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This work presents the mathematical modeling for the nonlinear vibration analysis of membrane and shell structures of arbitrary shape. These structures are usually the optimal form in many engineering applications. However, they may buckle under specific loading conditions and, in most cases, are sensitive to geometric imperfections. These structures when liable to unstable buckling present for load levels lower than the static critical load a multiwell potential functions, which has an underlying influence on the nonlinear dynamic behavior and stability of the structure in a dynamic environment. The energy barrier of each well is a key factor, and depending on the force control parameters and initial conditions, intra-well and cross-well motions may occur. Also coexisting attractors are the norm, influencing the structure's dynamic integrity. In these structures, escape from a potential well is preceded by global and …