Research of James Meiss
|Much of the research listed below has received support from the National Science Foundation, most recently under grants DMS-0707659, DMS-1211350, CMMI-1447440, CMMI 1553297, and DMS-1812481 Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF.|
|In 1994 Moser generalized Hénon's famous quadratic map to the four dimensional case. Moser's
quadratic sympletic map has at most four fixed points, and they are organized by a codimension three bifurcation that creates four fixed points at a single point in phase space. In a paper with Arnd Bäcker we study this quadfurcation, and show that it also occurs a when an accelerator mode is created in a four dimensional Froeschlé map.
The figure shows a three-dimsional slice through the 4D phase space. For this case there are four fixed points, two are doubly elliptic (red spheres) and two are elliptic-hyperbolic (green spheres). Invariant two-tori typically intersect the slice in a pair of rings. One such torus is shown projected from 4D, with the fourth dimension indicated by the color scale shown.
In a second paper we discuss the differences between the generic case and that of a weakly coupled pair of Hénon maps.
|Transport in Chirikov's area-preserving Standard Map appears to be ``quasilinear", that is
described by a random walk, when the parameter k is much larger than one: the action of an ensemble of initial conditions diffuses.
However, this normal diffusion fails dramatically when upon certain saddle-center bifurcations that lead to accelerator modes.
These new orbits create sticky, stable islands that accelerate, and drag chaotic orbits along, leading to super-diffusive behavior.
In a recent paper with Narcis Miguel, Carles Simo, and Arturo Vieiro, we studied the generic form of such accelerator modes in three-dimensional volume-preserving maps. We consider the case of a map with two angle variables, and one action. We show that the local form of a bubble can be described by a quadratic VP map, a special case of that derived in a paper with Lomeli. We discuss the trapping statistics for orbits near the bubble of stabilty, showing that there is a power-law decay similar to that seen in the area-preserving case, and to that we saw in a map with Mullowney and Julien.
The picture shows one example of orbits trapped near 3D bubble, outlining a family of invariant two-tori.