A Rigorous ODE Solver and Smale’s 14th Problem
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 attra...
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| Veröffentlicht in: | Foundations of computational mathematics 2002, Vol.2 (1), p.53-117 |
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| creator | Tucker, Warwick |
| description | 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. [PUBLICATION ABSTRACT] |
| doi_str_mv | 10.1007/s002080010018 |
| format | Article |
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| subjects | Algorithms Computation Differential equations Lorenz Curve Lorenz equations Mathematical analysis Mathematical models Mathematical problems Ordinary differential equations Perturbation methods Solvers |
| title | A Rigorous ODE Solver and Smale’s 14th Problem |
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