The structure of a stable manifold for fully implicit schemes

An analog of the Hadamard-Perron theorem on the existence of a local stable manifold in a neighborhood of a fixed hyperbolic-type point for implicit mappings is proved. This result allows one to constructively study the structure of a manifold for a finite-difference approximation in time in the case of quasilinear parabolic-type equations and to prove that, in terms of the integral metric, the manifold of the nonlinear problem exists in an unbounded ellipsoid. Several theoretical estimates are given. A number of numerical results are discussed.