Universal pattern underlying the recurrence of Kolmogorov-Arnol’d-Moser tori

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Abstract
The novel phenomenon of exchange of stability among periodic orbits is studied with a view of understanding the recurrence of Kolmogorov-Arnol’d-Moser tori in Hamiltonian systems. This leads to a very complex phase diagram in the extended Frenkel-Kontorova model with cascades of first-order transitions separating nearby competing commensurate ground states. The Farey route to a given rotation number allows a very simple universal additive rule for constructing the phase diagram and understanding the self-similar structure underlying the nobel incommensurabilities.