A new memristive chaotic system with a plane and two lines of equilibria. (English) Zbl 1469.34070
Summary: 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.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 94C60 | Circuits in qualitative investigation and simulation of models |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
Software:
ALOReferences:
| [1] | Arecchi, F., Meucci, R., Puccioni, G. & Tredicce, J. [1982] “ Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a \(q\)-switched gas laser,” Phys. Rev. Lett.49, 1217. |
| [2] | Bao, B., Bao, H., Wang, N., Chen, M. & Xu, Q. [2017a] “ Hidden extreme multistability in memristive hyperchaotic system,” Chaos Solit. Fract.94, 102-111. · Zbl 1373.34069 |
| [3] | 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. |
| [4] | 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. |
| [5] | 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. |
| [6] | 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. · Zbl 1434.92015 |
| [7] | Barati, K., Jafari, S., Sprott, J. C. & Pham, V. -T. [2016] “ Simple chaotic flows with a curve of equilibria,” Int. J. Bifurcation and Chaos26, 1630034-1-6. · Zbl 1352.34015 |
| [8] | Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L. V. & Sciuto, G. [2012] “ Memristive chaotic circuits based on cellular nonlinear networks,” Int. J. Bifurcation and Chaos22, 1250070-1-13. |
| [9] | Chen, G. & Dong, X. [1998] From Chaos to Order: Methodologies, Perspectives and Applications, Vol. 24 (World Scientific). · Zbl 0908.93005 |
| [10] | Chen, G. & Ueta, T. [1999] “ Yet another chaotic attractor,” Int. J. Bifurcation and chaos9, 1465-1466. · Zbl 0962.37013 |
| [11] | Chen, G., Moiola, J. L. & Wang, H. O. [2000] “ Bifurcation control: Theories, methods, and applications,” Int. J. Bifurcation and Chaos10, 511-548. · Zbl 1090.37552 |
| [12] | Chen, B., Zhu, Y., Hu, J. & Principe, J. C. [2013] System Parameter Identification: Information Criteria and Algorithms (Newnes). |
| [13] | 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. · Zbl 1430.94107 |
| [14] | 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. |
| [15] | Chua, L. [1971] “ Memristor-the missing circuit element,” IEEE Trans. Circuit Th.18, 507-519. |
| [16] | Chua, L. O. & Kang, S. M. [1976] “ Memristive devices and systems,” Proc. IEEE64, 209-223. |
| [17] | Danca, M.-F., Kuznetsov, N. & Chen, G. [2017] “ Unusual dynamics and hidden attractors of the Rabinovich-Fabrikant system,” Nonlin. Dyn.88, 791-805. |
| [18] | Feudel, U. [2008] “ Complex dynamics in multistable systems,” Int. J. Bifurcation and Chaos18, 1607-1626. |
| [19] | 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. · Zbl 1243.93036 |
| [20] | Gotthans, T., Sprott, J. C. & Petrzela, J. [2016] “ Simple chaotic flow with circle and square equilibrium,” Int. J. Bifurcation and Chaos26, 1650137-1-8. · Zbl 1345.34016 |
| [21] | Hilborn, R. C.et al. [2000] Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (Oxford University Press on Demand). · Zbl 1015.37001 |
| [22] | Itoh, M. & Chua, L. O. [2008] “ Memristor oscillators,” Int. J. Bifurcation and Chaos18, 3183-3206. · Zbl 1165.94300 |
| [23] | 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. · Zbl 1303.37014 |
| [24] | 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. · Zbl 1280.65072 |
| [25] | 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. · Zbl 1277.93030 |
| [26] | 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 Chaos24, 1450134-1-18. · Zbl 1302.34027 |
| [27] | 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. |
| [28] | Kundu, S., Majhi, S., Muruganandam, P. & Ghosh, D. [2018] “ Diffusion induced spiral wave chimeras in ecological system,” Eur. Phys. J. Special Topics227, 983-993. |
| [29] | 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. |
| [30] | 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. · Zbl 1398.37090 |
| [31] | 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. |
| [32] | Li, C. & Sprott, J. C. [2014] “ Multistability in the Lorenz system: A broken butterfly,” Int. J. Bifurcation and Chaos24, 1450131-1-7. · Zbl 1302.34015 |
| [33] | Li, Q., Zeng, H. & Li, J. [2015a] “ Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria,” Nonlin. Dyn.79, 2295-2308. |
| [34] | 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. · Zbl 1360.93638 |
| [35] | Lorenz, E. N. [1963] “ Deterministic nonperiodic flow,” J. Atmosph. Sci.20, 130-141. · Zbl 1417.37129 |
| [36] | Lorenz, E. [2000] “ The butterfly effect,” The Chaos Avant-Garde Memories of the Early Days of Chaos Theory, , Vol. 39 (World Scientific), pp. 91-94. · Zbl 1001.37089 |
| [37] | Lü, J., Chen, G. & Cheng, D. [2004] “ A new chaotic system and beyond: The generalized Lorenz-like system,” Int. J. Bifurcation and Chaos14, 1507-1537. · Zbl 1129.37323 |
| [38] | 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 ONE13, e0191120. |
| [39] | Majhi, S., Perc, M. & Ghosh, D. [2017] “ Chimera states in a multilayer network of coupled and uncoupled neurons,” Chaos27, 073109. |
| [40] | 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. · Zbl 1453.37073 |
| [41] | Mazumder, P., Kang, S.-M. & Waser, R. [2012] “ Memristors: Devices, models, and applications,” Proc. IEEE100, 1911-1919. |
| [42] | Mirjalili, S. [2015] “ The ant lion optimizer,” Adv. Engin. Softw.83, 80-98. |
| [43] | Panahi, S., Shirzadian, T., Jalili, M. & Jafari, S. [2019] “ A new chaotic network model for epilepsy,” Appl. Math. Comput.346, 395-407. · Zbl 1428.92050 |
| [44] | Peng, Y., Sun, K., He, S. & Yang, X. [2018] “ Parameter estimation of a complex chaotic system with unknown initial values,” Eur. Phys. J. Plus133, 1-13. |
| [45] | 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. B33, 1950041. |
| [46] | Peng, Y., Sun, K., He, S. & Peng, D. [2019b] “ Parameter identification of fractional-order discrete chaotic systems,” Entropy21, 27. |
| [47] | Pham, V.-T., Volos, C. & Gambuzza, L. V. [2014] “ A memristive hyperchaotic system without equilibrium,” Sci. World J.2014, 368986. |
| [48] | 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. Plus133, 1-12. |
| [49] | 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. |
| [50] | 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. |
| [51] | 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. |
| [52] | 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. |
| [53] | Sprott, J. C. [2010] Elegant Chaos: Algebraically Simple Chaotic Flows (World Scientific). · Zbl 1222.37005 |
| [54] | 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 Chaos27, 1730038-1-18. · Zbl 1379.34045 |
| [55] | Strogatz, S. H. [2018] Nonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and Engineering (CRC Press). |
| [56] | Vanderplaats, G. N. [2001] Numerical Optimization Techniques for Engineering Design (Vanderplaats Research & Development, Colorado Springs). · Zbl 0613.90062 |
| [57] | 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. Papers66, 4767-4779. |
| [58] | Wei, Z. [2011] “ Dynamical behaviors of a chaotic system with no equilibria,” Phys. Lett. A376, 102-108. · Zbl 1255.37013 |
| [59] | Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. [1985] “ Determining Lyapunov exponents from a time series,” Physica D16, 285-317. · Zbl 0585.58037 |
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