A theoretical model of a nonlinear hyperbolic metamaterial is presented in the form of a stack of subwavelength layers of linear plasmonic and nonlinear dielectric materials. A broad picture of the properties of evanescent waves (high-k modes) in this stack is investigated by plotting global transmission diagrams. The presence of nonlinearity strongly modifies these diagrams. The emergence and modification of nonlinear evanescent waves is observed. Some signatures of nonlinear phenomenon such as formation of orbits and trajectories around fixed points are also seen in our work.