Geodesic
Hidden attractors of some multistable systems with infinite number of equilibria
Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 18-33.

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It is well known that mathematically simple systems of nonlinear differential equations can exhibit chaotic behavior. Detection of attractors of chaotic systems is an important problem of nonlinear dynamics. Results of recent researches have made it possible to introduce the following classification of periodic and chaotic attractors depending on the presence of neighborhood of equilibrium into their basin of attraction – self-excited and hidden attractors. The presence of hidden attractors in dynamical systems has received considerable attention to both theoretical and applied research of this phenomenon. Revealing of hidden attractors in real engineering systems is extremely important, because it allows predicting the unexpected and potentially dangerous system response to perturbations in its structure. In the past three years after discovering by S. Jafari and J .C. Sprott chaotic system with a line and a plane of equilibrium with hidden attractors there has been much attention to systems with uncountable or infinite equilibria. In this paper it is offered new models of control systems with an infinite number of equilibrium possessing hidden chaotic attractors: a piecewise-linear system with a locally stable segment of equilibrium and a system with periodic nonlinearity and infinite number of equilibrium points. The original analytical-numerical method developed by the author is applied to search hidden attractors in investigated systems. Bibliography: 31 titles.
Keywords: Piecewise-linear system, segment of equilibria, infinite number of equilibria, cycle, hidden attractor, analytical-numerical method. 
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I. M. Burkin. Hidden attractors of some multistable systems with infinite number of equilibria. Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 18-33. http://geodesic.mathdoc.fr/item/CHEB_2017_18_2_a2/

[1] Shilnikov L. P., “A case of the existence of a countable number of periodic motions”, Sov. Math. Docklady, 169:3 (1965), 558–561

[2] Lorenz E. N., “Deterministic nonperiodic flow”, J. Atmos. Sci., 20 (1963), 65–75

[3] Rössler O. E., “An Equation for Continuous Chaos”, Physics Letters A, 57:5 (1976), 397–398 | DOI | MR

[4] Chua L. O., “A zoo of Strange Attractors from the Canonical Chua's Circuits”, Proc. Of the IEEE 35th Midwest Symp. on Circuits and Systems, Cat. No. 92CH3099-9 (Washington, 1992), v. 2, 916–926

[5] Aizerman M. A., “On a problem concerning the stability in the large of dynamical systems”, Uspekhi Mat. Nauk, 4:4(32) (1949), 187–188 | MR | Zbl

[6] Kalman R. E., “Physical and Mathematical mechanisms of instability in nonlinear automatic control systems”, Transactions of ASME, 79:3 (1957), 553–566 | MR

[7] Pliss V. A., Nonlocal problems of the theory of oscillations, Nauka Publ., M., 1964, 367 pp. | MR

[8] Leonov G. A., “On stability in the large of nonlinear systems in the critical case of two zero roots”, Pricl. Mat. Mekh., 45:4 (1981), 752–755 | MR | Zbl

[9] Leonov G. A., “Effective methods for periodic oscillations search in dynamical systems”, Appl. Math. Mech., 74:1 (2010), 24–50 | MR | Zbl

[10] Burkin I. M., “The Buffer Phenomenon in Multidimensional Dynamical Systems”, Diff. Equations, 38:5 (2002), 615–625 | MR | Zbl

[11] Burkin I. M., Soboleva D. V., “On structure of global attractor of MIMO automatic control systems”, Izvestiya TulGU. Estestvenniye nauki, 2012, no. 1, 5–16

[12] Burkin I. M., Nguen N. K., “Analytical-Numerical Methods of Finding Hidden Oscillations in Multidimensional Dynamical Systems”, Diff. Equations, 50:13 (2014), 1695–1717 | MR | Zbl

[13] Burkin I. M., “Method of “Transition into Space of Derivatives”: 40 Years of Evolution”, Diff. Equations, 51:13 (2015), 1717–1751 | MR | Zbl

[14] Andrievsky B. R., Kuznetsov N. V., Leonov G. A., Seledzhi S. M., “Hidden oscillations in stabilization system of flexible launcher with saturating actuators”, IFAC Proceedings Volumes, 19:1 (2013), 37–41 | DOI | MR

[15] Wang B., Zhou S., Zheng X., Zhou C., Dong J., Zhao J., “Image watermarking using chaotic map and DNA coding”, Optik, 126 (2015), 4846–4851 | DOI

[16] Liu H., Kadir A., Li Y., “Audio encryption scheme by confusion and diffusion based on multi-scroll chaotic system and one-time keys”, Optik, 127 (2016), 7431–7438 | DOI

[17] Bragin V. O., Vagaitsev V. I., Kuznetsov N. V., Leonov G. A., “Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua's circuits”, J. Comput. Syst. Sci. Int., 50:4 (2011), 511–543 | MR | Zbl

[18] Leonov G. A., Kuznetsov N. V., “Hidden attractors in dynamical systems: From hidden oscillation in Hilbert–Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits”, Int. J. Bifurcation and Chaos, 23:1 (2013), 1330002 | DOI | MR | Zbl

[19] Jafari S., Sprott J. C., Golpayegani S. M. R. H., “Elementary quadratic chaotic flows with no equilibria”, Phys. Lett. A, 377 (2013), 699–702 | DOI | MR

[20] Molaie M., Jafari S., Sprott J. C., Golpayegani S. M. R. H., “Simple chaotic flows with one stable equilibrium”, Int. J. Bifurcation and Chaos, 23:11 (2013), 1350188 | DOI | MR | Zbl

[21] Wangand X., Chen G., “A chaotic system with only one stable equilibrium”, Communications in Nonlinear Science and Numerical Simulation, 17:3 (2012), 1264–1272 | DOI | MR

[22] Wei Z., “Dynamical behaviors of a chaotic system with no equilibria”, Phys. Lett. A, 376 (2011), 102–108 | DOI | MR | Zbl

[23] Wei Z., “Delayed feedback on the 3-D chaotic system only with two stable node-foci”, Comput. Math. Appl., 63 (2011), 728–738 | DOI | MR

[24] Wang X., Chen G., “Constructing a chaotic system with any number of equilibria”, Nonlinear Dyn., 71 (2013), 429–436 | DOI | MR

[25] Jafari S., Sprott J. C., “Simple chaotic flows with a line equilibrium”, Chaos Solitons Fractals, 57 (2013), 79–84 | DOI | MR | Zbl

[26] Pham V.-T., Jafari S., Volos C., Vaidyanathan S., Kapitaniak T., “A chaotic system with infinite equilibria located on a piecewise linear curve”, Optik, 127 (2016), 9111–9117 | DOI

[27] Jafari S., Sprott J. C., Malihe Molaie, “A Simple Chaotic Flow with a Plane of Equilibria”, International Journal of Bifurcation and Chaos, 26:6 (2016), 1650098 | DOI | MR | Zbl

[28] Viet-Thanh Phama, Jafari S., Volos C., “A novel chaotic system with heart-shaped equilibrium and its circuital implementation”, Optik, 131 (2017), 343–349 | DOI

[29] Wang X., Viet-Thanh Pham, Volos C., “Dynamics, Circuit Design, and Synchronization of a New Chaotic System with Closed Curve Equilibrium”, Complexity, 2017 (2017), 7138971, 9 pp. | DOI | MR | Zbl

[30] Gelig A. Kh., Leonov G. A., Yakubovich V. A., The stability of nonlinear systems with nonunique equilibrium, Nauka Publ., M., 1978, 400 pp. | MR

[31] Holodniok M., Klíč A., Kubíček M., Marek M., Methods of the analysis of nonlinear dynamic models, Mir, M., 1991, 364 pp.