1,147 research outputs found
Symbolic codes for rotational orbits
Symbolic codes for rotational orbits and “islands-around-islands” are constructed for the
quadratic, area-preserving H´enon map. The codes are based upon continuation from an antiintegrable
limit, or alternatively from the horseshoe. Given any sequence of rotation numbers
we obtain symbolic sequences for the corresponding elliptic and hyperbolic rotational orbits.
These are shown to be consistent with numerical evidence. The resulting symbolic partition
of the phase space consists of wedges constructed from images of the symmetry lines of the
map
Homoclinic Bifurcations for the Henon Map
Chaotic dynamics can be effectively studied by continuation from an
anti-integrable limit. We use this limit to assign global symbols to orbits and
use continuation from the limit to study their bifurcations. We find a bound on
the parameter range for which the Henon map exhibits a complete binary
horseshoe as well as a subshift of finite type. We classify homoclinic
bifurcations, and study those for the area preserving case in detail. Simple
forcing relations between homoclinic orbits are established. We show that a
symmetry of the map gives rise to constraints on certain sequences of
homoclinic bifurcations. Our numerical studies also identify the bifurcations
that bound intervals on which the topological entropy is apparently constant.Comment: To appear in PhysicaD: 43 Pages, 14 figure
Rotation numbers of invariant manifolds around unstable periodic orbits for the diamagnetic Kepler problem
In this paper, a method to construct topological template in terms of
symbolic dynamics for the diamagnetic Kepler problem is proposed. To confirm
the topological template, rotation numbers of invariant manifolds around
unstable periodic orbits in a phase space are taken as an object of comparison.
The rotation numbers are determined from the definition and connected with
symbolic sequences encoding the periodic orbits in a reduced Poincar\'e
section. Only symbolic codes with inverse ordering in the forward mapping can
contribute to the rotation of invariant manifolds around the periodic orbits.
By using symbolic ordering, the reduced Poincar\'e section is constricted along
stable manifolds and a topological template, which preserves the ordering of
forward sequences and can be used to extract the rotation numbers, is
established. The rotation numbers computed from the topological template are
the same as those computed from their original definition.Comment: 8 figures, 1 tabl
Steve Smale and Geometric Mechanics
Thus, one can say-perhaps with only a slight danger of oversimplification-
that reduction theory synthesises the work of Smale, Arnold (and their
predecesors of course) into a bundle, with Smale as the base and Arnold as
the fiber. This bundle has interesting topology and carries mechanical connections (with associated Chern classes and Hannay-Berry phases) and has interesting singularities (Arms, Marsden, and Moncrief, Guillemin and Sternberg, Atiyab, and otbers). We will describe some of these features later
Nonlinear rotations on a lattice
We consider a prototypical two-parameter family of invertible maps of
, representing rotations with decreasing rotation number. These
maps describe the dynamics inside the island chains of a piecewise affine
discrete twist map of the torus, in the limit of fine discretisation. We prove
that there is a set of full density of points which, depending of the parameter
values, are either periodic or escape to infinity. The proof is based on the
analysis of an interval-exchange map over the integers, with infinitely many
intervals.Comment: LaTeX, 34 pages with 4 figure
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