298 research outputs found
Heteroclinic orbits and transport in a perturbed integrable Suris map
Explicit formulae are given for the saddle connection of an integrable family
of standard maps studied by Y. Suris. When the map is perturbed this connection
is destroyed, and we use a discrete version of Melnikov's method to give an
explicit formula for the first order approximation of the area of the lobes of
the resultant turnstile. These results are compared with computations of the
lobe area.Comment: laTex file with 6 eps figure
Generating Forms for Exact Volume-Preserving Maps
We study the group of volume-preserving diffeomorphisms on a manifold. We
develop a general theory of implicit generating forms. Our results generalize
the classical formulas for generating functions of symplectic twist maps.Comment: laTeX, 20 pages, 1 figur
Rotation Vectors for Torus Maps by the Weighted Birkhoff Average
In this paper, we focus on distinguishing between the types of dynamical
behavior that occur for typical one- and two-dimensional torus maps, in
particular without the assumption of invertibility. We use three fast and
accurate numerical methods: weighted Birkhoff averages, Farey trees, and
resonance orders. The first of these allows us to distinguish between chaotic
and regular orbits, as well as to calculate the frequency vectors for the
regular case to high precision. The second method allows us to distinguish
between the periodic and quasiperiodic orbits, and the third allows us to
distinguish among the quasiperiodic orbits to determine the dimension the
resulting attracting tori. We first consider the well-studied Arnold circle
map, comparing our results to the universal power law of Jensen and Ecke. We
next consider quasiperiodically forced circle maps, inspired by models
introduced by Ding, Grebogi, and Ott. We use the Birkhoff average to
distinguish between "strong" chaos (positive Lyapunov exponents) and "weak"
chaos (strange nonchaotic attractors). Finally, we apply our methods to 2D
torus maps, building on the work of Grebogi, Ott, and Yorke. We distinguish
incommensurate, resonant, periodic, and chaotic orbits and accurately compute
the proportions of each as the strength of the nonlinearity grows. We compute
generalizations of Arnold tongues corresponding to resonances and to periodic
orbits, and we show that chaos typically begins before the map becomes
noninvertible. We show that the proportion of nonresonant orbits does not obey
a universal power law like that seen in the 1D case.Comment: Keywords: Circle maps, Quasiperiodic forcing, Arnold tongues,
Resonance, Birkhoff averages, Strange nonchaotic attractor
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
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
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
Birkhoff Averages and the Breakdown of Invariant Tori in Volume-Preserving Maps
In this paper, we develop numerical methods based on the weighted Birkhoff average for studying two-dimensional invariant tori for volume-preserving maps. The methods do not rely on symmetries, such as time-reversal symmetry, nor on approximating tori by periodic orbits. The rate of convergence of the average gives a sharp distinction between chaotic and regular dynamics and allows accurate computation of rotation vectors for regular orbits. Resonant and rotational tori are distinguished by computing the resonance order of the rotation vector to a given precision. Critical parameter values, where tori are destroyed, are computed by a sharp decrease in convergence rate of the Birkhoff average. We apply these methods for a threedimensional generalization of Chirikov’s standard map: an angle-action map with two angle variables. Computations on grids in frequency and perturbation amplitude allow estimates of the critical set. We also use continuation to follow tori with fixed rotation vectors. We test three conjectures for cubic fields that have been proposed to give locally robust invariant tori, but are not able to provide compelling evidence that one of these three fields is more robust than the other two.</p
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
Symmetry Reduction by Lifting for Maps
We study diffeomorphisms that have one-parameter families of continuous
symmetries. For general maps, in contrast to the symplectic case, existence of
a symmetry no longer implies existence of an invariant. Conversely, a map with
an invariant need not have a symmetry. We show that when a symmetry flow has a
global Poincar\'{e} section there are coordinates in which the map takes a
reduced, skew-product form, and hence allows for reduction of dimensionality.
We show that the reduction of a volume-preserving map again is volume
preserving. Finally we sharpen the Noether theorem for symplectic maps. A
number of illustrative examples are discussed and the method is compared with
traditional reduction techniques.Comment: laTeX, 31 pages, 5 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
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