The generalized form of the non-homogeneous Mathieu differential equation is analyzed in this paper. This type of differential equation arises from dynamic behavior of a pendulum subjected to the butterfly support motion. The Lindstedt-Poincare’s technique is considered in order to obtain the analytical solutions. The transition curves in some special cases are presented and their related periodic solutions with periods of 2π and 4π are obtained. Numerical simulation is carried out for some typical points in ε-δ plane.

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