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.

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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.

Invariant Tori surrounding a pair of doubly elliptic fixed points in Moser's map'
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.

Bubble of stability near an Accelerator Mode

Bibliographic Information

  1. My Vita (pdf file)
  2. My Erdös number is at most 4
  3. ORCID
  4. Researcher ID
  5. Google Scholar
  6. zbMath

Books, Pedagogy and Reviews


Pedagogical Articles

Fields of Research

Computational Topology

Fluid Dynamics

Hamiltonian Dynamics

Plasma Physics

Classes of Dynamical Systems

Area-Preserving Maps

Symplectic Maps

Volume-Preserving Maps

Phemomena and Methods


Invariant Tori

Piecewise Smooth Bifurcations

Polynomial Maps


Transitory Dynamics


Twistless Bifurcations

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revised Dec 12, 2018

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