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Chaos and Instability in Integration Algorithms Model Documents

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Main Document

Chaos and Instability in Integration Algorithms Model 

written by Fernando Silva Fernandes

The Chaos and Instability in Integration Algorithms Model uses the the nonlinear and non-autonomous differential equation dx/dt = t - x^2 to show how careful one must be when choosing a differential equation solver, even for an apparently simple problem.  The chaotic behaviors here detected do not seem an intrinsic property of the differential equation, though the patterns are startling similar to the May-Feigenbaum's standard map. Indeed they disappear with more robust integration algorithms. The chaos here invoked only means that for some algorithms and integration times the trajectories are not random, but are unpredictable. Apparently, in such cases, there is also no dependence on initial conditions, a typical characteristic of intrinsic chaotic systems. This application illustrates that using an adaptive algorithm is not only a question of speed and CPU use, but also of precision and stability and the care that must be taken to avoid wrong conclusions, especially when using computer packages as "black-boxes."

The Chaos and Instability in Integration Algorithms Model was developed using version 5 of the Easy Java/JavaScript Simulations (EjsS) modeling tool.  It is distributed as a ready-to-run (compiled) Java archive.   Double clicking the jar file will run the program if Java is installed.  You can modify this simulation if you have EjsS installed by right-clicking within the map and selecting "Open Ejs Model" from the pop-up menu item.

Published September 1, 2014
Last Modified September 2, 2014

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Source Code Documents

Chaos and Instability in Integration Algorithms Source Code 

The source code zip archive contains an XML representation of the Chaos and Instability in Integration Algorithms Model.   Unzip this archive in your EjsS workspace to compile and run this model using EjsS.

Last Modified September 1, 2014

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