Differential equations and Optimal control

About isochrons of strong focus

A. E. Rudenok, M. N. Vasilevich

Belarusian State University, Minsk

Abstract: The article analyses the methods for studying the isochronism of autonomous systems on a plane for different types of singular points: centre, weak focus and strong focus. The difficulty of determining a weak isochronous focus is explained. Examples of systems in the Poincaré – Dulac normal form, which show that the requirement to be isochronous imposes weak restrictions on a system with a weak focus, are given. A theorem on the form of a normalising change of a strongly isochronous strong focus is proved. Isochrones of a strongly isochronous strong focus of a system with quadratic nonlinearities are found. Phase portraits of the obtained isochronous foci and their isochrones are constructed.

Keywords: Сentre; weak focus; strong focus; isochronism; isochrones; Poincaré – Dulac normal form; system with quadratic nonlinearities.

UDC: 517.925.42