# Research of James Meiss

## Subjects

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.

## Books, Pedagogy and Reviews

### Books

• MacKay, R. S. and J.D. Meiss, Eds. (1987). Hamiltonian Dynamical Systems: a reprint selection. London, Adam-Hilgar Press, 784pp., ISBN 0-85274-205-3. (Buy from Amazon)
• Hazeltine, R. D. and J.D. Meiss (1991). Plasma Confinement. Redwood City, CA, Addison-Wesley, 394 pp., ISBN 0201-53353-5.
• R.D. Hazeltine and J.D. Meiss, Plasma Confinement, (2003) 2nd Edition, Dover Press, 480 pp., ISBN 0486432424. (Buy from Amazon)
• J.D. Meiss, Differential Dynamical Systems, (2007) SIAM, Philadelphia 412 pp., ISBN 978-0-899816-35-1.
• J.D. Meiss, Differential Dynamical Systems: Revised Edition. (2017) SIAM, Philadelphia, 392 pp., ISBN 978-1-61197-463-8.

## Fields of Research

### Plasma Physics

• A. Aydemir, R.D. Hazeltine, J.D. Meiss, and M. Kotschenreuther, "Destabilization of Alfvén-Resonant Modes by Resistivity and Diamagnetic Drifts", Physics of Fluids 30 4-6 (1987).
• J.D. Meiss, "Transport Near the Onset of Chaos", Physics Today, Physics News of 1986, January (1987).
• A. Y. Aydemir, R.D. Hazeltine, M. Kotschenreuther, J.D. Meiss, P.J. Morrison, D.W Ross, F. L. Waelbroeck, J.C. Wiley, "Nonlinear MHD Studies in Toroidal Geometry", Plasma Physics and Controlled Nuclear Fusion Research 1988, Lausanne, Switzerland (International Atomic Energy Agency, Vienna, 1989), 131-143.
• J.D. Meiss, "Comment on Microwave Ionization of H-atoms: breakdown of classical dynamics for high frequencies", Phys. Rev. Lett 62 1576 (1989).
• J.D. Meiss and R.D. Hazeltine, "Canonical Coordinates for Guiding Center Particles", Physics of Fluids, B2 2563-2567 (1990).
• Hayashi, T., T. Sato, H.J. Gardner and J.D. Meiss, "Evolution of Magnetic Islands in a Heliac", Physics of Plasmas 2 752-759 (1994).
• J.L. Tennyson J.D. Meiss and P.J. Morrison, "Self-Consistent Chaos in the Beam-Plasma Instability", Physica D 71 1-17 (1994). (PDF reprint).