A megastable oscillator with two types of attractors

Document Type : Research Article

Authors

1 Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai, India

2 Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

3 Department of Computer Technology Engineering, College of Information Technology, Imam Ja'afar Al-Sadiq University, Baghdad, Iraq

4 Department of Mathematics, Visva-Bharati, Santiniketan-731235, India

5 - Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran - Health Technology Research Institute, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

Abstract

In this paper, a megastable system is designed with particular formation of attractors. It has two formations of attractors: the inner ones with a smaller amplitude, and the outer ones with the Eye of God nebula shape and larger amplitude. To the best of our knowledge, such a megastable oscillator with this special formation of attractors has not been studied before. Afterward, the oscillator is forced, and its attractors are discussed. Different dynamics of this new oscillator are investigated using tools such as bifurcation and Lyapunov exponent diagram, and basins for each attractor.

Keywords

References

References:
1. Sprott, J.C. "A proposed standard for the publication of new chaotic systems”, International Journal of Bifurcation and Chaos, 21(09), pp. 2391-2394 (2011). https://doi.org/10.1142/S021812741103009X. 
2. Xu, G., Shekofteh, Y., Akgül, A., et al. "A new chaotic system with a self-excited attractor: entropy measurement, signal encryption, and parameter estimation”, Entropy, 20(2), p. 86 (2018). https://doi.org/10.3390/e20020086.
3. Wang, C., Zhou, L., and Wu, R. "The design and realization of a hyper-chaotic circuit based on a fluxcontrolled memristor with linear memductance”, Journal of Circuits, Systems and Computers, 27(03), 1850038 (2018). https://doi.org/10.1142/S021812661850038X.
4. Liang, H., Wang, Z., Yue, Z., et al. "Generalized synchronization and control for incommensurate fractional unified chaotic system and applications in secure communication”, Kybernetika, 48(2), pp. 190- 205 (2012). http://eudml.org/doc/246442.
5. Zhou, M. and Wang, C. "A novel image encryption scheme based on conservative hyperchaotic system and closed-loop diffusion between blocks”, Signal Processing, 171, 107484 (2020). https://doi.org/10.1016/j.sigpro.2020.107484.
6. Cheng, G., Wang, C., and Xu, C., "A novel hyperchaotic image encryption scheme based on quantum genetic algorithm and compressive sensing”, Multimedia Tools and Applications, 79(39), pp. 29243- 29263 (2020). https://doi.org/10.1007/s11042-020-09542-w.
7. Deng, J., Zhou, M., Wang, C., et al. "Image segmentation encryption algorithm with chaotic sequence generation participated by cipher and multifeedback loops”, Multimedia Tools and Applications, 80(9), pp. 13821-13840 (2021). https://doi.org/10.1007/s11042-020-10429-z.
8. Wang, Z., Wei, Z., Sun, K., et al. "Chaotic flows with special equilibria”, The European Physical Journal Special Topics, 229(6), pp. 905-919 (2020). https://doi.org/10.1140/epjst/e2020-900239-2.
9. Wei, Z., Wang, R., and Liu, A. "A new finding of the existence of hidden hyperchaotic attractors with no equilibria”, Mathematics and Computers in Simulation, 100, pp. 13-23 (2014). https://doi.org/10.1016/j.matcom.2014.01.001.
10. Wei, Z. and Zhang, W. "Hidden hyperchaotic attractors in a modified Lorenz–Stenflo system with only one stable equilibrium”, International Journal of Bifurcation and Chaos, 24(10), 1450127 (2014). https://doi.org/10.1142/S0218127414501272.
11. Danca, M.-F., Bourke, P., and Kuznetsov, N. "Graphical structure of attraction basins of hidden chaotic attractors: The Rabinovich–Fabrikant system”, International Journal of Bifurcation and Chaos, 29(01), 1930001 (2019). https://doi.org/10.1142/S0218127419300015.
12. Danca, M.-F., Kuznetsov, N., and Chen, G. "Unusual dynamics and hidden attractors of the Rabinovich– Fabrikant system”, Nonlinear Dynamics, 88(1), pp. 791-805 (2017). https://doi.org/10.1007/s11071-016-3276-1.
13. Danca, M.-F. and Kuznetsov, N. "Hidden strange nonchaotic attractors”, Mathematics, 9(6), 652 (2021). https://doi.org/10.3390/math9060652.
14. Danca, M.-F. "Coexisting hidden and self-excited attractors in an economic model of integer or fractional order”, International Journal of Bifurcation and Chaos, 31(04), 2150062 (2021). https://doi.org/10.1142/S0218127421500620.
15. Bao, H., Hua, Z., Li, H., et al. "Discrete memristor hyperchaotic maps”, IEEE Transactions on Circuits and Systems I: Regular Papers, 68(11), pp. 4534-4544 (2021). https://doi.org/10.1109/TCSI.2021.3082895. 
16. Zhou, L., Wang, C., and Zhou, L. "A novel noequilibrium hyperchaotic multi‐wing system via introducing memristor”, International Journal of Circuit Theory and Applications, 46(1), pp. 84-98 (2018). https://doi.org/10.1002/cta.2339.
17. Li, C., Hu, W., Sprott, J.C., et al. "Multistability in symmetric chaotic systems”, The European Physical Journal Special Topics, 224(8), pp. 1493-1506 (2015). https://doi.org/10.1140/epjst/e2015-02475-x.
18. Wang, N., Li, C., Bao, H., et al. "Generating multi-scroll Chua’s attractors via simplified piecewise-linear Chua’s diode”, IEEE Transactions on Circuits and Systems I: Regular Papers, 66(12), pp. 4767-4779 (2019). https://doi.org/10.1109/TCSI.2019.2933365.
19. Lai, Q. and Chen, S. "Generating multiple chaotic attractors from Sprott B system”, International Journal of Bifurcation and Chaos, 26(11), 1650177 (2016). https://doi.org/10.1142/S0218127416501777.
20. Xu, Q., Liu, T., Feng, C.-T., et al. "Continuous nonautonomous memristive Rulkov model with extreme multistability”, Chinese Physics B, 30(12), 128702 (2021). https://doi.org/10.1088/1674-1056/ac2f30.
21. Feudel, U. and Grebogi, C. "Why are chaotic attractors rare in multistable systems?”, Physical Review Letters, 91(13), 134102 (2003). https://doi.org/10.1103/PhysRevLett.91.134102.
22. Marmillot, P., Kaufman, M., and Hervagault, J.F. "Multiple steady states and dissipative structures in a circular and linear array of three cells: Numerical and experimental approaches”, The Journal of Chemical Physics, 95(2), pp. 1206-1214 (1991). https://doi.org/10.1063/1.461151.
23. Meucci, R., Marc Ginoux, J., Mehrabbeik, M., et al. "Generalized multistability and its control in a laser”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(8), 083111 (2022). https://doi.org/10.1063/5.0093727.
24. Hammel, S., Jones, C., and Moloney, J.V. "Global dynamical behavior of the optical field in a ring cavity”, JOSA B, 2(4), pp. 552-564 (1985). https://doi.org/10.1364/JOSAB.2.000552.
25. Canavier, C., Baxter, D., Clark, J., et al. "Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of postsynaptic activity”, Journal of Neurophysiology, 69(6), pp. 2252-2257 (1993). https://doi.org/10.1152/jn.1993.69.6.2252.
26. Prengel, F., Wacker, A., and Schöll, E. "Simple model for multistability and domain formation in semiconductor superlattices”, Physical Review B, 50(3), p. 1705 (1994). https://doi.org/10.1103/PhysRevB.50.1705.
27. Bao, B., Jiang, T., Xu, Q., et al. "Coexisting infinitely many attractors in active band-pass filter-based memristive circuit”, Nonlinear Dynamics, 86(3), pp. 1711-1723 (2016). https://doi.org/10.1007/s11071-016-2988-6.
28. Li, C., Sprott, J.C., Hu, W., et al. "Infinite multistability in a self-reproducing chaotic system”, International Journal of Bifurcation and Chaos, 27(10), 1750160 (2017). https://doi.org/10.1142/S0218127417501607.
29. Li, C., Thio, W. J.-C., Sprott, J.C., et al. "Constructing infinitely many attractors in a programmable chaotic circuit”, IEEE Access, 6, pp. 29003-29012 (2018). https://doi.org/10.1109/ACCESS.2018.2824984.
30. Karami, M., Ramakrishnan, B., Hamarash, I.I., et al. "Investigation of the simplest megastable chaotic oscillator with spatially triangular Wave Damping”, International Journal of Bifurcation and Chaos, 32(07), 2230016 (2022). https://doi.org/10.1142/S0218127422300166.
31. Ramakrishnan, B., Ahmadi, A., Nazarimehr, F., et al. "Oyster oscillator: a novel mega-stable nonlinear chaotic system”, The European Physical Journal Special Topics, pp. 1-9 (2021). https://doi.org/10.1140/epjs/s11734-021-00368-7.
32. Tang, Y., Abdolmohammadi, H.R., Khalaf, A.J.M., et al. "Carpet oscillator: A new megastable nonlinear oscillator with infinite islands of self-excited and hidden attractors”, Pramana, 91(1), pp. 1-6 (2018). https://doi.org/10.1007/s12043-018-1581-6.
33. Kahn, P.B. and Zarmi, Y., Nonlinear Dynamics: Exploration Through Normal Forms, Courier Corporation (2014). 
34. Wolf, A., Swift, J.B., Swinney, H.L., and Vastano, J.A. "Determining Lyapunov exponent from a time series", Physica D., 16, pp. 285-317 (1985). https://doi.org/10.1016/0167-2789(85)90011-9.
35. Karami, M., Ramamoorthy, R., Ali, A.M.A., et al. "Jagged-shape chaotic attractors of a megastable oscillator with spatially square-wave damping”, The European Physical Journal Special Topics, Springer, 231(11), pp. 2445-2454 (2022). https://doi.org/10.1140/epjs/s11734-021-00373-w.
36. Wang, Z., Hamarash, I.I., Shabestari, P.S., et al. "A new megastable oscillator with rational and irrational parameters”, International Journal of Bifurcation and Chaos, 29(13), 1950176 (2019). https://doi.org/10.1142/S0218127419501761.
Scientia Iranica
Volume 32, Issue 5
Transactions on Computer Science & Engineering and Electrical Engineering
March and April 2025 Article ID:6909

Files

History

  • Receive Date: 19 June 2022
  • Revise Date: 03 September 2022
  • Accept Date: 19 December 2022

Share

How to cite

Statistics