| Sumario: | In this article, we study the primary resonances of van der Pol systems with parametric excitation using the multiple scales method (MSM) and the homotopy analysis method (HAM). First, we study the nonlinear dynamic response of a coupled system with parametric excitation when the ratio of internal resonances are different, and obtain the four-dimensional average equation of the rectangular coordinate form using the MSM, thereby periodic motions are found in the system. Second, using the HAM, we obtain the four periodic solutions, in which there are two sets of in-phase periodic solutions and two sets of out-of-phase periodic solutions. Finally, we obtain the frequency response curves using the MSM and the HAM, in which it is found that the differences could be ignored.
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