engleski

Introduction to fractal analysis of orbits of dynamical systems

In this talk I give the initial results concerning analysis of epsilon-neighborhoods of orbits of dynamical systems The idea comes from the fractal geometry, while the motivation comes from the 16th Hilbert problem. It is of interest to determine how many limit cycles can bifurcate from a given limit periodic set in a generic unfolding. The cyclicity is classically obtained by studying the multiplicity of fixed points of the Poincare map. We establish a relation between the cyclicity of a limit periodic set of a planar system and the leading term of the asymptotic expansion of area of "- neighborhoods of the Poincare map of an orbit. A natural idea is that higher density of orbits reveals higher cyclicity. The box dimension could be read from the leading term of the asymptotic expansion of area of "- neighborhood. In this talk I will concentrate on weak focus as a simplest case for the study. Furthermore, shortly I will talk about different directions of research coming from that idea: classi cations of Dulac maps, slow-fast systems, oscillatory integrals and fractal zeta functions.

box dimension, multiplicity, bifurcation

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