Iteration of continuous functions and dynamics of solutions for some boundary value problems
Abstract
It is known there are boundary value problems investigation of which is directly reduced to study of iteration of functions. This allows to carry on detailed and deep analysis of original problems. We consider such a class of simple (in form) problems, solutions of which demonstrate, nevertheless, very complicated behavior, make possible the simulation of self-birthing structures, including self-similar ones, and self-stochasticity phenomenon.
References
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2. A.N. Sharkovskii, How complicated can be one dimensional dynamical systems: descriptive estimates of sets, in Dynamical systems and Ergodic theory, Banach Center Publ. 23 (1989), 447-453.
3. A.N. Sharkovsky, S.P. Kolyada, A.G. Sivak, V.V. Fedorenko, Dynamics of One-Dimensional Mappings, Naukova Dumka, Kiev 1989 (in Russian); Kluwer Academic Publ., Dordrecht 1997.
4. A.N. Sharkovsky, Yu.L. Maistrenko, E.Yu. Romanenko, Difference Equations and Their Applications, Naukova Dumka, Kiev 1986 (in Russian); Kluwer Academic Publ., Dordrecht 1993.
5. E.Yu. Romanenko, A.N. Sharkovsky, From one-dimensional to infinite-dimensional dynamical systems: ideal turbulence, Ukrain. Math. J. 48 (1996), 1604-1627 (in Ukrainian); Ukrain. Math. J. 48, 1817-1842.
6. E.Yu. Romanenko, A.N. Sharkovsky, Dynamics of solutions for simplest nonlinear boundary value problems, Ukrain. Math. J. 51 (1999), 810-826 (in Ukrainian).
7. A.N. Sharkovsky, E.Yu. Romanenko, Problems of turbulence theory and iteration theory, in Proc. of the European Conf. on Iteration Theory (ECIT-91), World Scientific, Singapore, 242-252.
8. A.N. Kolmogorov, V.M. Tikhomirov, ɛ-entropy and ɛ-capacity of sets in functional spaces, Uspekhi Mat. Nauk, 14 (1959), 3-86 (in Russian); in Selected Works of A.N. Kolmogorov, vol. III, Kluwer Academic Publ., Dordrecht 1993.
9. U. Burkart, A new proof of a Theorem of Sharkovskii, Lecture at the 1979 Intern. Symp. on Functional Equations, Abstract in Aequationes Math. 20 (1980), 300.
10. U. Burkart, Interval mapping graphs and periodic points of continuous functions, J. Combin. Theory, Ser. B 32 (1982), 57-68.
SharkovskyA. N. (1999). Iteration of continuous functions and dynamics of solutions for some boundary value problems. Annales Mathematicae Silesianae, 13, 243-255. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14153
Alexander N. Sharkovsky
Institute of Mathematics, National Academy of Sciences of Ukraine Ukraine
Institute of Mathematics, National Academy of Sciences of Ukraine Ukraine
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