Journal of Statistical Physics, VoL 25, No. 4, 1981 Stable and Unstable Manifolds of the H non Mapping Valter Franceschini 1 and Lucio Russo 1 Received May 12, 1980 By using a parametric representation of the stable and unstable manifolds, we prove that for some given values of the parameter (in particular in the case first investigated by H~non) the H~non mapping has a transversal homoclinic orbit. KEY WORDS: Henon mapping; strange attractor; stable and unstable manifolds; homoclinic point. 1, INTRODUCTION Since H~non's analysis, (0 considerable interest has been devoted to the mapping of the plane into itself: T(x, y) = (y + 1 - ax2,bx) (1.1) H6non studied T in the case a = 1.4, b -- 0.3; numerical investigations of (1.1)(1-3) have shown that for these and other values of the parameters the system (1.1) seems to exhibit a "chaotic behavior." This behavior was related via a theorem by Smale, to the existence of a transversal homoclinic orbit. (s) Smale's theorem (4) (see also Ref. 5) states that if a diffeomorphism F has a transversal homoclinic orbit, then there exists a Cantor set A in which, for some M, F M is topologically equivalent to the shift automor- phism. Curry, in Ref. 3, gave numerical evidence to the existence of a transversal homoclinic orbit in the case a = 1.4, b = 0.3. More recently Marotto (6) proved analytically that for a > 1.55 and b small enough such an orbit exists; however, Marotto's proof does not provide an explicit range of b values for which his results hold. Here, by using a parametric representation of the stable and unstable manifolds and with the aid of a 1 Istituto Matematico, Universit/t di Modena, 1-41100 Modena, Italy. 757 0022-4715/81/0800-0757503.00/0 9 PlenumPublishing Corporation