A Rigorous ODE Solver and Smale's 14th Problem

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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.

Original languageEnglish
Pages (from-to)53-117
Number of pages65
JournalFoundations of Computational Mathematics
Volume2
Issue number1
DOIs
Publication statusPublished - Feb 2002
Externally publishedYes

Keywords

  • Auto-validating algorithms
  • Dynamical systems
  • Lorenz attractor
  • Normal forms

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