Volume-preserving maps with an invariant

Abstract

Several families of volume-preserving maps on R(3) that have an integral are constructed using techniques due to Suris. We study the dynamics of these maps as the topology of the two-dimensional level sets of the invariant changes. (c) 2002 American Institute of Physics.

Merger of coherent structures in time-periodic viscous flows

“…The topological equivalence to spheres namely means that the invariant surfaces typically accommodate one ͑or more͒ isolated periodic points that, through connectivity of fluid parcels, each join with adjacent points into one ͑or more͒ periodic lines intersecting the invariant surfaces. 4,5 ͑Note that in the present configuration the period-1 line is implied by symmetry properties of the flow. 6 ͒ Nondegenerate periodic lines thus formed are typically partitioned into elliptic and hyperbolic segments that connect through isolated parabolic points ͑Fig.…”

“…͑Note the map is areapreserving in the proximity of the islands. 4 ͒ Hence, the tracer dynamics are locally governed by two constants of motion, namely F 1 ͑r , z͒ and H͑ , ͒, implying that ⌽ is locally a two-action map. Perturbation effectuates coalescence of the closed orbits into tubes that are parameterized by an adiabatic invariant 1 and transforms the local two-action map into a local one-action map in essentially the same way as the basic flow.…”

“…Studies on such one-action maps thus far performed primarily concern the dynamics within invariant surfaces rather than the response to perturbations and, moreover, involve highly idealized configurations. 4,5 Investigations on the response of the one-action state to perturbations appear nonexistent to date, though. This motivates the study hereafter on the response of an experimentally realizable one-action map involving invariant surfaces topologically equivalent to spheres to perturbation by inertial effects.…”

Nilpotent normal form for divergence-free vector fields and volume-preserving maps

“…Volume-preserving maps are a natural generalization of areapreserving maps to higher dimensions. They also arise as the normal form for certain homoclinic bifurcations for three-dimensional systems [15,14], and as integrators for incompressible flows [27,26,18,32] and thus have intrinsic mathematical interest [6,3,34,28,20,13,21,22].…”

Heteroclinic intersections between invariant circles of volume-preserving maps

“…However, for a special choice of h there is an integral. This example is related to the work of Suris [26,27] on area-preserving integrable maps, but is distinct from the threedimensional maps found in [28] …”