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.