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In a Hamiltonian system Chirikov's resonance overlap criterion approximately predicts the onset of chaotic behavior. Furthermore in a system where resonances overlap, the strengths of the resonances and their frequency differences can be used to approximate diffusion coefficients (as explored by Chirikov in '79). The overlap criterion is easy to estimate and so often used for physical systems. I was surprised to hear that there are non-linear systems that appear to satisfy a resonance overlap criterion but do not exhibit chaotic behavior.

Is there a simple example of such a system?
What are the properties of such systems?

The site above refers to the Toda lattice but I am not gaining any intuition from it.

Some background ---

One can describe the KAM theorem in terms decaying Fourier coefficients for the perturbation and iterative perturbation theory for the Hamiltonian system. At some level in the perturbation theory the Fourier coefficients are sufficiently small that they no longer overlap and so the perturbation expansion must converge (and so you get an integrable model or tori). The focus here is on the existence of tori not on the onset of chaos.

The reason for my interest is I am tempted to try and classify N-body systems based on width of analyticity of their perturbations (setting the decay rates of their Fourier coefficients) but the easiest way I know to do this is to count resonances and devise a way to estimate when they fill phase space.

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1 Answer 1

This is a very partial answer to your question but let's see if this helps

So from a very heuristic perspective in the KAM-case the onset of chaos is a result of the smale horseshoes generated by the homoclinic tangles. Of course not any homoclinic tangle has the generating mechanism for the smale horseshoe and this is where the Chirikov condition comes in. If the underlying system is constructed in such a way that the system satisfies (many) resonances but these resonances are actually symmetries of the system then you can `fool' the Chirikov condition in thinking that chaotic dynamics takes place while the system on the whole might be integrable.

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