


Dynamical Systems / Systèmes dynamiques (Org: Michael A. Radin, Rochester Institute of Technology)
 BERNARD BROOKS, Rochester Institute of Technology
Rumour Propagation Modeled as a Dynamical System on a Network

Mathematical models of rumour propagation have traditionally used a
`rumour as epidemic' approach. Such models are loosely based on a
reactiondiffusion system and grossly simplify the connections between
the individual people in question. In most cases they are thought of
as all being infinitely and continuous connected. Instead, we will
consider a population connected together in a given network
architecture. How does the architecture of the network itself affect
propagation? The population will have two subgroups, each with a
different set of values such as Liberals and Conservatives. How does
the distribution of these subgroups on the network affect rumour
propagation?
 SHARENE BUNGAY, University of Waterloo, Waterloo, Ontario
Numerical analysis of a bidirectional ring neural network
with delayed feedback

Many physiological systems involve rings of similar neurons. Such
neural networks can be mathematically modelled as a system of delay
differential equations, where the coupling between neurons is often
modelled as simple step functions or nonlinear sigmoidal functions.
Here, we investigate a threedimensional bidirectional symmetric ring
neural network with delayed coupling and self feedback. This model
was numerically analyzed to identify and compute the different types
of periodic solutions arising from equivariant Hopf bifurcations.
This work was performed with the bifurcation package DDEBIFTOOL which
also allowed the computation of the stability of the various periodic
solutions. Secondary bifurcations and multistability near codimension
two bifurcation points are also investigated. These results are
complemented by numerical simulations of the system using XPPAUT.
 MONICA COJOCARU, University of Guelph, Guelph, ON
Projected Dynamical Systems and Evolutionary Variational
Inequalities with Applications to Dynamic Traffic Networks

In this talk we make explicit the connection between projected
dynamical systems on Hilbert spaces and evolutionary variational
inequalities. We give a novel formulation that unifies the underlying
constraint sets for such inequalities, which arise in timedependent
traffic network and spatial price equilibrium problems. We provide a
traffic network numerical example in which we compute the curve of
equilibria.
This is joint work with Patrizia Daniele, University of Catania, and
Anna Nagurney, Isenberg School of Management, University of
Massachusetts.
 MARSHALL HAMPTON, University of Minnesota
On Smale's 6th Problem: A solution in the fourbody case

In 1998 Smale formulated 18 problems for the 21st century, the sixth
of which was: prove that there are finitely many relative equilibria
for positive masses in the planar nbody problem. In joint work
with Richard Moeckel, we have proven this result for n = 4.
 KRIS HEIDLER, University of Guelph, Guelph, Ontario N1G 2W1
Using the Collage Method on the Lorenz System

It is well known that the Lorenz system
generates a "butterfly attractor" for the standard parameter value
choice of a=10, b=[ 8/3], and r=28. Given a dataset of
observed solution component values, with parameters in this chaotic
realm, the inverse problem of approximating the parameter values is
considered. In this talk, we investigate how well the collage method
handles this problem.
 HERBERT KUNZE, University of Guelph
Using Collage Coding to Solve Inverse Problems

Broad classes of inverse problems in differential and integral
equations can be viewed as seeking to approximate a target x of a
metric space X by fixed points of contraction maps on X. Each
contraction map depends on the parameters of the underlying
systemfor example, chemical reaction rates, population interaction
rates, Hooke's constantsand such problems frequently appear in the
parameter estimation literature. The "collage method" attempts to
solve such inverse problems by finding a map T_{c} that sends the
target as close as possible to itself. In this talk, after briefly
introducing the framework which surrounds the collage method, I will
discuss some recent applications, as well as some technical issues.
 TCHAVDAR MARINOV, Department of Mathematical Sciences, University of Alberta,
Edmonton
SolitaryWave Solutions Identification of Boussinesq and
Kortewegde Vries Equations as an Inverse Problem

A special numerical technique has been developed for identification of
solitary wave solutions of Boussinesq and Kortewegde Vries
equations. Stationary localized waves are considered in the frame
moving to the right. The original illposed problem is transferred
into a problem of the unknown coefficient from overposed boundary
data in which the trivial solution is excluded. The Method of
Variational Imbedding is used for solving the inverse problem. The
generalized sixth order Boussinesq equation is considered for
illustration.
 MICHAEL A. RADIN, Rochester Institute of Technology
Trichotomy behavior of solutions of a rational difference equation

We will investigate a particular rational difference equation:
y[n+1] = (py[n1]+y[n2])/(q+y[n2]) 

and discover how the properties of the given equation and the long
term behavior of the positive solutions depends on the relationship
between the parameters and not on the initial conditions.
 WEIGUANG YAO, York University, Department of Mathematics and Statistics
Immune System Memory Realization in A Lattice Population Model

A general process of the immune system consists of effector stage and
memory stage. It is not fully understood how the memory stage forms
and how the system switches from the effector stage to the memory
stage. Existing mathematical models of the immune system can describe
either the effector stage or the memory stage, but not both. We
formulate a mathematical model based on a recently developed cellular
automata model for influenza A virus, and show that these two stages
can be smoothly realized in our model. The numerical simulations of
our model agree with the clinical observation very well.

