Higher-order approximation of subharmonics close to strong resonances in the forced oscillators

In this paper, we elaborate a refined analytical approach to study the subharmonic solutions as well as their bifurcation curves close to strong resonance points of order q = 1, 2, and 3 in the forced nonlinear oscillators. Therefore, we propose to develop the higher-order generalized averaging method to construct the approximations up to third-order, giving explicit formulas in the general case. A process permitting the study of several resonances simultaneously is proposed. Furthermore, for a class of forced nonlinear oscillators, the existence possibility of two cycles of order q near resonance points, which appear by saddle node bifurcations is shown. The theoretical predictions of these bifurcations are tested by numerical integrations and very good agreement is found.