New megastable system with 2-D strip of hidden attractors and analytical solutions
https://doi.org/10.22405/2226-8383-2021-22-4-361-369
Abstract
Many real dynamical systems are characterized by the presence of a set coexisting attractors.
This property of systems is called multistability. In multistable systems, a sudden transition to unwanted or unknown attractors can occur. Such a transition can lead to catastrophic events. It
turned out that multistability is also associated with the emergence of unpredictable attractors, which are called hidden attractors. An attractor is called hidden if its area of attraction does not intersect with small neighborhoods of an unstable fixed point. One of the defining reasons for studying multistable chaotic systems with different characteristics is a wide range of their potential engineering applications - in control theory, computer science, cryptology, artificial neural networks, image encryption, secure communication, and weak signal detection. In recent years, researchers have turned to developing methods for artificially designing systems with desired dynamics. In this case, the main efforts are focused on creating systems with an infinite
number of coexisting attractors - extremely multistable and megastable systems. It turned out that such systems open up new possibilities for solving some applied problems, for example, for realizing control of the signal amplitude and polarity in engineering systems or for creating new image encryption systems. In this paper, a new smooth three-dimensional dynamical system is constructed, reversible in time, containing an analytical solution and a strange multifractal hidden attractor. The basin of attraction of the attractor includes almost all three-dimensional space, and its dimension is "almost 3". By replacing one of the variables of the system with a periodic function of this variable, a system is constructed that has a 1-D strip of hidden chaotic attractors of dimension "almost 3"and, at the same time, an infinite number of analytical solutions. A special transformation of the latter system allows us to design a megastable system with a 2-D strip of hidden attractors.
About the Authors
Igor’ Mikhailovich Burkin
Tula State Univetsity
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Oksana Igorevna Kuznetsova
Tula State Univetsity
Russian Federation
Russian Federation
postgraduate student
References
1. Rossler, O., Adryaman Y., Shaukat S. & all. 2020, “Chaos Theory and Applications in applied sciences and engineering“, An interdisciplinary journal of nonlinear science, vol. 2, no. 1.
2. Pisarchik, A. N. & Feudel, U. 2014, “Control of multistability“, Phys. Rep., vol. 540. pp. 167-218.
3. Leonov, G. A. & Kuznetsov, N. V. 2013, “Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic
4. attractors in Chua circuits“, International Journal of Bifurcation and Chaos, vol. 23, no. 1. art. no. 1330002.
5. Burkin, I. & Nguen, N. K. 2014, “Analytical-Numerical Methods of Finding Hidden Oscillations in Multidimensional Dy-namical Systems“, Diff. Equations, vol. 50, no. 13, pp. 1695–1717.
6. Wei, Z. & Zhang, W. 2014, “Hidden Hyperchaotic Attractors in a Modified Lorenz-Stenflo System with Only One Stable Equilibrium“, Int. J. Bifurcation Chaos, vol. 24, no. 10. art. no.
7.
8. Guanrong, C. 2021, “Chaos Theory and Applications: A New Trend“, Chaos Theory and Application, vol. 3, no. 1, pp. 1–2.
9. Burkin, I. M. 2017, “Hidden attractors of some multistable systems with infinite number of equilibria“, Chebyshevskii Sb., vol. 18, no. 2, pp. 18–33.
10. Pham, V-T., Volos, C., Jafari, S.,Wei, Z. &Wang, X. 2014, “Constructing a novel no-equilibrium chaotic system“, Int. J. Bifurcation Chaos, vol. 24, no. 5, art. no. 1450073.
11. Faghani, Z., Nazarimehr, F., Jafari, S. & Sprott J. C. 2019, “Simple Chaotic Systems with Specific Analytical Solutions“, Int. J. Bifurcation Chaos, vol. 29, no. 9, art. no. 1950116.
12. Li, C., Lu, T., Chen, G. & Xing, H. 2019, “Doubling the coexisting attractors“, Chaos, vol. 29, art. no. 051102.
13. Zhang, X. & Chen, G. 2017, “Constructing an autonomous system with infinitely many chaotic attractors“, Chaos, vol. 27, no. 7, art. no. 071101.
14. Li, C., Sprott, J. C., Hu, W. & Xu, Y. 2017, “Infinite multistability in a self-reproducing chaotic system“, Int J Bifurc. Chaos, vol. 27(10), art. no. 1750160.
15. Burkin, I. M. & Kuznetsova, O. I. 2021, “Designing megastable systems with multidimensional lattice of chaotic attractors “, Chebyshevskii Sb., vol. 22, no. 1, pp. 105–117.
16. Burkin, I. M. & Kuznetsova, O. I. 2109, “On some methods for generating extremely multistable systems“, J. Phys.: Conf. Ser., 1368 042050.
17. Burkin, I. M. & Kuznetsova, O. I. 2019, “Generation of Extremely Multistable Systems Based on Lurie Systems“, Vestnik St. Petersburg University, Mathematics, vol. 52, no. 4, pp. 342–348.
18. Sprott, J. C., 2020, “Do We Need More Chaos Examples“, Chaos Theory and Applications, vol. 2, no. 2, pp. 1–3.
19. Sprott, J. C., Hoover, W. & Hoover, C. 2014, “Heat Condition, and the Lack Thereof, in Time-Reversible Dynamical Systems: Generalized Nos’e-Hoover Oscillators with a Temperature
20. Gradient“, Phys. Rev. E., vol. 89, 042914
21. Leonov, G. A. 2016, “Analytical analysis of a Nose–Hoover generator“, Dokl. Phys., vol. 6, pp. 340–342.
Review
For citations:
Burkin I.M., Kuznetsova O.I. New megastable system with 2-D strip of hidden attractors and analytical solutions. Chebyshevskii Sbornik. 2021;22(4):361-369. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-4-361-369
Views:
282
Email this article 