ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On Morse Spectrum for Homoclinic Tangency Case


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.



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