PapersNo Access

A New Megastable Oscillator with Rational and Irrational Parameters

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

Search for more papers by this author

Ibrahim Ismael Hamarash

Department of Computer Science and Engineering, University of Kurdistan-Hewler, Iraq

Search for more papers by this author

Payam Sadeghi Shabestari

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

Search for more papers by this author

, and

Sajad Jafari

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

Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

E-mail Address: sajad.jafari@tdtu.edu.vn

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218127419501761Cited by:15 (Source: Crossref)

Abstract

In this paper, a new two-dimensional nonlinear oscillator with an unusual sequence of rational and irrational parameters is introduced. This oscillator has endless coexisting limit cycles, which make it a megastable dynamical system. By periodically forcing this system, a new system is designed which is capable of exhibiting an infinite number of coexisting asymmetric torus and strange attractors. This system is implemented by an analog circuit, and its Hamiltonian energy is calculated.

Keywords:

References

Ai, W., Sun, K. & Fu, Y. [2018] “ Design of multiwing-multiscroll grid compound chaotic system and its circuit implementation,” Int. J. Mod. Phys. C 29, 1850049. Link, Web of Science, Google Scholar
Aram, Z., Jafari, S., Ma, J., Sprott, J. C., Zendehrouh, S. & Pham, V.-T. [2017] “ Using chaotic artificial neural networks to model memory in the brain,” Commun. Nonlin. Sci. Numer. Simulat. 44, 449–459. Crossref, Web of Science, Google Scholar
Bao, B., Jiang, T., Xu, Q., Chen, M., Wu, H. & Hu, Y. [2016a] “ Coexisting infinitely many attractors in active band-pass filter-based memristive circuit,” Nonlin. Dyn. 86, 1711–1723. Crossref, Web of Science, Google Scholar
Bao, B.-C., Xu, Q., Bao, H. & Chen, M. [2016b] “ Extreme multistability in a memristive circuit,” Electron. Lett. 52, 1008–1010. Crossref, Web of Science, Google Scholar
Bao, B., Jiang, T., Wang, G., Jin, P., Bao, H. & Chen, M. [2017] “ 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., Wang, N., Bao, B., Chen, M., Jin, P. & Wang, G. [2018] “ Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria,” Commun. Nonlin. Sci. Numer. Simulat. 57, 264–275. 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
Borah, M. & Roy, B. K. [2017] “ An enhanced multi-wing fractional-order chaotic system with coexisting attractors and switching hybrid synchronisation with its nonautonomous counterpart,” Chaos Solit. Fract. 102, 372–386. Crossref, Web of Science, Google Scholar
Bouali, S., Buscarino, A., Fortuna, L., Frasca, M. & Gambuzza, L. [2012] “ Emulating complex business cycles by using an electronic analogue,” Nonlin. Anal.: Real World Appl. 13, 2459–2465. Crossref, Web of Science, Google Scholar
Buscarino, A., Fortuna, L., Frasca, M. & Sciuto, G. [2011] “Design of time-delay chaotic electronic circuits,” IEEE Trans. Circuits Syst.-I: Reg. Papers 58, 1888–1896. Google Scholar
Buscarino, A., Fortuna, L., Frasca, M., Sciuto, G. et al. [2014] A Concise Guide to Chaotic Electronic Circuits (Springer). Crossref, Google Scholar
Cafagna, D. & Grassi, G. [2014] “ Chaos in a new fractional-order system without equilibrium points,” Commun. Nonlin. Sci. Numer. Simulat. 19, 2919–2927. Crossref, Web of Science, Google Scholar
Chen, M., Sun, M., Bao, B., Wu, H., Xu, Q. & Wang, J. [2018] “ Controlling extreme multistability of memristor emulator-based dynamical circuit in flux-charge domain,” Nonlin. Dyn. 91, 1395–1412. Crossref, Web of Science, Google Scholar
Gotthans, T. & Petržela, J. [2015] “ New class of chaotic systems with circular equilibrium,” Nonlin. Dyn. 81, 1143–1149. Crossref, Web of Science, Google Scholar
He, S.-B., Sun, K.-H. & Zhu, C.-X. [2013] “ Complexity analyses of multi-wing chaotic systems,” Chinese Phys. B 22, 050506. Crossref, Web of Science, Google Scholar
He, S., Li, C., Sun, K. & Jafari, S. [2018] “ Multivariate multiscale complexity analysis of self-reproducing chaotic systems,” Entropy 20, 556. Crossref, Web of Science, Google Scholar
He, S., Sun, K. & Wang, H. [2019] “ Dynamics and synchronization of conformable fractional-order hyperchaotic systems using the homotopy analysis method,” Commun. Nonlin. Sci. Numer. Simulat. 73, 146–164. Crossref, Web of Science, Google Scholar
Jafari, S. & Sprott, J. C. [2013] “ Simple chaotic flows with a line equilibrium,” Chaos Solit. Fract. 57, 79–84. Crossref, Web of Science, Google Scholar
Jafari, S., Sprott, J. C. & Golpayegani, S. M. R. H. [2013] “ Elementary quadratic chaotic flows with no equilibria,” Phys. Lett. A 377, 699–702. Crossref, Web of Science, Google Scholar
Jafari, S., Sprott, J. C., Pham, V.-T., Volos, C. & Li, C. [2016a] “ Simple chaotic 3D flows with surfaces of equilibria,” Nonlin. Dyn. 86, 1349–1358. Crossref, Web of Science, Google Scholar
Jafari, S., Sprott, J. C. & Molaie, M. [2016b] “ A simple chaotic flow with a plane of equilibria,” Int. J. Bifurcation and Chaos 26, 1650098. Link, Web of Science, Google Scholar
Kapitaniak, T. & Leonov, G. [2015] “ Multistability: Uncovering hidden attractors,” Eur. Phys. J. Special Topics 224, 1405–1408. Crossref, Web of Science, Google Scholar
Kundu, S., Majhi, S., Karmakar, P., Ghosh, D. & Rakshit, B. [2018] “ Augmentation of dynamical persistence in networks through asymmetric interaction,” EPL (Europhys. Lett.) 123, 30001. Crossref, Google Scholar
Kuznetsov, N. V., Leonov, G. A. & Vagaitsev, V. I. [2010] “ Analytical-numerical method for attractor localization of generalized Chua’s system,” IFAC Proc. Vol. 43, 29–33. Crossref, Google Scholar
Kuznetsov, N., Mokaev, T. & Vasilyev, P. [2014] “ Numerical justification of Leonov conjecture on Lyapunov dimension of Rössler attractor,” Commun. Nonlin. Sci. Numer. Simulat. 19, 1027–1034. Crossref, Web of Science, Google Scholar
Lai, Q. & Wang, L. [2016] “ Chaos, bifurcation, coexisting attractors and circuit design of a three-dimensional continuous autonomous system,” Optik 127, 5400–5406. Crossref, Web of Science, Google Scholar
Lai, Q. & Chen, S. [2016a] “ Generating multiple chaotic attractors from Sprott b system,” Int. J. Bifurcation and Chaos 26, 1650177-1–13. Link, Web of Science, Google Scholar
Lai, Q. & Chen, S. [2016b] “ Research on a new 3D autonomous chaotic system with coexisting attractors,” Optik 127, 3000–3004. Crossref, Web of Science, Google Scholar
Leonov, G. A. & Kuznetsov, N. V. [2007] “ Time-varying linearization and the Perron effects,” Int. J. Bifurcation and Chaos 17, 1079–1107. Link, Web of Science, Google Scholar
Li, X. & Fu, X. [2011] “ Synchronization of chaotic delayed neural networks with impulsive and stochastic perturbations,” Commun. Nonlin. Sci. Numer. Simulat. 16, 885–894. Crossref, Web of Science, Google Scholar
Li, X. & Rakkiyappan, R. [2013] “ Impulsive controller design for exponential synchronization of chaotic neural networks with mixed delays,” Commun. Nonlin. Sci. Numer. Simulat. 18, 1515–1523. Crossref, Web of Science, Google Scholar
Li, C., Wu, L., Li, H. & Tong, Y. [2013] “ A novel chaotic system and its topological horseshoe,” Nonlin. Anal.: Model. Contr. 18, 66–77. Crossref, Web of Science, Google Scholar
Li, C., Hu, W., Sprott, J. C. & Wang, X. [2015a] “ Multistability in symmetric chaotic systems,” Eur. Phys. J. Special Topics 224, 1493–1506. Crossref, Web of Science, Google Scholar
Li, C., Sprott, J. C., Yuan, Z. & Li, H. [2015b] “ Constructing chaotic systems with total amplitude control,” Int. J. Bifurcation and Chaos 25, 1530025-1–14. Link, Web of Science, Google Scholar
Li, C., Sprott, J. C., Hu, W. & Xu, Y. [2017a] “ Infinite multistability in a self-reproducing chaotic system,” Int. J. Bifurcation and Chaos 27, 1750160-1–11. Link, Web of Science, Google Scholar
Li, C., Sprott, J. C. & Mei, Y. [2017b] “ An infinite 2D lattice of strange attractors,” Nonlin. Dyn. 89, 2629–2639. Crossref, Web of Science, Google Scholar
Li, C.-B., Thio, W. J.-C., Sprott, J. C., Zhang, R.-X. & Lu, T.-A. [2017c] “ Linear synchronization and circuit implementation of chaotic system with complete amplitude control,” Chinese Phys. B 26, 120501. Crossref, Web of Science, Google Scholar
Li, C., Thio, W. J.-C., Sprott, J. C., Iu, H. H.-C. & Xu, Y. [2018] “ Constructing infinitely many attractors in a programmable chaotic circuit,” IEEE Access 6, 29003–29012. Crossref, Web of Science, Google Scholar
Liu, Z., Wu, F., Alzahrani, F. & Ma, J. [2018] “ Control of multi-scroll attractors in a memristor-coupled resonator via time-delayed feedback,” Mod. Phys. Lett. B 32, 1850399. 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. & Ghosh, D. [2018] “ Alternating chimeras in networks of ephaptically coupled bursting neurons,” Chaos 28, 083113. Crossref, Web of Science, Google Scholar
Majhi, S., Bera, B. K., Ghosh, D. & Perc, M. [2018] “ Chimera states in neuronal networks: A review,” Phys. Life Rev. 28, 100–121. Crossref, Web of Science, Google Scholar
Majhi, S., Ghosh, D. & Kurths, J. [2019a] “ Emergence of synchronization in multiplex networks of mobile Rössler oscillators,” Phys. Rev. E 99, 012308. Crossref, Web of Science, Google Scholar
Majhi, S., Kapitaniak, T. & Ghosh, D. [2019b] “ Solitary states in multiplex networks owing to competing interactions,” Chaos 29, 013108. Crossref, Web of Science, Google Scholar
Molaie, M., Jafari, S., Sprott, J. C. & Golpayegani, S. M. R. H. [2013] “ Simple chaotic flows with one stable equilibrium,” Int. J. Bifurcation and Chaos 23, 1350188-1–7. Link, Web of Science, Google Scholar
Panahi, S., Aram, Z., Jafari, S., Ma, J. & Sprott, J. [2017] “ Modeling of epilepsy based on chaotic artificial neural network,” Chaos Solit. Fract. 105, 150–156. Crossref, Web of Science, Google Scholar
Peng, D., Sun, K., He, S. & Alamodi, A. O. [2019] “ What is the lowest order of the fractional-order chaotic systems to behave chaotically?,” Chaos Solit. Fract. 119, 163–170. Crossref, Web of Science, Google Scholar
Pham, V.-T., Jafari, S., Wang, X. & Ma, J. [2016] “ A chaotic system with different shapes of equilibria,” Int. J. Bifurcation and Chaos 26, 1650069-1–5. Web of Science, Google Scholar
Pisarchik, A. N. & Feudel, U. [2014] “ Control of multistability,” Phys. Rep. 540, 167–218. Crossref, Web of Science, Google Scholar
Preissl, H., Lutzenberger, W. & Pulvermüller, F. [1996] “ Is there chaos in the brain?,” Behav. Brain Sci. 19, 307–308. Crossref, Web of Science, Google Scholar
Singh, J. P., Roy, B. K. & Jafari, S. [2018] “ New family of 4D hyperchaotic and chaotic systems with quadric surfaces of equilibria,” Chaos Solit. Fract. 106, 243–257. Crossref, Web of Science, Google Scholar
Sprott, J. C. [2010] Elegant Chaos: Algebraically Simple Chaotic Flows (World Scientific). Link, Google Scholar
Sprott, J. C., Jafari, S., Khalaf, A. J. M. & Kapitaniak, T. [2017] “ Megastability: Coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping,” Eur. Phys. J. Special Topics 226, 1979–1985. Crossref, Web of Science, Google Scholar
Tang, Y., Abdolmohammadi, H. R., Khalaf, A. J. M., Tian, Y. & Kapitaniak, T. [2018a] “ Carpet oscillator: A new megastable nonlinear oscillator with infinite islands of self-excited and hidden attractors,” Pramana 91, 11. Crossref, Web of Science, Google Scholar
Tang, Y.-X., Khalaf, A. J. M., Rajagopal, K., Pham, V.-T., Jafari, S. & Tian, Y. [2018b] “ A new nonlinear oscillator with infinite number of coexisting hidden and self-excited attractors,” Chinese Phys. B 27, 040502. Crossref, Web of Science, Google Scholar
Wang, X. & Chen, G. [2012] “ A chaotic system with only one stable equilibrium,” Commun. Nonlin. Sci. Numer. Simulat. 17, 1264–1272. Crossref, Web of Science, Google Scholar
Wang, X. & Chen, G. [2013] “ Constructing a chaotic system with any number of equilibria,” Nonlin. Dyn. 71, 429–436. Crossref, Web of Science, Google Scholar
Wang, Z., Abdolmohammadi, H. R., Alsaadi, F. E., Hayat, T. & Pham, V.-T. [2018] “ A new oscillator with infinite coexisting asymmetric attractors,” Chaos Solit. Fract. 110, 252–258. Crossref, Web of Science, 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
Wei, Z. & Wang, Z. [2013] “ Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium,” Kybernetika 49, 359–374. Web of Science, Google Scholar
Wei, Z., Tang, Y., Chen, H. & Pehlivan, I. [2013] “ Adaptive reduced-order function projective synchronization and circuit design of hyperchaotic DLE with no equilibria,” Optoelectron. Adv. Mater.-Rapid Commun. 7, 984–990. Web of Science, Google Scholar
Wei, Z., Pham, V.-T., Khalaf, A. J. M., Kengne, J. & Jafari, S. [2018] “ A modified multistable chaotic oscillator,” Int. J. Bifurcation and Chaos 28, 1850085-1–9. Link, Web of Science, Google Scholar
Wu, F., Zhou, P., Alsaedi, A., Hayat, T. & Ma, J. [2018] “ Synchronization dependence on initial setting of chaotic systems without equilibria,” Chaos Solit. Fract. 110, 124–132. Crossref, Web of Science, Google Scholar
Wu, F., Ma, J. & Zhang, G. [2019] “ A new neuron model under electromagnetic field,” Appl. Math. Comput. 347, 590–599. Crossref, Web of Science, Google Scholar
Zhang, G., Wu, F., Wang, C. & Ma, J. [2017a] “ Synchronization behaviors of coupled systems composed of hidden attractors,” Int. J. Mod. Phys. B 31, 1750180. Link, Web of Science, Google Scholar
Zhang, L., Sun, K., He, S., Wang, H. & Xu, Y. [2017b] “ Solution and dynamics of a fractional-order 5D hyperchaotic system with four wings,” Eur. Phys. J. Plus 132, 31. Crossref, Web of Science, Google Scholar
Zhang, G., Wang, C., Alsaedi, A., Ma, J. & Ren, G. [2018] “ Dependence of hidden attractors on non-linearity and Hamilton energy in a class of chaotic system,” Kybernetika 54, 648–663. Web of Science, Google Scholar
Zhou, P. & Huang, K. [2014] “ A new 4D non-equilibrium fractional-order chaotic system and its circuit implementation,” Commun. Nonlin. Sci. Numer. Simulat. 19, 2005–2011. Crossref, Web of Science, Google Scholar
Zhou, P. & Yang, F. [2014] “ Hyperchaos, chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points,” Nonlin. Dyn. 76, 473–480. Crossref, Web of Science, Google Scholar
Zhou, P. & Ke, M. [2017] “ A new 3D autonomous continuous system with two isolated chaotic attractors and its topological horseshoes,” Complexity 2017, 4037682. Crossref, Web of Science, Google Scholar