  We show the existence of a local stable manifold for a bidirectional
discrete-time nondiffeomorphic nonlinear Hamiltonian dynamics. This is the case
where zero is a closed loop eigenvalue and therefore the Hamiltonian matrix is
not invertible. In addition, we show the eigenstructure and the symplectic
properties of the mixed direction nonlinear Hamiltonian dynamics. We extend the
Local Stable Manifold Theorem for the nonlinear discrete-time Hamiltonian map
with a hyperbolic fixed point. As a consequence, we show the existence of a
local solution to the Dynamic Programming Equations, the equations
corresponding to the discrete-time optimal control problem.
