Algebraične metode v dinamičnih sistemih

Algebraic Methods in Dynamical Systems. In this monograph the theory of dynamical systems meets abstract algebra. From the dynamical point of view we are focused to center and cyclicity problems. From the ring theory point of view we consider polynomial rings and their ideals. We first consider Gröbner bases, the connection idealvariety and minimal decompositions of varieties. The basic information on Singular is included. In the second chapter the most important concepts like singularities, limit cycles, central variety, the center problem and the cyclicity from dynamical theory are considered. The computation and analysis of focus quantities and corresponding (polynomial) ideals represents the connection to chapter one. We use Bautin’s (Darboux and similar) method(s) to determine necessary (sufficient) conditions for the center problem. In third chapter we present and analyze some recent results in continuous (planar and 3D systems of ODEs) and discrete systems, mostly with quadratic, cubic and quartic perturbations added. In both cases the problem of center and cyclicity is considered. Examples concerning discrete systems are limited to dynamics of a real function whose square (in sense of function composition) is a near identity function arising from Żołądek equation. The cyclicity is analyzed by a generalization of Christopher’s theorem.