485 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
Characterizing and modeling the dynamics of online popularity
Online popularity has enormous impact on opinions, culture, policy, and
profits. We provide a quantitative, large scale, temporal analysis of the
dynamics of online content popularity in two massive model systems, the
Wikipedia and an entire country's Web space. We find that the dynamics of
popularity are characterized by bursts, displaying characteristic features of
critical systems such as fat-tailed distributions of magnitude and inter-event
time. We propose a minimal model combining the classic preferential popularity
increase mechanism with the occurrence of random popularity shifts due to
exogenous factors. The model recovers the critical features observed in the
empirical analysis of the systems analyzed here, highlighting the key factors
needed in the description of popularity dynamics.Comment: 5 pages, 4 figures. Modeling part detailed. Final version published
in Physical Review Letter
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
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
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
Transport in Transitory, Three-Dimensional, Liouville Flows
We derive an action-flux formula to compute the volumes of lobes quantifying
transport between past- and future-invariant Lagrangian coherent structures of
n-dimensional, transitory, globally Liouville flows. A transitory system is one
that is nonautonomous only on a compact time interval. This method requires
relatively little Lagrangian information about the codimension-one surfaces
bounding the lobes, relying only on the generalized actions of loops on the
lobe boundaries. These are easily computed since the vector fields are
autonomous before and after the time-dependent transition. Two examples in
three-dimensions are studied: a transitory ABC flow and a model of a
microdroplet moving through a microfluidic channel mixer. In both cases the
action-flux computations of transport are compared to those obtained using
Monte Carlo methods.Comment: 30 pages, 16 figures, 1 table, submitted to SIAM J. Appl. Dyn. Sy
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Chaotic Advection and Reaction During Engineered Injection and Extraction in Heterogeneous Porous Media
During in situ remediation of contaminated groundwater, a treatment solution is often injected into the contaminated region to initiate reactions that degrade the contaminant. Degradation reactions only occur where the treatment solution and the contaminated groundwater are close enough that mixing will bring them together. Degradation is enhanced when the treatment solution is spread into the contami- nated region, thereby increasing the spatial extent of mixing and degradation reactions. Spreading results from local velocity variations that emerge from aquifer heterogeneity and from spatial variations in the external forcings that drive flow. Certain patterns in external forcings have been shown to create chaotic advection, which is known to enhance spreading of solutes in groundwater flow and other laminar flows. This work uses numerical simulations of flow and reactive transport to investigate how aquifer heterogene- ity changes the qualitative and quantitative aspects of chaotic advection in an aquifer, and the extent to which these changes enhance contaminant degradation. We generate chaotic advection using engineered injection and extraction (EIE), an approach that uses sequential injection and extraction of water in wells surrounding the contaminated region to create time-dependent flow fields that promote plume spreading. We demonstrate that as the degree of heterogeneity increases, both plume spreading and contaminant degradation increase; however, the increase in contaminant degradation is small relative to the increase in plume spreading. Our results show that the combined effects of EIE and heterogeneity produce substan- tially more stretching than either effect separately
Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps
Invariant tori play a fundamental role in the dynamics of symplectic and
volume-preserving maps. Codimension-one tori are particularly important as they
form barriers to transport. Such tori foliate the phase space of integrable,
volume-preserving maps with one action and angles. For the area-preserving
case, Greene's residue criterion is often used to predict the destruction of
tori from the properties of nearby periodic orbits. Even though KAM theory
applies to the three-dimensional case, the robustness of tori in such systems
is still poorly understood. We study a three-dimensional, reversible,
volume-preserving analogue of Chirikov's standard map with one action and two
angles. We investigate the preservation and destruction of tori under
perturbation by computing the "residue" of nearby periodic orbits. We find tori
with Diophantine rotation vectors in the "spiral mean" cubic algebraic field.
The residue is used to generate the critical function of the map and find a
candidate for the most robust torus.Comment: laTeX, 40 pages, 26 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
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