In contrast with the two cases of multi-frequency excitations and forced oscillations of the Duffing equation in which the excitations appear as in-homogeneities in the governing equation, the excitations that appear as coefficients in the governing equations are considered here. Such excitations are called parametric excitations and the simplest possible equation that describes the parametric excitations of a system having a single degree of freedom is the Mathieu equation. The asymptotic solutions and transition curves for the generalized form of the non-homogeneous Mathieu differential equation are investigated in this paper. This type of governing differential equation of motion arises from the dynamic behavior of a pendulum undergoing a butterfly type end support motion. The strained parameter technique is used to obtain periodic asymptotic solutions. The transition curves for some special cases are presented and their corresponding periodic solutions with the periods of 2π and 4π are evaluated. The stability analyses of those transition curves in the ε - δ plane are carried out, analytically, using the multiple scales method - is the perturbation parameter. The numerical simulations for some typical points in the ε - δ plane are performed and the dynamic characteristics of the resulting phase plane trajectories are discussed.

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