Question about the KAM theorem

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In summary, the conversation discussed chaotic motion in a Hamiltonian system with a non-linear perturbation attached to it. It was found that a perturbation parameter of 0.2 leads to chaotic motion at a certain initial condition. The speaker is looking to determine the minimum perturbation parameter needed for chaos analytically, and asked if the KAM theorem could be used for this calculation. The other person mentioned the overlap criterion developed by Chirikov, which involves transforming to action-angle coordinates and expanding the perturbation in Fourier series.
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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.
 

FAQ: Question about the KAM theorem

1. What is the KAM theorem and what does it prove?

The KAM (Kolmogorov-Arnold-Moser) theorem is a mathematical theorem that proves the existence of quasi-periodic solutions in dynamical systems. It states that for a certain class of Hamiltonian systems, there exists a set of initial conditions for which the solutions will remain quasi-periodic (i.e. have multiple frequencies) for a long period of time.

2. Who developed the KAM theorem?

The KAM theorem was first proved by three mathematicians: Andrey Kolmogorov, Vladimir Arnold, and Jürgen Moser. Each of them contributed to the theorem in different ways, and therefore it is sometimes referred to as the KAM theorem or the KAM theory.

3. What are some applications of the KAM theorem?

The KAM theorem has many applications in physics, astronomy, and engineering. It has been used to study the stability of the solar system, the motion of celestial bodies, and the behavior of pendulum clocks. It also has applications in plasma physics, fluid dynamics, and nonlinear optics.

4. Can the KAM theorem be applied to all dynamical systems?

No, the KAM theorem only applies to a certain class of Hamiltonian systems, which are systems that conserve energy. It does not apply to dissipative systems, which lose energy over time, or to non-Hamiltonian systems.

5. Are there any open problems or limitations to the KAM theorem?

Yes, there are still some open problems and limitations to the KAM theorem. One limitation is that it only guarantees the existence of quasi-periodic solutions for a certain set of initial conditions, and not for all initial conditions. There are also some open problems related to the stability and persistence of these solutions. Additionally, the KAM theorem does not apply to chaotic systems, which have unpredictable and sensitive behavior.

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