Abstract
An approach for deriving the topological entropy of dynamical systems generated by one-dimensional piecewise-continous and piecewise-monotonous maps of an interval is proposed. The technique is based on the application of kneading theory and allows to reduce the calculations for finding a root of some polynomial which is the kneading determinant. Examples of computing the topological entropy including those from the experimental data are presented; the comparison with the Lyapunov exponent is given.
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References
R.L. Adler, A.G. Konheim, M.H. McAndrew. Topological entropy. Trans. Am. Math. Soc., 1965, 114, 2, 309–319.
R. Bowen. Methods of Symbolic Dynamics. M.: Mir, 1979 (in Russian).
U.N. Kornfeld, Ya.G. Sinai, S.V. Fomin. The Ergodic Theory. M.: Nauka, 1080 207 (in Russian).
J. Guckenheimer. The growth of topological entropy for one-dimensional maps. Lect; Notes in Math., 1980, 819, 216–223.
M.I. Maikin. On the continuity of the entropy of discontinuous maps of an interval In: Methods of Qualitative Theory of Differential Equations. Gorky Univ., 1982, 35–38 (in Russian).
J. Milnor, W. Thurston. On Iterated Maps of Interval. Preprint, Princeton Univ., 1977.
M.I. Maikin. Methods of Symbolic Dynamics in the Theory of One-Dimensional Discontinuous Maps. Cand. Sci. Dissertation, Gorky, 1985 (in Russian).
D. Ruelle. An inequality for the entropy of differentiate maps. Bol. Soc. Bras. Math 1978, 9, 83–87.
L.A. Komraz. Dynamic models of Gipp’s balance regulator. Zh. Prikl. Mat. Fiz., 1971, 35, 1, 148–162 (in Russian).
M.I. Malkin, A.L. Zheleznyak, I.L. Zheleznyak. Computational Aspects of the Entropy Theory of One-Dimensional Dynamical Systems. Preprint of the Institute of Appl. Physics N222, Gorky, 1988.
H.L. Swinney. Observations of order and chaos in nonlinear systems. Physica D, 1983, 70, 1–3, 3–15.
A.M. Belyantsev, A.L. Zheleznyak et al. Dynamics of avalanche-like electron heating oscillations and chaotic response behaviour of multilayer heterostructures. 3rd Conf. on Physics and Technology of GaAs and Other III-V Semiconductors. Abstracts CSSR, 1988, p. 51.
I. Shimada, T. Nagashima. A numerical approach to ergodic problem of dissipative dynamical systems. Prog. Teor. Phys., 1979, 61, 6, 1605–1616.
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© 1990 Springer-Verlag Berlin, Heidelberg
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Zheleznyak, A.L. (1990). An Approach to the Computation of the Topological Entropy. In: Gaponov-Grekhov, A.V., Rabinovich, M.I., Engelbrecht, J. (eds) Nonlinear Waves 3. Research Reports in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75308-4_28
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DOI: https://doi.org/10.1007/978-3-642-75308-4_28
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