Models Methods Software

Dan Hughes

References for Chaos Part 0

The literature references cited in Chaos Part 0 are listed in this post. Maybe this will become a Pages.

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April 28, 2007 Posted by | Chaos and Lorenz | , , , | Leave a comment

Chaos Part 0: ODE Chaotic Response is Numerical Noise

In this post we look at a every important issue relative to the foundations of chaotic response as inferred from numerical calculations with systems of ordinary differential equations (ODEs). The relationship of the contents of these notes to calculations with numerical weather prediction (NWP) and general climate models (GCMs) might be discussed in future posts.

Summary Statement
Converged numerical solutions of systems of ordinary differential equations that exhibit chaotic response have never been published. Converged numerical solutions have yet to be obtained for such equation systems. The preceding statements need to be qualified by limiting them to long ranges of values of the independent variable. And of course the statements apply only to the situations for which a given equation system exhibits chaotic response. Additionally, the statements are based on what I will denote as standard numerical solution methods. Standard numerical solution methods include just about all methods that are widely used. Solution methods other than standard, denoted as self-validating, will be specified later in these notes.

Given that chaotic response of complex dynamical systems at its foundation is based almost entirely on numerical solutions of the ODE systems, this situation is almost beyond comprehension. The recently-published paper reviewed here and here on this blog clearly shows the lack of convergence as the step size is refined.

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April 28, 2007 Posted by | Uncategorized | , , | 2 Comments