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

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 $d$ 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