Normal forms, quasi-invariant manifolds, and bifurcations of nonlinear difference-algebraic equations

We study the existence of quasi-invariant manifolds in a neighborhood of a fixed point of the difference-algebraic equation(Delta AE) F(z(n), z(n+1)) = 0, where F : R-2m -> R-m is a smooth map satisfying F(0, 0) = 0. We demonstrate the existence of quasi-invariant manifolds on which one can de. ne forward and backward orbits of the Delta AE under mild assumptions on its linearization at the fixed point z = 0. Indeed, by assuming this linearization to be a regular matrix pencil, one obtains a functional equation satisfied by invariant manifolds which can be solved using an extension of the contraction mapping to spaces that satisfy an interpolation property. If the Delta AE under study is permitted to depend smoothly on a parameter, we then obtain a Neimark-Sacker bifurcation theorem as a corollary that can be deduced from the existence of a normal form for nonlinear Delta AEs.