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

For any polynomial Rayleigh-Duffing systems x = y, y = - Emi=0 aixi - L+ni=0 biyi+1 with m, n is an element of N, ai, bi is an element of R and ambn = 0, we characterize its dynamics near infinity via Poincare 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 Poincare disc that has a unique equilibrium which is a center. (c) 2023 Elsevier Inc. All rights reserved.