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Zbl 1101.37318
Severino, R.; Sharkovsky, A.; Sousa Ramos, J.; Vinagre, S.
Topological invariants in a model of a time-delayed Chua's circuit.
(English)
[J] Nonlinear Dyn. 44, No. 1-4, 81-90 (2006). ISSN 0924-090X; ISSN 1573-269X/e

Summary: In the last 30 years, some authors have been studying several classes of boundary value problems (BVP) for partial differential equations (PDE) using the method of reduction to obtain a difference equation with continuous argument which behavior is determined by the iteration of a one-dimensional (1D) map [see, for example, {\it E. Yu. Romanenko} and {\it A.~N. Sharkovsky}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, No. 7, 1285--1306 (1999; Zbl 0964.35165); {\it A. N. Sharkovsky}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, No. 5, 1419--1425 (1995; Zbl 0886.58099), Ann. Math. Sil. 13, 243--255 (1999; Zbl 0960.37014), New Progress in Difference Equations'', Proceedings of the ICDEA'2001, Augsburg, Germany 2001, 3--22 (2003; Zbl 1065.39040), {\it A. N. Sharkovsky, Ph. Deregel} and {\it L. O. Chua}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, No. 5, 1283--1302 (1995; Zbl 0886.58108), {\it A. N. Sharkovsky, Yu. L. Maistrenko} and {\it E. Yu. Romanenko}, Difference equations and their applications, Kluwer, Dordrecht (1993; Zbl 0881.58020)]. In this paper we consider the time-delayed Chua's circuit introduced by {\it A. N. Sharkovsky} [Int. J. Bifurcation Chaos Appl. Sci. Eng. 4, No. 2, 303--309 (1994; Zbl 0808.94029)], {\it A. N. Sharkovsky, Yu. L. Maistrenko, Ph. Deregel}, and {\it L. O. Chua} [J. Circuits, Syst. Comput. 3, No. 2, 645--668 (1993)] which behavior is determined by properties of one-dimensional map [see {\it A. N. Sharkovsky, Ph. Deregel, and L. O. Chua} (1995, loc. cit.); {\it Yu. L. Maistrenko, V. L. Maistrenko, S. I. Vikul}, and {\it L. O. Chua}, Int. J. Bifurcation Chaos 5, No. 3, 653--671 (1995); A. N. Sharkovsky (1994, loc. cit.); {\it A. N. Sharkovsky, Yu. L. Maistrenko, Ph. Deregel}, and {\it L. O. Chua} (1993, loc. cit.)]. To characterize the time-evolution of these circuits we can compute the topological entropy and to distinguish systems with equal topological entropy we introduce a second topological invariant.
MSC 2000:
*37N20 Dynamical systems in other branches of physics
94C05 Analytic circuit theory
78A55 Technical appl. of optics and electromagnetic theory
37E05 Maps of the interval
37B99 Topological dynamics

Keywords: boundary value problems; Chua's circuit; difference equations; one-dimensional maps; symbolic dynamics; topological invariants

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