Patrick Shipman

Department of Mathematics
Colorado State University


Applied Dynamics Lab


Spring 2011 Meeting Time and Place:
Tuesdays 2-3 in Weber 201

Upcoming Talks


Previous Talks

May 3, 2011
Savanna Modeling and Species Invasion: Two Dynamical Systems
Christopher Strickland 

In this talk, I summarize my research to date on modeling savanna ecosystems and on modeling species invasion. First, I will motivate and describe current efforts to model savanna dynamics while showing some details of a model currently under development by ecologists at CSU. I will then describe my contribution to this model, as well as my plans for further research this summer in Australia. Next, I will motivate and describe current efforts to model the spread of an invasive species, building up to our own model which is under development. In doing so, I will discuss relevant nonstandard diffusion models, including some new formulations that are particular to our problem. This talk will be non-technical, and hopefully will serve to describe several possible areas of mathematical research which are of current relevance to ecologists.

April 19, 2011
A Stochastic Model of Programmed Ribosomal Frameshifting in Viral Protein Synthesis
Brenae Bailey
The University of Arizona

Many viruses can produce different proteins from the same RNA sequence by encoding them in overlapping genes. The mechanism that causes the ribosomes of infected cells to decode both genes is called programmed ribosomal frameshifting (PRF). Although PRF has beenrecognized for 25 years, the mechanism is not well understood. I will present a model that treats RNA translation into proteins as a stochastic process in which the transition probabilities are based on the free energies of local molecular interactions. The model can reproduce observed translation rates and frameshift efficiencies, and can be used to predict the effects of mutations in the viral RNA sequence on the frameshift efficiency.

March 9 2011
Bargaining over Productivity and Wages when Technical Change is Induced: Implications for Growth, Distribution, and Employment
Daniele Tavani

In a simple one-sector economy operating at full capacity, workers and firms bargain a la [Nash (1950)] over wages and productivity gains taking into account the trade-o s faced by fi rms in choosing factor-augmenting technologies. The aggregate environment resulting from optimal decision rules on wages, productivity gains, savings and investment, is described by a two-dimensional dynamical system in the employment rate and output/capital ratio. The economy converges cyclically to a long-run equilibrium involving a Harrod-neutral pro file of technical change, a constant rate of employment of labor, and constant input shares. The type of oscillations predicted by the model is qualitatively consistent with the available data on the United States (1963-2003), replicates the dynamics found in earlier models of growth cycles such as [Goodwin (1967)] [Shah and Desai (1981)], [van der Ploeg (1987)], and is veri fied numerically in simulations. Institutional change, as captured by variations in workers' bargaining power, has a positive effect on the rate of growth of output per worker but a negative effect on employment. Economic policy can also aff ect the growth and distribution pattern through changes in unemployment compensations, which also have a positive impact on labor productivity growth but a negative impact on employment.

Feb 15 2010
Security from Chaos
Francis Motta

A degree-three polynomial with a single parameter is constructed and used to construct a discrete dynamical system that is topologically conjugate to the shift map on three indices. For each positive integer greater than 1, a piecewise continuous function on [0,1] with a single parameter is constructed and a topological conjugacy is demonstrated between the shift map on n symbols and this function. Using the correspondance for n = 10, a theoretical algorithm is developed to construct dynamic passwords which depend on time. The chaotic nature of the shift map and its sensititvity to initial conditions suggests a robust and difficult-to-break dynamic password system.

Feb 22 2011
Roundtable Disscusion:
Experimental Mathematics Laboratory

Friday, Dec. 3 2010
Storing Cycles in Continuous Asymmetric Hopfield-type Networks
Chuan Zhang

This report investigates conditions for cycles of binary state vectors which can be stored in an asymmetric Hopfield-type neural network, and analyzes bifurcations in the networks of a ring of neurons with nearest neighbor coupling. In a simplified case, a sufficient condition for admissible cycles is proposed and proved. Using a set of admissible cycles of the binary vectors, networks of a ring of neurons with nearest neighbor coupling are constructed using the pseudoinverse learning rule. Bifurcations in these networks are analyzed systematically, and the results show that networks of odd and even numbers have different bifurcation structures.

Friday, Dec. 10 2010.
Chemical Oscillators
Melody Dodd

The study of oscillating chemical reactions is an active area of research in both chemistry and mathematics. These reactions oscillate as they progress, producing cyclic color changes; the reaction mechanism can be modeled with a system of ordinary differential equations. This presentation includes an overview of the history and mathematics of chemical oscillators and a live demonstration of one such reaction.

Friday, Oct 8 2010
Perturbations and Asymptotic Methods:
The Mathematical Nature of the Boundary Layer
Iuliana Oprea

In 1904, Ludwig Prandtl, a little known physicist at that time, revolutionized fluid dynamic with his idea that the effects of the friction are experienced only very near an object moving through a fluid. His only eight pages seminal paper proved to be one of the most important fluid dynamics papers ever written.  We will discuss Prandtl's paradigm example of the harmonic oscillator to illustrate the fundamental idea of boundary layer theory and its mathematical nature, as well as to introduce the concept of singularly perturbed problems and basic math tools to approach them. The mathematical level will not go beyond an undergrad course on ordinary differential equations.

April 1, 2010
Fractals of the Degree-Two Standard Family of Circle Maps
Christopher Strickland

In this talk, I seek to characterize the family of one dimensional fractals formed by the infinite composition and rescaling of the degree-two standard family. I will first build up some theory on circle maps and their infinite composition while providing graphical examples from the degree-two standard family. Next, I will motivate the examination of critical points and their dynamics under infinite composition, which will result in a bifurcation diagram of critical point values on the circle. Time permitting, I will characterize the different regions of the bifurcation diagram for qualitative information about the fractals.

April 22, 2010
New results in nearly continuous Kakutani equivalence
Bethany Springer

Advisor: Daniel Rudolph
It was previously known that a conjugacy between induced systems on Polish probability spaces (X,T,\mu) and (Y,S, \nu) on nearly clopen subsets $A \subset X$ and $B \subset Y$ of same relative measure can be extended to a measurable orbit equivalence between the systems. This talk gives an algorithm for modifying the conjugacy in order to establish nearly continuous Kakutani equivalence. We also show that in the case $\mu(A) > \nu(B)$, we can establish conjugacy between $T$ and an induced system $S_{\bar B}$ where the systems are nearly uniquely ergodic.
Don't worry, I will also give an introduction to the type of dynamical systems I work with.

April 29, 2010
Steady State Hopf Mode Interaction in Anisotropic Systems
Jennifer Maple

We present a theoretical and numerical analysis of a system of four globally coupled complex Ginzburg-Landau equations modeling steady oblique-normal Hopf mode interaction observed in experiments in electroconvection of nematic liquid crystals.

March 4, 2010
Geometric Data Analysis: Techniques and Iterations
Josh Thompson

We will begin with a discussion of a few common methods of data analysis, and their motivations. We will end with an investigation into the dynamics of a not-so-common method, the Multivariate State Estimation Technique. In the middle hopefully we will learn something, have fun and see some interesting images.

March 11, 2010
Hopf Bifurcation in Anisotropic Reaction Diffusion Systems Posed in Large Rectangles:
Travis Olson

The oscillatory instability (Hopf bifurcation) for anisotropic reaction diffusion equations posed in large (but finite) rectangles is investigated. For the case considered, the solution of the reaction diffusion system is represented in terms of slowly modulated complex amplitudes of four wave-trains propagating in four oblique directions. While for the infinitely extended system the modulating amplitudes are independent dynamical variables, the finite size of the domain leads to relations between them induced by wave reflections at the boundaries. This leads to a single amplitude equation for a doubly periodic function that captures all four envelopes in different regions of its fundamental domain. The amplitude equation is derived by matching an asymptotic bulk solution to an asymptotic boundary layer solution. Preliminary numerical simulations show that the complexity of the solutions significantly increases when the rescaled control parameter is increased.

March 25, 2010
Summary of some important types of intermittency:
Yang Zou

Intermittency is usually referred as the occurrence of a signal alternating randomly between long regular phases and relatively short irregular bursts. And sometimes it can be the intermittent switching between two or more chaotic states. In this talk I will summaries some important types of intermittency, which includes Pomeau-Manneville Type I, II, and III intermittency, Type X intermittency, Type V intermittency, on-off intermittency, in-out intermittency, and some kinds of crisis-induced intermittency. Their mechanisms will be briefly described. And some classical examples generating them will be given. Statistical properties of these intermittencies will also be investigated finally.

Feburary 4, 2010
Patterns and Complexity: An Introduction
Juliana Oprea
An open problem involving minimal surfaces and phase separation
Patrick Shipman

Feburary 11, 2010
Tutorial on Symmetries and Differential Equations
Patrick Shipman

Feburary 18, 2010
Hurricane Model
Blake Rutherford

Feburary 25, 2010
Bubble Elasticity
Dan Brake

Fall 09 I at Universiteit Leiden, Holland, I studied foam elasticity in the Granular and Disordered Media group headed by Martin Van Hecke. We discovered that near the jamming transition, both in a Couette cell and constrained linearly, a submerged group of bubbles reacts nonlinearly to an applied compressive force.