Combined Homotopy and Galerkin Stability Analysis of Mathieu-Like Equations

We propose a homotopy analysis method in combination with Galerkin projections to obtain transition curves of Mathieu-like equations. While constructing homotopy, we think of convergence-control parameter as a function of embedding parameter and call it a convergence-control function. Homotopy analysis provides a relation between the parameters of the Mathieu equation that also includes free parameters arising from the convergence-control function. We generate extra nonlinear algebraic equations using Galerkin projections and solve numerically for arriving at transition curves. We demonstrate the usefulness of our method in the case of three distinct versions of linear Mathieu equations by carefully choosing nonlinear and auxiliary operators. Since homotopy analysis does not demand smallness of any of the parameters, our approach has a distinct advantage over perturbation methods in determining transition curves covering a large region of the parameter space. The method is applicable to a wide variety of parametrically excited oscillators.