Coexistence phenomenon in autoparametric excitation of two degree of freedom systems

Coexistence phenomenon refers to the absence of expected tongues of instability in parametrically excited systems. In this paper we obtain sufficient conditions for coexistence to occur in the generalized Ince equation (1+a1cost+a2cos2t)v¨+(b1sint+b2sin2t)v˙+(δ+c1cost+c2cos2t)v=0.The results are applied to the stability of motion of a non-linear normal mode, the x-mode, in a class of conservative two degree of freedom systems.