Foundations of Computational Mathematics

, Volume 2, Issue 1, pp 53–117

A Rigorous ODE Solver and Smale’s 14th Problem

  • Warwick Tucker
Article

DOI: 10.1007/s002080010018

Cite this article as:
Tucker, W. Found. Comput. Math. (2002) 2: 53. doi:10.1007/s002080010018

Abstract

We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic with directed rounding. As an application, we prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. This conjecture was recently listed by Steven Smale as one of several challenging problems for the twenty-first century. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. Furthermore, the flow of the equations admits a unique SRB measure, whose support coincides with the attractor. The proof is based on a combination of normal form theory and rigorous computations.

Key words

Lorenz attractorDynamical systemsAuto-validating algorithmsNormal forms

AMS Classification

Primary 37C10Secondary 37D45, 65G30

Copyright information

© SFoCM 2002

Authors and Affiliations

  • Warwick Tucker
    • 1
  1. 1.Department of Mathematics Malott Hall Cornell University Ithaca, NY 14853-4201, USA warwick@math.cornell.eduUSA