The effects of strong cubic nonlinearity on the existence of periodic solutions of the Mathieu-Dufng equation

Abstract

This paper deals with systems governed by the Mathieu-Dufng equation, with a time-dependent coefcient of the linear term and a constant, not necessarily small coefcient of the cubic term. This coefcient can be positive or negative. The method of strained parameters applied to a linear system governed by the Mathieu equation is extended to a strongly nonlinear system. As a result, the curves corresponding to the parameter values at which peri-odic solutions exist are obtained. It is shown that they strongly depend on the value of the coefcient of nonlinearity and the initial conditions. The corresponding parameter planes are plot-ted. Numerical integrations are carried out to conrm the analyti-cal results. © 2009 by ASME.