Mathematica and KAM tori reforming

28/06/2009

Some days ago I received an email from Wolfram Research asking to me to produce a demonstration for their demonstrations project based on my last proved theorem about KAM tori reforming (see here). WolframDemosBeing aware of the power of Mathematica I have found the invitation quite stimulating.  The idea behind these demonstrations is to use Mathematica’s command Manipulate that permits to have interactive presentations. A typical application is exactly in the area of differential equations where you can have some varying parameters. But the possibilities are huge for this method and, indeed, you can find almost 5000 demonstrations at that site. Indeed, in this way you are able to explore the behavior of mathematical models interactively and this appears as a really helpful tool. If you mean to send a demo of yours, be advised that it will undergo peer-review. So, such a publication has exactly the same value of other academic titles.

In a few days I prepared the demo and I have sent it to Wolfram. It was accepted for publication last friday. You can find it here. You can check by yourself the truthfulness of my theorem. The advantage to work in classical mechanics is that you can have immediately an idea of what is going on by numerics. Manipulate of Mathematica is a powerful tool in your hands to accomplish such an aim.

Finally, you can download the source code and modify it by yourself changing ranges, equations and so on. I tried the original Duffing oscillator without dissipation and, granted the validity of KAM theorem, one can verify an identical behavior with tori reforming for a very large perturbation. A shocking evidence without experiments, isn’t it?

Update: My demonstration has been updated (see here). The code has been improved, and so the presentation, due to a PhD student, Simon Tyler, that did this work. Thank you very much, Simon!


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