Tomas Johnson
Rigidity for infinitely renormalizable area-preserving maps.
Mon, 05/07/2012 - 15:01 — SomNambulAuthors: Denis Gaidashev, Tomas Johnson, Marco Martens
Subjects: Dynamical Systems
AbstractArea-preserving maps have been observed to undergo a universal
period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser
period doubling cascade in one-dimensional dynamics. A renormalization approach
has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of
existence of a conservative renormalization fixed point. Furthermore, it has
been shown that {\it infinitely renormalizable maps} in a neighborhood of this
fixed point admit invariant Cantor sets on which the dynamics is "stable" - the
Lyapunov exponents vanish on these sets.Rigorous Enclosures of Slow Manifolds.
Wed, 01/11/2012 - 15:01 — SomNambulAuthors: John Guckenheimer, Tomas Johnson, Philipp Meerkamp
Subjects: Dynamical Systems
AbstractSlow-fast dynamical systems have two time scales and an explicit parameter
representing the ratio of these time scales. Locally invariant slow manifolds
along which motion occurs on the slow time scale are a prominent feature of
slow-fast systems. This paper introduces a rigorous numerical method to compute
enclosures of the slow manifold of a slow-fast system with one fast and two
slow variables. A triangulated first order approximation to the two dimensional
invariant manifold is computed "algebraically".No elliptic islands for the universal area-preserving map.
Tue, 12/14/2010 - 16:01 — SomNambulAuthors: Tomas Johnson
Subjects: Dynamical Systems
AbstractA renormalization approach has been used in \cite{EKW1} and \cite{EKW2} to
prove the existence of a \textit{universal area-preserving map}, a map with
hyperbolic orbits of all binary periods. The existence of a horseshoe, with
positive Hausdorff dimension, in its domain was demonstrated in \cite{GJ1}. In
this paper the coexistence problem is studied, and a computer-aided proof is
given that no elliptic islands with period less than $18$ exist in the domain.
It is also shown that the area enclosed by elliptic islands is less than
$0.046$.Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Stable Sets.
Wed, 11/18/2009 - 16:01 — SomNambulAuthors: Denis Gaidashev, Tomas Johnson
Subjects: Dynamical Systems
AbstractIt is known that the famous Feigenbaum-Coullet-Tresser period doubling
universality has a counterpart for area-preserving maps of ${\fR}^2$. A
renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a
computer-assisted proof of existence of a "universal" area-preserving map $F_*$
-- a map with orbits of all binary periods $2^k, k \in \fN$. In this paper, we
consider {\it infinitely renormalizable} maps -- maps on the renormalization
stable manifold in some neighborhood of $F_*$ -- and study their dynamics.
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