On approximate projecting on a stable manifold

For an element of a Banach space that belongs to a neighborhood of a fixed point of the given resolving operator, the problem of projecting on the corresponding stable manifold is examined. The projector is specified by a basis that describes the admissible modifications. The original problem is reduced to solving a nonlinear equation of a special form. Under the conventional assumptions, the solvability of this equation is proved. It is shown that the proposed method is locally equivalent to the well-known methods for approximating the stable manifold. The high efficiency of the method is demonstrated by the numerical experiments. Their results for the two-dimensional Chafe–Infant equation are presented.