The destruction of tori in volume-preserving maps

Invariant tori are prominent features of symplectic and volume-preserving maps. From the point of view of chaotic transport the most relevant tori are those that are barriers, and thus have codimension one. For an n-dimensional volume-preserving map, such tori are prevalent when the map is nearly “integrable,” in the sense of having one action and n − 1 angle variables. As the map is perturbed, numerical studies show that the originally connected image of the frequency map acquires gaps due to resonances and domains of nonconvergence due to chaos. We present examples of a three-dimensional, generalized standard map for which there is a critical perturbation size, εc, above which there are no tori. Numerical investigations to find the “last invariant torus” reveal some similarities to the behavior found by Greene near a critical invariant circle for area preserving maps: the crossing time through the newly destroyed torus appears to have a power law singularity at εc, and the local phase space near the critical torus contains many high-order resonances.

► Codimension-one invariant tori are robust features of smooth volume-preserving maps.
► Resonances in the rotation vector (frequency) map govern the structure of the dynamics.
► Perturbation causes the image of the frequency map to move and collapse to a Cantor set.
► Invariant tori exist in regular regions of the frequency map.
► The destruction of tori is determined finding orbits that are either confined or cross through regions of phase space.