Implicit approximation of a stable saddle manifold generated by a two-dimensional quadratic map

Consider the Henon's map T-b : (x --> 1 - ax(2) + y, y --> bx), the paramet ers (a, b) being such that \b\ < 1, with the existence of an attracting set A. This paper deals with an approximate implicit analytical representation of the stable manifold W-S(q(1)) of the saddle fixed point q(1) belonging to the basin boundary of the attracting set A. A method of successive appro ximations of iterative type is used from the definition of a "generating ap proximation" g(0) (x, y) = 0 (approximation of order zero). In the case of absence of homoclinic points to q(1), a generating approximation is defined from the two parabolas constituting the degenerate stable manifold in the (x, y) plane when b = 0. Formally the result is extended when homoclinic po ints to q(1) are created for b = 0, i.e. the degenerate stable manifold W-S (q(1)) is made up of infinitely many parabolas.