chaotic systems,” Entropy 21, 27. Crossref, Web of Science, Google Scholar
Pham, V.-T., Volos, C. & Gambuzza, L. V. [2014] “ A memristive hyperchaotic system without equilibrium,” Sci. World J. 2014, 368986. Crossref, Web of Science, Google Scholar
Ruan, J., Sun, K., Mou, J., He, S. & Zhang, L. [2018] “ Fractional-order simplest memristor-based chaotic circuit with new derivative,” The Europ. Phys. J. Plus 133, 1–12. Crossref, Web of Science, Google Scholar
Shekofteh, Y., Jafari, S., Sprott, J. C., Golpayegani, S. M. R. H. & Almasganj, F. [2015] “ A Gaussian mixture model based cost function for parameter estimation of chaotic biological systems,” Commun. Nonlin. Sci. Numer. Simulat. 20, 469–481. Crossref, Web of Science, Google Scholar
Shekofteh, Y., Jafari, S. & Rajagopal, K. [2019a] “ Cost function based on hidden Markov models for parameter estimation of chaotic systems,” Soft Comput. 23, 4765–4776. Crossref, Web of Science, Google Scholar
Shekofteh, Y., Jafari, S., Rajagopal, K. & Pham, V.-T. [2019b] “ Parameter identification of chaotic systems using a modified cost function including static and dynamic information of attractors in the state space,” Circuits Syst. Sign. Process. 38, 2039–2054. Crossref, Web of Science, Google Scholar
Sinha, A., Malo, P. & Deb, K. [2017] “ A review on bilevel optimization: From classical to evolutionary approaches and applications,” IEEE Trans. Evolution. Comput. 22, 276–295. Crossref, Web of Science, Google Scholar
Sprott, J. C. [2010] Elegant Chaos: Algebraically Simple Chaotic Flows (World Scientific). Link, Google Scholar
Stankevich, N. V., Kuznetsov, N. V., Leonov, G. A. & Chua, L. O. [2017] “ Scenario of the birth of hidden attractors in the Chua circuit,” Int. J. Bifurcation and Chaos 27, 1730038-1–18. Link, Web of Science, Google Scholar
Strogatz, S. H. [2018] Nonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and Engineering (CRC Press). Crossref, Google Scholar
Vanderplaats, G. N. [2001] Numerical Optimization Techniques for Engineering Design (Vanderplaats Research & Development, Colorado Springs). Google Scholar
Wang, N., Li, C., Bao, H., Chen, M. & Bao, B. [2019] “Generating multi-scroll Chua’s attractors via simplified piecewise-linear Chua’s diode,” IEEE Trans. Circuits Syst.-I: Reg. Papers 66, 4767–4779. Google Scholar
Wei, Z. [2011] “ Dynamical behaviors of a chaotic system with no equilibria,” Phys. Lett. A 376, 102–108. Crossref, Web of Science, Google Scholar
Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. [1985] “ Determining Lyapunov exponents from a time series,” Physica D 16, 285–317. Crossref, Web of Science, Google Scholar