We construct a model for a Hamiltonian three degrees of freedom system which on one hand contains chaos and on the other hand is simple enough that we can make analytic approximations for the geometry, the location in the phase space, the development scenario and the decay scenario of the fundamental normally hyperbolic invariant manifold ( NHIM ). This NHIM is the central ingredient of the chaotic set of the dynamics. We compare the analytic approximations with numerical results. Thereby we obtain new insight into the development and decay scenario of NHIMs under parameter changes of the dynamics and we learn ideas for their analytic approximations. In particular and irrespective of the latter, we identify the decay front of the NHIM with its outermost invariant substructure for which the normal instability still dominates the tangential instability at all of its points. Thereby, the location of the decay front is partially determined by a nonlocal mechanism.