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 thevariational approach is proposed for describing and predicting analytical approximations of a conservativeoscillator with strong odd-nonlinearity. the new technique does not depend upon small parameterassumptions, and incorporates salient features of both methods of homotopy perturbation and thevariational approach. the cubicquintic duffing oscillator is analyzed to illustrate the usefulness andeffectiveness of the proposed technique. four approximate formulas for the frequency are established forsmall, as well as large, amplitudes of motion. the results of applying this procedure to the cubicquinticduffing equation are compared to those analytical and exact solutions, in order to substantiate the accuracyand correctness of the approximate analytical approach. the approach can be easily extended to othernonlinear oscillators.