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Research of James Meiss

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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.

Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps

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 critical function: the maximum value of k for which there is an invariant circle of a given rotation number

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.

Rottation number as a function of y and k for the standard map


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

Books

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

Anti-Integrability

Invariant Tori

Piecewise Smooth Bifurcations

Polynomial Maps

Synchronization

Transitory Dynamics

Transport

Twistless Bifurcations


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revised May 31, 2020

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