64 research outputs found

### Efficient Computation of Invariant Tori in Volume-Preserving Maps

In this paper we implement a numerical algorithm to compute codimension-one
tori in three-dimensional, volume-preserving maps. A torus is defined by its
conjugacy to rigid rotation, which is in turn given by its Fourier series. The
algorithm employs a quasi-Newton scheme to find the Fourier coefficients of a
truncation of the series. This technique is based upon the theory developed in
the accompanying article by Blass and de la Llave. It is guaranteed to converge
assuming the torus exists, the initial estimate is suitably close, and the map
satisfies certain nondegeneracy conditions. We demonstrate that the growth of
the largest singular value of the derivative of the conjugacy predicts the
threshold for the destruction of the torus. We use these singular values to
examine the mechanics of the breakup of the tori, making comparisons to
Aubry-Mather and anti-integrability theory when possible

### Statistics of the Island-Around-Island Hierarchy in Hamiltonian Phase Space

The phase space of a typical Hamiltonian system contains both chaotic and
regular orbits, mixed in a complex, fractal pattern. One oft-studied phenomenon
is the algebraic decay of correlations and recurrence time distributions. For
area-preserving maps, this has been attributed to the stickiness of boundary
circles, which separate chaotic and regular components. Though such dynamics
has been extensively studied, a full understanding depends on many fine details
that typically are beyond experimental and numerical resolution. This calls for
a statistical approach, the subject of the present work. We calculate the
statistics of the boundary circle winding numbers, contrasting the distribution
of the elements of their continued fractions to that for uniformly selected
irrationals. Since phase space transport is of great interest for dynamics, we
compute the distributions of fluxes through island chains. Analytical fits show
that the "level" and "class" distributions are distinct, and evidence for their
universality is given.Comment: 31 pages, 13 figure

### Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series

Topology based analysis of time-series data from dynamical systems is
powerful: it potentially allows for computer-based proofs of the existence of
various classes of regular and chaotic invariant sets for high-dimensional
dynamics. Standard methods are based on a cubical discretization of the
dynamics and use the time series to construct an outer approximation of the
underlying dynamical system. The resulting multivalued map can be used to
compute the Conley index of isolated invariant sets of cubes. In this paper we
introduce a discretization that uses instead a simplicial complex constructed
from a witness-landmark relationship. The goal is to obtain a natural
discretization that is more tightly connected with the invariant density of the
time series itself. The time-ordering of the data also directly leads to a map
on this simplicial complex that we call the witness map. We obtain conditions
under which this witness map gives an outer approximation of the dynamics, and
thus can be used to compute the Conley index of isolated invariant sets. The
method is illustrated by a simple example using data from the classical H\'enon
map.Comment: laTeX, 9 figures, 32 page

### Moser's quadratic, symplectic map

In 1994, J\"urgen Moser generalized H\'enon's area-preserving quadratic map
to obtain a normal form for the family of four-dimensional, quadratic,
symplectic maps. This map has at most four isolated fixed points. We show that
the bounded dynamics of Moser's six parameter family is organized by a
codimension-three bifurcation, which we call a quadfurcation, that can create
all four fixed points from none.
The bounded dynamics is typically associated with Cantor families of
invariant tori around fixed points that are doubly elliptic. For Moser's map
there can be two such fixed points: this structure is not what one would expect
from dynamics near the cross product of a pair of uncoupled H\'enon maps, where
there is at most one doubly elliptic point. We visualize the dynamics by escape
time plots on 2D planes through the phase space and by 3D slices through the
tori.Comment: 12 pages, 6 figures. For videos see
https://www.comp-phys.tu-dresden.de/supp

### Nonexistence of Invariant Tori Transverse to Foliations: An Application of Converse KAM Theory

Invariant manifolds are of fundamental importance to the qualitative
understanding of dynamical systems. In this work, we explore and extend
MacKay's converse KAM condition to obtain a sufficient condition for the
nonexistence of invariant surfaces that are transverse to a chosen 1D
foliation. We show how useful foliations can be constructed from approximate
integrals of the system. This theory is implemented numerically for two models,
a particle in a two-wave potential and a Beltrami flow studied by Zaslavsky
(Q-flows). These are both 3D volume-preserving flows, and they exemplify the
dynamics seen in time-dependent Hamiltonian systems and incompressible fluids,
respectively. Through both numerical and theoretical considerations, it is
revealed how to choose foliations that capture the nonexistence of invariant
tori with varying homologies.Comment: 25 pages, 18 figure

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