# Examples when Chirikov's Resonance Overlap Criterion fails to predict the onset of Chaos

In a Hamiltonian system Chirikov's resonance overlap criterion approximately predicts the onset of chaotic behavior. In a system where resonances overlap, the strengths of the resonances and their frequency differences can be used to approximate diffusion coefficients. The overlap criterion is easy to estimate and so often used to gain intuition on physical systems.

I was surprised to hear that there are dynamical systems that appear to satisfy a resonance overlap criterion but do not exhibit chaotic behavior.

Are there simple clear examples of such systems? I am curious as to what types of systems these might be --- it would be handy if I could show that any particular system is not likely to be in this class.

I posted on math.se a few days ago but unfortunately have no responses.

-
Do you mean examples of the completely integrable systems mentioned in the Scholarpedia article? – j.c. Aug 18 '11 at 18:30
The article mentions the Toda lattice. Taking the periodic Toda lattice (in Flashka variables), you obtain a finite dimensional non-linear system. Then one needs to realize that it is integrable (= easy), and find an orbit where the criterion fails. My guess is that any non-constant orbit will do. – Helge Aug 18 '11 at 18:46
@jc It seems obvious that a completely integrable system cannot be chaotic. Maybe what I should have asked is how a completely integrable system can be made to look like a resonance overlap criterion is satisfied. I think it is possible to formulate a resonance overlap criterion in terms of Fourier coefficients at any order of perturbation theory. Are there examples of non-integrable systems where the overlap criterion fails? Are there any simple examples? I am not really sure what I should be asking... – Alice Aug 18 '11 at 18:50
@Helge Based on your comment I found a very nice review article [here][1] showing me how to do this! Thanks, I better understand this example. [1]: lvov.weizmann.ac.il/Course/Toda-lattice.pdf – Alice Aug 18 '11 at 19:32