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Русская версия

**Georgii Sergeevich Osipenko**

Moscow State M.Lomonosov University,Sebastopol branch

299001, Sebastopol, geroev Sevastopolia str. 7

**N. Ampilova**

Russia, 198504, Petergof, Universitetski pr.28

Saint-Peterburg State University

Faculty of Mathematics and Mechanics

The Morse spectrum is defined as the limit set of Lyapunov exponents and is one of main characteristics of dynamical systems.It is important for system having infinitely many trajectories with long periods. G. Osipenko supposed a practical approach to the Morse spectrum computation, which is based on the symbolic image method. Symbolic image of a dynamical system is an oriented graph presenting the dynamics of the transformation of the system phase space. As it was shown by G. Osipenko, the Morse spectrum of labeled symbolic image is an approximation of a given system spectrum. In this work we study the structure of the Morse spectrum when there is homoclinic tangency of stable and unstable manifolds of a fixed hyperbolic point. We prove that the spectrum contains the segment for which initial and end points are defined by stable and unstable Lyapunov exponents of this point.

- homoclinic tangency
- invariant manifolds
- Lyapunov exponents
- Morse spectrum

- V. Avrutin, P. Levi, M. Schanz, D. Fundinger, G. Osipenko, Investigation of dynamical systems using symbolic image: efficient implementation and application. Int. J. of Bifurcation and Chaos, v. 16 (2006), no. 12, 3451-3496
- R. Bowen. Symbolic Dynamics. Ann. Math. Soc., Providence, R. I., v. 8 (1982)
- Y. Cao, S. Luzzatto, I. Rios, Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: horseshoes with internal tangencies, Discrete and continuous dynamical systems, v. 15 (2006), no. 1, 61-71
- F. Colonius and W. Kliemann. The Dynamics of Control, Burkhauser, 2000
- C. Conley. Isolated Invariant set and the Morse Index. CBMS Regional Conference Series, v. 38 (1978), Amer. Math. Soc., Providence
- A. Katok, B. Hasselblat. Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995
- D. Lind, B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, 1995
- G. S. Osipenko. On a symbolic image of dynamical systems. In Boundary value problems, Interuniv. Collect. Sci. Works, (1983), p. 101 - 105
- G. S. Osipenko. Spectrum of a dynamical system and applied symbolic dynamics. Journal of Mathematical Analysis and Applications, v. 252 (2000), no. 2, p. 587 - 616
- G. S. Osipenko, J. V. Romanovsky, N. B. Ampilova, and E. I. Petrenko. Computation of the Morse Spectrum. Journal of Mathematical Sciences, v. 120 (2004), no. 2, 1155 - 1166
- George Osipenko. Dynamical systems, Graphs, and Algorithms. Lectures Notes in Mathematics, 1889, Springer, Berlin, 2007
- George Osipenko. Symbolic image and invariant measures of dynamical systems. Ergodic Theory and Dynamical Systems, v. 30 (2010), 1217 - 1237
- M. Shub, Stabilite globale de systems denamiques, Asterisque, v. 56, 1978, 1-21