engleski

Epsilon-NEIGHBORHOODS OF ORBITS OF DYNAMICAL SYSTEMS

In this talk I give the results concerning analysis of epsilon-neighborhoods of orbits of dynamical systems, obtained by scientific group at University of Zagreb and the collaborators. The idea for that analysis comes from the fractal geometry, while the motivation comes from the 16th Hilbert problem. The problem asks for an upper bound of the number of limit cycles of polynomial vector fields in the plane, as a function of the degree of the vector field. It is of interest to determine how many limit cycles can bifurcate from a given limit periodic set in a generic unfolding. This number is called the cyclicity of the limit periodic set. 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 epsilon-neighborhoods of the Poincar´e 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 epsilon-neighborhood. It was discovered that the box dimension of an orbit signals the bifurcation. The generic bifurcations of 1-dimensional discrete systems are characterized by the box dimension of orbits. Hopf and Hopf-Takens bifurcations could be studied using bifurcation of discrete system defined by the Poincare map. The Poincare map near weak focus and limit cycle is differentiable, which is crucial for relating the box dimension and the cyclicity of a limit periodic set. The problem in hyperbolic polycycle case is that the Poincare map, usually called Dulac map, is not differentiable. We introduce the appropriate generalizations of box dimension, depending on a particular scale for a given problem. The cyclicity was concluded in the case of saddle loops. If we study more terms in the asymptotic expansion of area of epsilon- neighborhood, we can obtain more information about the Dulac map, including formal normal forms and formal classification.

epsilon-neighborhood, multiplicity, bifurcation

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