Singular periodic solutions of polynomial differential equations

A. N. Saharov

Nizhnii Novgorod State Agricultural Academy

Abstract: We consider the problem of periodic solutions of the equation
$$\dot z\,=\,z^m+a_1(t)z^{m-1}+\cdots+a_{m-1}(t)z+a_m(t),\quad z\in \mathbb{C},$$
with coefficients $a_k(t),\,k=1,\ldots,m$ periodic in $t$. It is known that equations of this type can have, in addition to ordinary periodic solutions, also special periodic solutions that have a finite number of discontinuities in the period. The compactification procedure for the phase space of an equation makes it possible to determine the conditions that limit the number of ordinary periodic solutions, as well as to describe the mechanism for changing the structure of periodic solutions in terms of rotation numbers.

Keywords: compactification of phase space, periodic solutions, singular periodic solutions, rotation numbers, modromy mapping.

UDC: 517.45

MSC: 35B10