The Effects of Strong Cubic Nonlinearity on the Existence of Periodic Solutions of the Mathieu–Duffing Equation

Abstract

This paper deals with systems governed by the Mathieu–Duffing equation, with a time-dependent coefficient of the linear term and a constant, not necessarily small coefficient of the cubic term. This coefficient 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 periodic solutions exist are obtained. It is shown that they strongly depend on the value of the coefficient of nonlinearity and the initial conditions. The corresponding parameter planes are plotted. Numerical integrations are carried out to confirm the analytical results.