Break-up of invariant surfaces in action–angle–angle maps and flows

An important class of three-dimensional volume-preserving maps and flows arises as a perturbation from integrable action–action–angle maps and flows. The properties of this class of maps and flows are discussed. While action–angle–angle volume-preserving maps admit an analog of the KAM theorem, general results on nonexistence of two-dimensional invariant manifolds of action–action–angle maps are proven here. Nonexistence of such two-dimensional invariant manifolds means possibility of global transport and a mechanism for such transport — the local mechanism of resonance-induced dispersion [Phys. Rev. Lett. 75 (1995) 3669] — is studied perturbatively. Resonance-induced dispersion is shown to arise from the existence of periodic orbits that survive perturbation at places where two-dimensional invariant manifolds break down.