A Refined Global Poincaré–Bendixson Annulus with the Limit Cycle of the Rayleigh System

Abstract

New methods for constructing two Dulac–Cherkas functions are developed using which a better, depending on the parameter \(\lambda >0\), inner boundary of the Poincaré–Bendixson annulus \(A(\lambda ) \) is found for the Rayleigh system. A procedure is proposed for directly finding a polynomial whose zero level set contains a transversal oval used as the outer boundary of \(A(\lambda )\). An interval for \(\lambda \) is specified with which the best outer boundary of the annulus \( A(\lambda )\) is a closed contour composed of two arcs of the constructed oval and two arcs of unclosed curves of the zero level set of one of the Dulac–Cherkas functions. Thus, a refined global Poincaré–Bendixson annulus for the limit cycle of the Rayleigh system is presented.