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, DMS-1812481 and AGS-2001670. 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. Support from the Simons Foundation, grant #601972, "Hidden Symmetries and Fusion Energy" is also gratefully acknowledged.|
|The dynamics of an integrable Hamiltonian or volume-preserving system consists of periodic and quasi-periodic motion on invariant tori. When such a system is smoothly perturbed, Kolmogorov-Arnold-Moser (KAM) theory implies that some of these tori persist and some are replaced by isolated periodic orbits, islands, or chaotic regions. On each KAM torus, the dynamics is conjugate to a rigid rotation with some fixed frequency vector. Typically, as the perturbation grows the proportion of chaotic orbits increases and more of the tori are destroyed.
In a paper with Evelyn Sander, we explore an alternative technique, based on windowed Birkhoff averages, to distinguish between chaotic, resonant, and quasiperiodic dynamics for area preserving maps. We applied this technique, in to find two-dimensional tori for 3D volume-preserving maps. An important question in both these situations is: how can one distinguish between ``irrational" and ``rational" numbers numerically. We show how an answer to question can be computed if it is reformulated: what is the smallest period rational within a given tolerance. This leads, in the invariant circle case, to a method based on the Farey tree expansion. In higher dimensions, a similar method can be applied to find commensurabilities
Recently, Nathan Duignan and I, applied these methods to flows. We show how the super-convergence of the weighted Birkhoff average also applies to the case of a smooth flow when the rotation vector is Diophantine, generalizing earlier work of Das, Sander & Yorke— for the map case.
We applied these methods to distinguish regular and chaotic regions for one-and-a-half degree of freedom Hamiltonian systems, using the two-wave model (that we also studied usng converse KAM methhods), and a simple model for magnetic field line flow. We also show that it can distinguish chaotic orbits in a "strange-nonchaotic-attractor" (SNC) first studied by Grebogi, Ott, Pelikan and Yorke. The interesting aspect of these orbits is that they lie on geometrically strange attractors, but have zero maximal Lyapunov exponents.
The picture (right) shows results for the two-wave model: the color indicates the number of digits computed in the average (up to 18-yellow) as a function of initial condition (varying p along a line x = 0.0). Orbits on the Poincare section that are chaotic (digits less than 5) are blue.
|The concept of anti-integrability was introduced by Aubry and Abramovicci in 1983 for the standard map, viewed as a linear chain of particles connected by springs in a periodic potential. They reasoned that the integrable limit corresponded to vanishing potential energy, so that the springs dominated giving equal spacing at equilibrium. By contrast, anti-integrability corresponds to vanishing kinetic energy, so that particles sit at critical points of the potential. What is most interesting about this limit is that it is relatively easy, using a contraction mapping style argument, to show that AI states persist, and this gives conjugacy to a shift on a symbolic dynamics.
In the paper, Amanda Hampton and I generalize these ideas to the family of quadratic three-dimensional diffeomorphisms that were obtained in Lomelí & Meiss . We write the map as a third difference equation, and scale to isolate the nonlinear terms. A unique feature of this study is that the AI limit corresponds to a quadratic correspondence---a quadratic curve that corresponds to a one-dimensional dynamical system. We show that there are a number of parameter values for which a full shift on two-symbols exists at the AI limit and that these orbits can be continued away from the limit.
The figure at the right shows an orbit of a 3D quadratic map. continued away from the AI limit. At the limit, the orbit falls on the intersection of the two elliptic cylinders. As we move away from this limit, the orbit maintains some of this structure.
More recently, Hampton and I studied bifurcations that create strange attractors for a special case of this family that can be thought of as a 3D version of Henon's map.