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.|
|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. We applied a smoothed version of time averaging—based on earlier work of Das, Sander & Yorke—to accurately determine whether an orbit of an area-preserving map is chaotic or not, and when it is regular to compute its rotation number. The picture (right) shows results for the standard map: the color indicates the number of digits
computed in the average (up to 18-red) as a function of initial condition (varying y along a line x = 0.321), and of the parameter k. Orbits that are chaotic (digits less than 5) are black. We use this method to construct the so-called
Invariant tori can also be found numerically by taking limits of periodic orbits and by iterative methods based on the conjugacy to rotation. In these methods, one fixes a frequency vector and attempts to find invariant sets on which the dynamics has this frequency. In the current paper we do not fix the rotation vector in advance, so this method permits one to accurately compute the rotation vector for each initial condition that lies on a regular orbit. As such the method is analogous to Laskar’s frequency analysis, which uses a windowed Fourier transform to compute rotation numbers.