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We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrodinger equation  where the potential  approximates a two-step function. The term  generalizes the typical -power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincare map. We discuss the periodic and the Neumann boundary conditions. The value of the term  although small, can be explicitly estimated.

Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential