Cycles of dynamical systems on the plane and Rolle's theorem

Abstract

In this paper it is shown that cycles of planar dynamical systems with polynomial vector fields are in many ways similar to ovals of algebraic curves, and more generally to "separating solutions" of such systems, on general algebraic curves. For example, for separating solutions the analog of Bezout's theorem is valid. The proofs are straightforward. They are based on the version of Rolle's theorem and Bezout's theorem for planar algebraic curves offered below. We note that according to Hilbert's conjecture the total number of limit cycles of a dynamical system with a polynomial vector field is bounded above by the degree of the field. From the validity of the conjecture and the results of the present note it follows that the curve consisting of all the limit cycles of a polynomial dynamical system resembles an algebraic curve. At the present time, even the finiteness of the number of limit cycles has not been proved: Ii'yashenko proved that Dyudaktsproof [i] contains an unfillable gap (cf. [2]) . I. Rolle's Theorem for Dynamical Systems We consider a smooth dynamical system on the plane