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Chaos generation using binary hysteresis. (English) Zbl 0608.58033
A two dimensional dynamical system with hysteresis is considered. It is shown that an axis return map is continuous and has a periodic point of period three (for appropriate values of parameters). By Sharkovskij’s theorem (the authors refer to the later paper of Li and Yorke) the dynamics of such a system has some chaotic features. The experimental results showing that such phenomena may occur in electronic circuits are given. In the authors’ view, the pumping of the heart is of similar nature.
Reviewer: M.Lyubich

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
94C05 Analytic circuit theory
Full Text: DOI
[1] T.-Y. Li and J. A. Yorke, Period three implies chaos,Amer. Math. Monthly,82, 985–992, 1975. · Zbl 0351.92021
[2] O. E. Rössler, Continuous chaos–four prototype equations, inBifurcation Theory and Applications in Scientific Disciplines (O. Gurel and O. E. Rössler, eds), pp. 376–392, Annals of the New York Academy of Sciences, Vol. 316, 1979.
[3] R. W. Newcomb and N. El-Leithy, A binary hysteresis chaos generator,Proceedings of the 1984IEEE International Symposium on Circuits and Systems, Montreal, pp. 856–859, 1984.
[4] R. W. Newcomb and N. El-Leithy, Trajectory calculations and chaos existence in a binary hysteresis chaos generator,Proceedings of the Fifth International Symposium on Network Theory, Sarajevo, 1984.
[5] V. I. Arnold,Ordinary Differential Equations, MIT Press, Cambridge, MA, 1973. · Zbl 0296.34001
[6] A. A. Andronov, A. A. Vitt, and S. E. Khaikin,Theory of Oscillators (translated by F. Immirzi and edited by W. Fishwick), Pergamon Press, Oxford, 1966. · Zbl 0188.56304
[7] G.-Q. Zhong and F. Ayrom, Periodicity and chaos in Chua’s circuit,IEEE Trans. Circuits and Systems,32, 501–503, 1985.
[8] R. W. Newcomb and S. Sathyan, An RC-op-amp chaos generator,IEEE Trans. Circuits and Systems,30, 54–56, 1983.
[9] T.-Y. Li and J. A. Yorke, Ergodic transformations from an interval into itself,Trans. Amer. Math. Soc,235, 183–192, 1978. · Zbl 0371.28017
[10] G. Pianigiani, On the existence of invariant measures, inApplied Nonlinear Analysis (V. Lakshmikantham, ed.), pp. 299–307, Academic Press, NY, 1979.
[11] A. Lasota, A solution of Ulam’s conjecture on the existence of invariant measures and its applications, inDynamical Systems, Vol. 2 (L. Cesari, ed.), pp. 47–55, Academic Press, NY, 1976. · Zbl 0333.28010
[12] N. Wiener, The homogeneous chaos, inCollected Works (P. Masani, ed), pp. 572–613 (including comments by L. Gross), MIT Press, Cambridge, MA, 1976.
[13] S. Ito, S. Tanaka, and H. Nakada, On umimodal linear transformations and chaos, II,Tokyo J. Math. 2, 241–259, 1979. · Zbl 0461.28017
[14] T. Saito, On a hysteresis chaos generator,Proceedings of the 1985 International Symposium on Circuits and Systems, Vol. 2, Kyoto, pp. 847–849, 1985.
[15] Y. S. Tang, A. I. Mees, and L. O. Chua, Synchronization and chaos,IEEE Trans. Circuits and Systems,30, 620–626, 1983.
[16] R. E. Plant, The role of direct feedback in the cardiacpacemaker, inApplied Nonlinear Analysis (V. Lakshmikantham, ed.) pp. 309–321, Academic Press, NY, 1979.
[17] A. S. Pikovsky and M. I. Rabinovich, Stochastic oscillations in dissipative systems,Phys. D,2, 8–24, 1981. · Zbl 1194.37062
[18] P. Stefan, A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line,Comm. Math. Phys.,54, 237–248, 1977. · Zbl 0354.54027
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