Published: Oct. 22, 2020

Amanda Hampton, Department of Applied Mathematics, University of Colorado Boulder

Anti-integrability for Quadratic Volume Preserving Maps

The dynamics of volume preserving maps can model a variety of mixing problems ranging from microscopic granular mixing, to dispersion of pollutants over our planet's atmosphere. We study a general quadratic volume preserving map using a concept first introduced thirty years ago in the field of solid-state physics: the anti-integrable (AI) limit. In the AI limit, orbits degenerate to a sequence of symbols and the dynamics reduces to the shift operator on the symbols. Such symbolic dynamics is a pure form of chaos. An advantage of this approach is that one can prove the existence of infinitely many orbits and, using a version of the contraction mapping theorem, find orbits that continue away from the limit to become deterministic orbits of the original system. A  novelty of the AI limit in our case is that one often needs to use contraction arguments to find orbits at the AI limit, as well as to determine necessary conditions for these orbits to persist for added perturbations. At the AI limit we can visualize orbits using one-dimensional maps (or actually “relations”). Upon perturbation these become full orbits in 3-dimensions with intriguing Cantor-like structures. A future goal is to continue these orbits to a (nearly) integrable case to understand how chaotic structures undergo bifurcation to regularity.