Question about the KAM theorem

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The discussion focuses on using the KAM theorem to analyze a Hamiltonian system with a nonlinear perturbation that exhibits chaotic motion at a specific perturbation parameter. The user seeks to determine the critical value of the perturbation parameter analytically for chaos to occur. It is suggested to utilize the overlap criterion developed by Chirikov in this analysis. The general approach involves transforming to action-angle coordinates and expanding the perturbation in Fourier series. This method aims to provide a clearer understanding of the conditions under which chaos emerges in the system.
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I have an Hamiltonian with a non linear perturbation attached to it. When the perturbation parameter equals .2 the system at a certain initial condition exhibits chaotic motion. I found this out graphically. I would like to calculate how large my perturbation parameter has to be analytically for the system to exhibit chaos. Would I use the KAM theorem for such a calculation? Any help will be much appreciated.
 
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There is the overlap criterion developed by Chirikov: link.

The general procedure is to go to the action-angle coordinates of the integrable system, and then having written the perturbation in terms of these variables expand it in Fourier series.
 
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