A family of discontinuous symplectic maps on the cylinder is considered. This family arises naturally in the study of nonsmooth Hamiltonian dynamics and in switched Hamiltonian systems. The transformation depends on two parameters and is a canonical model for the study of bounded and unbounded behavior in discontinuous area-preserving mappings due to nonlinear resonances. This paper provides a general description of the map and points out its connection with another map considered earlier by Kesten. In one special case, an unbounded orbit is explicitly constructed.
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