187 research outputs found
Simultaneous Border-Collision and Period-Doubling Bifurcations
We unfold the codimension-two simultaneous occurrence of a border-collision
bifurcation and a period-doubling bifurcation for a general piecewise-smooth,
continuous map. We find that, with sufficient non-degeneracy conditions, a
locus of period-doubling bifurcations emanates non-tangentially from a locus of
border-collision bifurcations. The corresponding period-doubled solution
undergoes a border-collision bifurcation along a curve emanating from the
codimension-two point and tangent to the period-doubling locus here. In the
case that the map is one-dimensional local dynamics are completely classified;
in particular, we give conditions that ensure chaos.Comment: 22 pages; 5 figure
Generic Twistless Bifurcations
We show that in the neighborhood of the tripling bifurcation of a periodic
orbit of a Hamiltonian flow or of a fixed point of an area preserving map,
there is generically a bifurcation that creates a ``twistless'' torus. At this
bifurcation, the twist, which is the derivative of the rotation number with
respect to the action, vanishes. The twistless torus moves outward after it is
created, and eventually collides with the saddle-center bifurcation that
creates the period three orbits. The existence of the twistless bifurcation is
responsible for the breakdown of the nondegeneracy condition required in the
proof of the KAM theorem for flows or the Moser twist theorem for maps. When
the twistless torus has a rational rotation number, there are typically
reconnection bifurcations of periodic orbits with that rotation number.Comment: 29 pages, 9 figure
Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps
Resonance tongues are mode-locking regions of parameter space in which stable
periodic solutions occur; they commonly occur, for example, near Neimark-Sacker
bifurcations. For piecewise-smooth, continuous maps these tongues typically
have a distinctive lens-chain (or sausage) shape in two-parameter bifurcation
diagrams. We give a symbolic description of a class of "rotational" periodic
solutions that display lens-chain structures for a general -dimensional map.
We then unfold the codimension-two, shrinking point bifurcation, where the
tongues have zero width. A number of codimension-one bifurcation curves emanate
from shrinking points and we determine those that form tongue boundaries.Comment: 27 pages, 6 figure
Resonance Zones and Lobe Volumes for Volume-Preserving Maps
We study exact, volume-preserving diffeomorphisms that have heteroclinic
connections between a pair of normally hyperbolic invariant manifolds. We
develop a general theory of lobes, showing that the lobe volume is given by an
integral of a generating form over the primary intersection, a subset of the
heteroclinic orbits. Our definition reproduces the classical action formula in
the planar, twist map case. For perturbations from a heteroclinic connection,
the lobe volume is shown to reduce, to lowest order, to a suitable integral of
a Melnikov function.Comment: ams laTeX, 8 figure
Ulam method for the Chirikov standard map
We introduce a generalized Ulam method and apply it to symplectic dynamical
maps with a divided phase space. Our extensive numerical studies based on the
Arnoldi method show that the Ulam approximant of the Perron-Frobenius operator
on a chaotic component converges to a continuous limit. Typically, in this
regime the spectrum of relaxation modes is characterized by a power law decay
for small relaxation rates. Our numerical data show that the exponent of this
decay is approximately equal to the exponent of Poincar\'e recurrences in such
systems. The eigenmodes show links with trajectories sticking around stability
islands.Comment: 13 pages, 13 figures, high resolution figures available at:
http://www.quantware.ups-tlse.fr/QWLIB/ulammethod/ minor corrections in text
and fig. 12 and revised discussio
Integrability and Ergodicity of Classical Billiards in a Magnetic Field
We consider classical billiards in plane, connected, but not necessarily
bounded domains. The charged billiard ball is immersed in a homogeneous,
stationary magnetic field perpendicular to the plane. The part of dynamics
which is not trivially integrable can be described by a "bouncing map". We
compute a general expression for the Jacobian matrix of this map, which allows
to determine stability and bifurcation values of specific periodic orbits. In
some cases, the bouncing map is a twist map and admits a generating function
which is useful to do perturbative calculations and to classify periodic
orbits. We prove that billiards in convex domains with sufficiently smooth
boundaries possess invariant tori corresponding to skipping trajectories.
Moreover, in strong field we construct adiabatic invariants over exponentially
large times. On the other hand, we present evidence that the billiard in a
square is ergodic for some large enough values of the magnetic field. A
numerical study reveals that the scattering on two circles is essentially
chaotic.Comment: Explanations added in Section 5, Section 6 enlarged, small errors
corrected; Large figures have been bitmapped; 40 pages LaTeX, 15 figures,
uuencoded tar.gz. file. To appear in J. Stat. Phys. 8
Chaos and Semiclassical Limit in Quantum Cosmology
In this paper we present a Friedmann-Robertson-Walker cosmological model
conformally coupled to a massive scalar field where the WKB approximation fails
to reproduce the exact solution to the Wheeler-DeWitt equation for large
Universes. The breakdown of the WKB approximation follows the same pattern than
in semiclassical physics of chaotic systems, and it is associated to the
development of small scale structure in the wave function. This result puts in
doubt the ``WKB interpretation'' of Quantum Cosmology.Comment: 14 pages in LaTex (RevTex), 6 figure
Quadratic Volume Preserving Maps
We study quadratic, volume preserving diffeomorphisms whose inverse is also
quadratic. Such maps generalize the Henon area preserving map and the family of
symplectic quadratic maps studied by Moser. In particular, we investigate a
family of quadratic volume preserving maps in three space for which we find a
normal form and study invariant sets. We also give an alternative proof of a
theorem by Moser classifying quadratic symplectic maps.Comment: Ams LaTeX file with 4 figures (figure 2 is gif, the others are ps
Molecular dynamics approach: from chaotic to statistical properties of compound nuclei
Statistical aspects of the dynamics of chaotic scattering in the classical
model of -cluster nuclei are studied. It is found that the dynamics
governed by hyperbolic instabilities which results in an exponential decay of
the survival probability evolves to a limiting energy distribution whose
density develops the Boltzmann form. The angular distribution of the
corresponding decay products shows symmetry with respect to angle. Time
estimated for the compound nucleus formation ranges within the order of
s.Comment: 11 pages, LaTeX, non
Visual Explorations of Dynamics: the Standard Map
The Macintosh application \textit{StdMap} allows easy exploration of many of
the phenomena of area-preserving mappings. This tutorial explains some of these
phenomena and presents a number of simple experiments centered on the use of
this program.Comment: Corrections in a couple of equations, and updated to the latest
version of StdMa
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