Topological Invariants in a Model of a Time-Delayed Chua's Circuit |
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Authors: | R Severino A Sharkovsky J Sousa Ramos S Vinagre |
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Institution: | 1. Departamento de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal 2. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska str., 3, 01601, Kiev, Ukraine 3. Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1, 1049-001, Lisboa, Portugal 4. Departamento de Matemática, Universidade de évora, Rua Rom?o Ramalho, 59, 7000-671, évora, Portugal
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Abstract: | 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, Romanenko, E. Yu. and Sharkovsky, A. N., International Journal of Bifurcation and Chaos 9(7), 1999, 1285–1306; Sharkovsky, A. N., International Journal of Bifurcation and Chaos 5(5), 1995, 1419–1425; Sharkovsky, A. N., Analysis Mathematica Sil 13, 1999, 243–255; Sharkovsky, A. N., in “New Progress in Difference Equations”, Proceedings of the ICDEA'2001, Taylor and Francis, 2003, pp. 3–22; Sharkovsky, A. N., Deregel, Ph., and Chua, L. O., International Journal of Bifurcation and Chaos 5(5), 1995, 1283–1302; Sharkovsky, A. N., Maistrenko, Yu. L., and Romanenko, E. Yu., Difference Equations and Their Applications, Kluwer, Dordrecht, 1993.). In this paper we consider the time-delayed Chua's circuit introduced in (Sharkovsky, A. N., International Journal of Bifurcation and Chaos 4(5), 1994, 303–309; Sharkovsky, A. N., Maistrenko, Yu. L., Deregel, Ph., and Chua, L. O., Journal of Circuits, Systems and Computers 3(2), 1993, 645–668.) which behavior is determined by properties of one-dimensional map, see Sharkovsky, A. N., Deregel, Ph.,
and Chua, L. O., International Journal of Bifurcation and Chaos 5(5), 1995, 1283–1302; Maistrenko, Yu. L., Maistrenko, V. L., Vikul, S. I., and Chua, L. O., International Journal of Bifurcation and Chaos 5(3), 1995, 653–671; Sharkovsky, A. N., International Journal of Bifurcation and Chaos 4(5), 1994, 303–309; Sharkovsky, A. N., Maistrenko, Yu. L., Deregel, Ph., and Chua, L. O., Journal of Circuits, Systems and Computers 3(2), 1993, 645–668. 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. |
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Keywords: | boundary value problems Chua's circuit difference equations one-dimensional maps symbolic dynamics topological invariants |
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