Dynamics of polynomial Rayleigh-Duffing system near infinity and its global phase portraits with a center

Abstract

For any polynomial Rayleigh-Duffing systems x˙=y,y˙=−∑i=0maixi−∑i=0nbiyi+1 with m,n∈N, ai,bi∈R and ambn≠0, we characterize its dynamics near infinity via Poincaré compactification, and verify that all the necessary information is coded in terms of m,n and the sign of am,bn. Here, we provide a new treatment blowing up a degenerate equilibrium via a continued fraction of a rational number. Moreover, a necessary and sufficient condition is obtained for an equilibrium of the system to be a center. As a consequence, we classify all global phase portraits of the Rayleigh-Duffing system on the Poincaré disc that has a unique equilibrium which is a center.