Sufficient conditions for the existence of periodic solutions of the extended Duffing–Van der Pol oscillator

Abstract

In this paper, some aspects on the periodic solutions of the extended Duffing–Van der Pol oscillator are discussed. Doing different rescaling of the variables and parameters of the system associated with the extended Duffing–Van der Pol oscillator, we show that it can bifurcate one or three periodic solutions from a two-dimensional manifold filled by periodic solutions of the referred system. For each rescaling we exhibit concrete values for which these bounds are reached. Beyond that we characterize the stability of some periodic solutions. Our approach is analytical and the results are obtained using the averaging theory and some algebraic techniques.