Physica D 145 (2000) 13–24 Universal pattern for homoclinic and periodic points D.V. Bevilaqua ∗,1 , M. Bas´ ılio de Matos Depto. de F´ ısica Teórica, IF, Universidade Federal do Rio de Janeiro, CP 68528, 21945-970 Rio de Janeiro, Brazil Received 21 September 1999; received in revised form 21 March 2000; accepted 3 April 2000 Communicated by J.D. Meiss Abstract Novel results regarding the distribution of homoclinic and periodic points in quadratic conservative mappings are presented. For these calculations, normal forms, in order to provide a “natural” system of coordinates, are used. Very simple asymptotic formulas for the coordinates of periodic and homoclinic points are obtained, and the entire invariant set is reconstructed with these formulas. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Normal forms; Homoclinic orbits; Conservative Hénon map 1. Introduction Since the works of Poincaré [1], it is known that homoclinic orbits play a significant role in the understanding of chaotic systems. While trying to comprehend the intersection of the unstable and stable manifolds of a hyperbolic point of a map on the plane, he wrote: “The complexity of this figure will be striking, and I shall not even try to draw it. Nothing is more suitable for providing us with an idea of the complex nature of the three body-problem, and all the problems of dynamics in general”. A lot of advances have been made towards the understanding of the homoclinic tangle and related phenomena. At present, the mere existence of a transversal homoclinic orbit can be used as the criterion for characterizing a system as chaotic [2]. Over the last two decades a lot of work was developed, in particular: the study of homoclinic tangencies and bifurcations; splitting of separatrices; and in the characterization of the homoclinic tangle (e.g. [3–7] and references therein). This illustrates that the study of the homoclinic tangle continues to be an important subject matter in the field of dynamics. In general, if there exists a hyperbolic fixed point of a mapping, the stable and unstable manifolds do not connect smoothly, but intersect transversely. Once these manifolds intersect they should intersect each other an infinite number of times, thus creating the homoclinic tangle (see Fig. 1). Since both stable and unstable manifolds are invariant their intersections, at the forward and backward iterations, must stay on both manifolds. They are therefore bi-asymptotic to the fixed point. These intersection points are called homoclinic points. Smale demonstrated that the set of homoclinic points is a Cantor set, this set is hyperbolic and the dynamics on it is described by shifts of bi-infinite sequences of symbols. From his work, we learn that homoclinic orbits accumulate periodic and other homoclinic orbits [2]. ∗ Corresponding author. 1 Supported by a grant from CNPq. 0167-2789/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII:S0167-2789(00)00088-9