305 research outputs found
Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations
We study the dynamics of the five-parameter quadratic family of
volume-preserving diffeomorphisms of R^3. This family is the unfolded normal
form for a bifurcation of a fixed point with a triple-one multiplier and also
is the general form of a quadratic three-dimensional map with a quadratic
inverse. Much of the nontrivial dynamics of this map occurs when its two fixed
points are saddle-foci with intersecting two-dimensional stable and unstable
manifolds that bound a spherical ``vortex-bubble''. We show that this occurs
near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, at
least in its normal form, an elliptic invariant circle. We develop a simple
algorithm to accurately compute these elliptic invariant circles and their
longitudinal and transverse rotation numbers and use it to study their
bifurcations, classifying them by the resonances between the rotation numbers.
In particular, rational values of the longitudinal rotation number are shown to
give rise to a string of pearls that creates multiple copies of the original
spherical structure for an iterate of the map.Comment: 53 pages, 29 figure
Thirty Years of Turnstiles and Transport
To characterize transport in a deterministic dynamical system is to compute
exit time distributions from regions or transition time distributions between
regions in phase space. This paper surveys the considerable progress on this
problem over the past thirty years. Primary measures of transport for
volume-preserving maps include the exiting and incoming fluxes to a region. For
area-preserving maps, transport is impeded by curves formed from invariant
manifolds that form partial barriers, e.g., stable and unstable manifolds
bounding a resonance zone or cantori, the remnants of destroyed invariant tori.
When the map is exact volume preserving, a Lagrangian differential form can be
used to reduce the computation of fluxes to finding a difference between the
action of certain key orbits, such as homoclinic orbits to a saddle or to a
cantorus. Given a partition of phase space into regions bounded by partial
barriers, a Markov tree model of transport explains key observations, such as
the algebraic decay of exit and recurrence distributions.Comment: Updated and corrected versio
Human dynamics revealed through Web analytics
When the World Wide Web was first conceived as a way to facilitate the
sharing of scientific information at the CERN (European Center for Nuclear
Research) few could have imagined the role it would come to play in the
following decades. Since then, the increasing ubiquity of Internet access and
the frequency with which people interact with it raise the possibility of using
the Web to better observe, understand, and monitor several aspects of human
social behavior. Web sites with large numbers of frequently returning users are
ideal for this task. If these sites belong to companies or universities, their
usage patterns can furnish information about the working habits of entire
populations. In this work, we analyze the properly anonymized logs detailing
the access history to Emory University's Web site. Emory is a medium size
university located in Atlanta, Georgia. We find interesting structure in the
activity patterns of the domain and study in a systematic way the main forces
behind the dynamics of the traffic. In particular, we show that both linear
preferential linking and priority based queuing are essential ingredients to
understand the way users navigate the Web.Comment: 7 pages, 8 figure
Heteroclinic intersections between Invariant Circles of Volume-Preserving Maps
We develop a Melnikov method for volume-preserving maps with codimension one
invariant manifolds. The Melnikov function is shown to be related to the flux
of the perturbation through the unperturbed invariant surface. As an example,
we compute the Melnikov function for a perturbation of a three-dimensional map
that has a heteroclinic connection between a pair of invariant circles. The
intersection curves of the manifolds are shown to undergo bifurcations in
homologyComment: LaTex with 10 eps figure
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
New Class of Eigenstates in Generic Hamiltonian Systems
In mixed systems, besides regular and chaotic states, there are states
supported by the chaotic region mainly living in the vicinity of the hierarchy
of regular islands. We show that the fraction of these hierarchical states
scales as and relate the exponent to the
decay of the classical staying probability . This is
numerically confirmed for the kicked rotor by studying the influence of
hierarchical states on eigenfunction and level statistics.Comment: 4 pages, 3 figures, Phys. Rev. Lett., to appea
Quantum Breaking Time Scaling in the Superdiffusive Dynamics
We show that the breaking time of quantum-classical correspondence depends on
the type of kinetics and the dominant origin of stickiness. For sticky dynamics
of quantum kicked rotor, when the hierarchical set of islands corresponds to
the accelerator mode, we demonstrate by simulation that the breaking time
scales as with the transport exponent
that corresponds to superdiffusive dynamics. We discuss also other
possibilities for the breaking time scaling and transition to the logarithmic
one with respect to
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
Fractal Conductance Fluctuations in a Soft Wall Stadium and a Sinai Billiard
Conductance fluctuations have been studied in a soft wall stadium and a Sinai
billiard defined by electrostatic gates on a high mobility semiconductor
heterojunction. These reproducible magnetoconductance fluctuations are found to
be fractal confirming recent theoretical predictions of quantum signatures in
classically mixed (regular and chaotic) systems. The fractal character of the
fluctuations provides direct evidence for a hierarchical phase space structure
at the boundary between regular and chaotic motion.Comment: 4 pages, 4 figures, data on Sinai geometry added to Fig.1, minor
change
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
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