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 and CMMI-1447440.|
|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 chaotoic orbits along, leading to super-diffusive behavior.
In a recent paper, 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.
|Transport in Symplectic maps has been studied extensively in the area-preserving case, and is especially well-understood using the ideas of flux through cantori developed by MacKay, Meiss and Percival. The corresponding picture for higher-dimensional symlectic maps is still, largely, open. Nevertheless, we know a number of things. For nearly integrable maps there are typically many invariant tori, by the KAM theorem, and transport, due to Arnold's mechanism, is very slow, according to Nekhoroshev's theorem.
How do these restrictions apply to the volume-prerserving case? There is still a version of KAM theory that applies, so nearly integrable VP maps have many tori. But does Nekhoroshev's theorem apply? In a paper with Guillery, we show that it does not: even for maps with a positive definite twist, there can be rapid transport along resonance channels. The figure shows a 2D action-slice through a phase space of a four dimensional VP map. The grayscale is the FLI: dark gray regions correspond to small Lyapunov exponent, and many invariant tori. The white regions are chaotic due to resonances. The red dots show an orbit that drifts rapidly along a resonance channel, switching from one to another upon intersection.