Rigorously Shadowing Numerical ODE Integrations by Containment
Abstract:
This document summarizes the primary contributions of my Ph.D. thesis, and the tasks that remain to be accomplished. It is intentionally terse, without much background included or many references cited. Note: my Qualifying Depth Paper, in case you don't have a copy handy, can be browsed on the Web at "http://www.cs.utoronto.ca/~wayne/research/thesis/depth/". My M.Sc. thesis is nearby, at ": : : thesis/msc/". Finally, my Research Proposal is at ": : : thesis/RP.ps.gz". An exact trajectory of a dynamical system lying close to a numerical trajectory is called a shadow. We present a general-purpose algorithm for rigorously proving the existence of shadows of numerical ODE integrations of arbitrary dimension. Much of the rigor is provided automatically by interval arithmetic and validated ODE integration software that is freely available. The method is a generalization of a previously published containment process that was applicable only to two-dimensional maps. We extend it to handle maps of arbitrary dimension, and finally to ODEs. It involves the building of hypercubes around the discrete numerical trajectory through which the shadow is guaranteed to pass at appropriate times. The proof consists of two steps: first, the rigorous computational verification of a certain property; and second, a simple geometric argument showing that the computed property implies the existence of a shadow. The computational step is almost entirely automated, easily adaptable to any ODE problem, and asymptotically more efficient than previous rigorous methods since it requires a rigorous bound only on the error of the solution, rather than on its variational equation. The algorithm also allows rigorously for the "rescaling " of time, which is a necessary ingredient for successfully shadowing ODEs. Finally, the algorithm is "local", in the sense that it builds the shadow inductively, requiring information only from the most recent integration step, rather than information about "special " points arbitrarily far away on the numerical trajectory or bounds on resolvents along the entire trajectory. The resulting algorithm is capable of shadowing the Lorenz equations longer (to this author's knowledge) than any previously published results, with local errors of 10 \Gamma6
