On the Strange Attractor and Transverse Homoclinic Orbits for the Lorenz Equations

Abstract

For suitable neighborhoods of a periodic solution of the Lorenz equations, individual solutions on the stable manifold Ms or on the unstable manifold Mu can be represented by means of series expansions. A convergence proof is presented. For truncations of these series, expressions for remainder terms are given. Interval enclosures of the truncated series and their remainder terms yield starting intervals for applications of Lohner′s enclosure algorithm as applied to the solutions of the Lorenz equations. This allows discussions of the strange attractor and homoclinic (transverse) orbits. Selected results are presented.