Coupling of homotopy and the variational approach for a conservative oscillator with strong odd-nonlinearity

In this paper, a new approach combining the features of the homotopy concept with the variational approach is proposed for describing and predicting analytical approximations of a conservative oscillator with strong odd-nonlinearity. The new technique does not depend upon small parameter assumptions, and incorporates salient features of both methods of homotopy perturbation and the variational approach. The cubic–quintic duffing oscillator is analyzed to illustrate the usefulness and effectiveness of the proposed technique. Four approximate formulas for the frequency are established for small, as well as large, amplitudes of motion. The results of applying this procedure to the cubic–quintic duffing equation are compared to those analytical and exact solutions, in order to substantiate the accuracy and correctness of the approximate analytical approach. The approach can be easily extended to other nonlinear oscillators.