Tuesday, February 11, 2014 - 4:05pm - 6:00pm
- Parameterization Methods for Computing Normally Hyperbolic Invariant Tori
Marta Canadell (University of Barcelona)
In this poster we explain two numerical algorithms for the
computation of normally hyperbolic invariant tori (NHIT) in families of
discrete dynamical systems. The application of the parameterization method
leads to solving invariance equations for which we use a Newton-like
method adapted to the dynamics and the geometry of the invariant torus and
its invariant bundles.
The first method computes the NHIT and its internal dynamics, which is a
The second method computes NHIT in which the internal dynamics is a fixed
quasi periodic rotation, by adjusting parameters of the family.
We apply these methods to continue NHIT w.r.t. parameters, and to explore
different mechanisms of breakdown of NHIT.
This is a join work with Alex Haro.
- Computational Intersection Theory: Numerical Methods and Computer Assisted Proof
Jason Mireles-James (Rutgers, The State University Of New Jersey )
Understanding connecting orbits between equilibria and periodic orbits of differential equations is a fundamental step toward understanding the global dynamics of differential equations. Connecting orbits organize transport between different regions of phase space, force the existence of complicated dynamics, and form the basic objects of classical tools in nonlinear analysis such as Morse/Floer homology. By first studying local stable and unstable manifolds it is possible to express connecting orbits as solutions of certain finite time boundary value problems. One theme in my research is the development of mathematically rigorous computational tools for solving these boundary value problems.
- Exploring Multiparameter Dynamics in Biology I: Switching Models
Tomas Gedeon (Montana State University)
We present a new approach to nonlinear dynamics based on topological methods that allows for a priori choices in the resolution of the model, both with respect to the variables and the parameters. We apply this approach to switching networks to investigate a transcriptional network driving the cell cycle.
Joint work with B. Cummins, B. Fan, T. Gedeon, S. Harker, A. Goullet and K. Mischaikow, Dept. of Mathematical Sciences, Montana State University and Dept. of Mathematics, Rutgers University.
- Using Persistence for Exploring Equilibria of Delay Equations
Firas Khasawneh (State University of New York Institute of Technology)Elizabeth Munch (University of Minnesota, Twin Cities)
This poster describes an exploratory study of the possibility of using techniques from topological data analysis for studying datasets generated from dynamical systems described by stochastic delay equations. The dataset is generated using Euler-Maryuama simulation for two first order systems with stochastic parameters drawn from a normal distribution. The first system contains additive noise whereas the second one contains parametric or multiplicative noise. Using Taken’s embedding, the dataset is converted into a point cloud in a high-dimensional space. Persistent homology is then employed to analyze the structure of the point cloud in order to study equilibria and periodic solutions of the underlying system. Our results show that the persistent homology successfully differentiates between different types of equilibria. Therefore, we believe this approach will prove useful for automatic data analysis of vibration measurements.
- Persistent Homology on Tapped Granular Media
Sergio Ardanza-Trevijano (Universidad de Navarra)
We present some applications of persistent homology to the physics of granular media, which are materials made of inert particles that interact only by dissipative contacts. More concretely we study tapped granular media in a 2D vertical container. The graph of contacts between particles has been used as a mean to study properties of these materials in numerical simulations. However in experimental settings it is difficult to obtain the exact graph, since the contacts are not precisely defined. Starting with noisy data (the position of the particles) we construct a parametrized Vietoris-Rips complex for an appropriate range of the filtration parameter. The corresponding first Betti numbers are then used to characterize different physical states that previous approaches could not distinguish.
- Restricted Cech Complexes for Evenly-Spaced Points on a Circle
Henry Adams (University of Minnesota, Twin Cities)
Consider the restricted Cech complex for evenly-spaced points around a circle. The restricted Cech complex is built not from balls in the plane but instead from balls in the circle, i.e. circular arcs. Since the intersection of two such circular arcs need not be contractible, the nerve lemma need not apply. The following two families of homotopy types are obtainable as the restricted Cech complex for evenly-spaced points on a circle. First, for any two nonnegative integers t and n, one can obtain a t-fold wedge sum of copies of the 2n-dimensional sphere. Second, for any n, one can obtain a single copy of the (2n+1)-dimensional sphere. These homotopy types are closely related to the Vietoris-Rips complexes for evenly-spaced points on a circle, which are studied by Michal Adamaszek in Clique complexes and graph powers. This is joint work with Christopher Peterson and Corrine Previte.
- The Globo-Toroid
Nikola (Nick) Samardzija (Emerson Process Management)
There is perhaps no simpler mathematical object that unifies forces of the nature better than the globo-toroid. The dynamics of this object can be modeled by, also, a very simple 3-dimensional ODE, which in an abstract sense may be viewed as the nature’s dynamo that packages and releases energy in our universe. Its signatures are found in the natural processes of creation, life and destruction. In this poster presentation the aim is to introduce the globo-toroid concept, and to lay the ground for future work. In doing so the simulated big data is used to show how the well-known mathematical notions of the periodic attractors, Poincaré sections, Lypunov exponents, Riemann sphere, slow and fast manifolds, as well as physics concept of the wormhole, are used to explain and illustrate the extraordinary behavior of the globo-toroid dynamics.
- Detecting Morse Decompositions of the Global Attractor of Regulatory Networks by Time Series Data
Hiroe Oka (Ryukoku University)
This is a joint work with B. Fiedler (Berlin), A. Mochizuki & G. Kurosawa (RIKEN), H. Kokubu (Kyoto)
The outline is as follows:
(1) I will introduce the notion determining node, feedback vertex set (FVS) for Regulatory network, which is given by
B. Fiedler, A. Mochizuki, G. Kurosawa, D. Saito, J. Dynam. Diff Eqns, 25 (2013)
(2) We will generalize these idea to dynamical time series analysis for Morse decomposition
(3) Example: Mirsky’s model of mammalian circadian rhythm
- Rigorously Computing the Persistent Homology of Functions
Jonathan Jaquette (Rutgers, The State University Of New Jersey )
When studying a differentiable function, it is difficult to determine the topology of sublevel sets corresponding to points near critical values of the function. Nevertheless, by studying the persistent homology of these sublevel sets, the importance of any one sublevel set is dwarfed by the global behavior of the function. We generalize the algorithm for rigorously computing the homology of two dimensional nodal domains [Day, Kalies, and Wanner 2009] to handle higher dimensional sublevel sets. We then describe how to rigorously compute persistent homology and present computational results.
- Efficient Computation of Invariant Tori in Volume-Preserving Maps
Adam Fox (Georgia Institute of Technology)
This poster details the implementation of 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. 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.
- Pacemakers in a Large Array of Oscillators with Nonlocal Coupling
Gabriela Jaramillo (University of Minnesota, Twin Cities)
We model pacemaker effects in a 1 dimensional array of oscillators with nonlocal coupling via an algebraically localized heterogeneity. We assume the oscillators obey simple phase dynamics and that the array is large enough so that it can be approximated by a continuous nonlocal evolution equation. We concentrate on the case of heterogeneities with negative average and show that steady solutions to the nonlocal problem exist. In particular, we show that these heterogeneities act as a wave source, sending out waves in the far field. This effect is not possible in 3 dimensional systems, such as the complex Ginzburg-Landau equation, where the wavenumber of weak sources decays at infinity. To obtain our results we use a series of isomorphisms to relate the nonlocal problem to the viscous eikonal equation. The linearization about the constant solution results in an operator, L, which is not Fredholm in regular Sobolev spaces. We show that when viewed in the setting of Kondratiev spaces the operator, L, is Fredholm. These spaces can be described as Sobolev spaces with algebraic weights that increase in degree with each derivative.
- Excess One Covering and Design of the Dynamics
Han Wang (University of Illinois at Urbana-Champaign)
We present our study on the topology of the space of the coverings. The configuration space Cov(n,r) of coverings is defined as the space of collections of n-element subsets of a metric space X forming an r-net in X. Here we consider the covering spaces over 2D grid domains, or some metric trees. We focus on the excess one covering (meaning that one needs at least n-1 metric r-balls to cover the whole space), and determine the homotopy type of the space. As an application of our study, the feedback control algorithms for repeated coverage are considered.
- Geometric Phase in the Hopf Bundle - An Evans Function Method
Colin Grudzien (University of North Carolina, Chapel Hill)
Evans function analysis has become a standard method of calculating the stability of non-linear waves for PDE's, and building on the machinery of the Evans function, we have proven a related but alternative form of analysis. The Hopf bundle has an embedding in complex space, and locally is represented by the cross product of a circle and complex projective space - the dynamical system associated with the linearized operator for a PDE can thus induce a winding number through parallel transport in the fibre. Our method uses parallel transport to count the multiplicity of eigenvalues contained within a loop in the spectral plane.
- Hadwiger Integration and Applications
Matthew Wright (University of Minnesota, Twin Cities)
The intrinsic volumes generalize both Euler characteristic and volume, quantifying the “size” of a set in various ways. Lifting the intrinsic volumes from sets to functions over sets, we obtain the Hadwiger Integrals, a family of integrals that generalize both the Euler integral and the Lebesgue integral. The classic Hadwiger Theorem says that the intrinsic volumes form a basis for the space of all valuations on sets. An analogous result holds for valuations on functions: with certain assumptions, any valuation on functions can be expressed in terms of Hadwiger integrals. These integrals provide various notions of the size of a function, which are potentially useful for analyzing data arising from sensor networks, cell dynamics, image processing, and other areas. This poster provides an overview of the intrinsic volumes, Hadwiger integrals, and possible applications.
- Comparison of different complexes for analysis of Point Cloud Data
Vimala Ramani (Anna University)
Analysis of point cloud data using simplicial complexes is a relatively new concept.
This has been developed and implemented mainly by the Stanford Computational topology group.
We analyzed the synthesized PCD of a hollow torus using the three simplicial complexes,
namely Vietoris-Rips, Witness and Lazy witness complexes with the help of the open source
software javaplex. We compared the three complexes for their computational time complexity
and their efficiency for identifying the real feature. According to our analysis the witness
complex is more efficient both in terms of computation time and feature identification.
- 3D Symmetric Tensor Fields: Topological Analysis and Editing
Eugene Zhang (Oregon State University)
Tensor fields appear in a wide range of science, engineering, and medical applications. The topology of a tensor field is complicated and intriguing. We show recent advances in understanding the topology of 3D symmetric tensor fields, with applications in scientific visualization and computer graphics.
- Certain Integral Inequalities Involving Pathway Fractional Operators
Praveen Agarwal (Anand International College of Engineering)
Abstract. A remarkably large number of inequalities involving the fractional inte-
gral operators have been investigated in the literature by many authors. Very recently,
Dumitru et al. [Chinese J. Math. (2013)], gave certain interesting fractional integral
inequalities involving the Gauss hypergeometric functions. Using the same technique,
in this paper, we present some(presumably) new fractional integral inequalities involv-
ing Pathway type fractional operators, whose special cases are shown to yield corre-
sponding inequalities associated with Saigo, Erdelyi-Kober and Riemann-Liouville type
fractional integral operators. Relevant connections of the results presented here with
those earlier ones are also pointed out.