Multiplicity of limit cycles and analytic m-solutions for planar differential systems

摘要

This work deals with limit cycles of real planar analytic vector fields. It is well known that given any limit cycle Gamma of an analytic vector field it always exists a real analytic function f(0)(x, y), defined in a neighborhood of Gamma, and such that Gamma is contained in its zero level set. In this work we introduce the notion of f(0)(x, y) being an m-solution, which is a merely analytic concept. Our main result is that a limit cycle Gamma is of multiplicity m if and only if f(0)(x, y) is an m-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles. (c) 2007 Elsevier Inc. All rights reserved.