  We prove structural stability under perturbations for a class of
discrete-time dynamical systems near a non-hyperbolic fixed point. We
reformulate the stability problem in terms of the well-posedness of an
infinite-dimensional nonlinear ordinary differential equation in a Banach space
of carefully weighted sequences. Using this, we prove existence and regularity
of flows of the dynamical system which obey mixed initial and final boundary
conditions. The class of dynamical systems we study, and the boundary
conditions we impose, arise in a renormalisation group analysis of the
4-dimensional weakly self-avoiding walk and the 4-dimensional n-component
$|\varphi|^4$ spin model.
