


Dynamical Systems
Org: Sue Ann Campbell (Waterloo), Yuming Chen (Wilfrid Laurier) and Huaiping Zhu (York) [PDF]
 JULIEN ARINO, McMaster University, 1280 Main Street West, Hamilton, Ontario
L8S 4K1
Mathematical aspects of metapopulation disease models
[PDF] 
Metapopulation disease models describe the spread of an infectious
disease in a population that is, typically, spatially fragmented.
Such models consist of systems of differential equations that are
embedded in graphs. The resulting large dimensionality renders their
mathematical analysis difficult. I will present some of the problems
that arise when dealing with this type of models, and some of the
solutions that were proposed to these problems.
 CHRIS BAUCH, University of Guelph, 50 Stone Road East, Guelph, ON N1G 2W1
Dynamic games with imitation predict vaccinating behaviour
[PDF] 
There exists an interplay between vaccine coverage, disease
prevalence, and the vaccinating behaviour of individuals. Moreover,
because of herd immunity, there is also a strategic interaction
between individuals when they are deciding whether or not to
vaccinate, since the probability that an individual becomes infected
depends upon how many other individuals are vaccinated. To understand
this potentially complex interplay, a dynamic game theory model is
developed in which individuals adopt strategies according to an
imitation dynamic (a learning process), and base vaccination decisions
on disease prevalence and perceived risks of vaccines and disease.
The model predicts that oscillations in vaccine uptake are more likely
in populations where individuals imitate others more readily or where
vaccinating behaviour is more sensitive to changes in disease
prevalence. Oscillations are also more likely when the perceived risk
of vaccines is high. The model reproduces salient features of the
time evolution of vaccine uptake and disease prevalence during the
wholecell pertussis vaccine scare in England & Wales during the
1970s. This suggests that using dynamic game theoretical models to
predict, and even manage, the population dynamics of vaccinating
behaviour may be feasible.
 ELENA BRAVERMAN, University of Calgary, 2500 University Drive N.W., Calgary,
AB T2N 1N4
On stability of equations with several delays and Mackey
Glass equation with variable coefficients
[PDF] 
In the first part of the talk, some new results on stability of linear
delay equations with several delays and variable delays and
coefficients are presented. These results can be applied to the local
stability of nonlinear equations. As an example, we consider the
MackeyGlass equation with variable coefficients and a nonconstant
delay N¢ = [(r(t)N(g(t)))/(1+(N(g(t))^{g})]  b(t)N(t) which
models white blood cell production. Other qualitative properties of
this equation, such as boundedness of solutions, persistence and
oscillation, are also discussed. It is also demonstrated that with
two delays the equation does not keep the persistence property.
 MONICA COJOCARU, University of Guelph, 50 Stone Road E., 548 MacNaughton
Building, Guelph, ON
Recent advancements in the theory of projected dynamical
systems
[PDF] 
We present some recent advancements in the theory of projected
dynamical systems, with reference to some of their applications.
 ADELA COMANICI, University of Houston, Dept. of Mathematics, 651 Philip
G. Hoffman Hall, Houston, TX 772043008
Forced Symmetry Breaking from SO(3) to SO(2) for
Rotating Waves on the Sphere
[PDF] 
Geometrical imperfections (e.g., the shape of a heart is not
exactly a sphere) and localized inhomogeneities (e.g., small
blood vessels) inherent in the cardiac tissue can influence the
dynamics of the spiral waves. In both cases, it can be considered
that the spherical symmetry SO(3) is broken to SO(2) symmetry.
In this talk, we consider an esmall perturbation of a
reactiondiffusion system on the sphere of radius r, which is only
SO(2)equivariant for e > 0, but SO(3)equivariant for
e = 0. The effects of forced symmetry breaking for rotating
waves on the sphere of radius r are presented.
Namely, for e = 0, we consider a normally hyperbolic relative
equilibrium SO(3) u_{0} with trivial isotropy. It persists to an
SO(2)invariant normally hyperbolic flowinvariant manifold
M(e). We study the dynamics on M(e) using the orbit
space reduction methods and PoincaréBendixson theorem on the
sphere. The problem reduces to the study of the following
differential equations on the unit sphere S^{2}:
x¢ = [X_{0} + eg^{S} (x,e)]x, e ³ 0 small, 
 (1) 
where X_{0} Î so(3), g^{S} : S^{2} ×[0,e_{0})® so(3). We analyze the differential equations
(1) using the Implicit function theorem and
the Poincaré map.
Then, we obtain that depending on the frequency vectors of the
rotating waves that form SO(3) u_{0}, these rotating waves (up to
SO(2)) will give either SO(2)orbits of rotating waves or
SO(2)orbits of modulated rotating waves (if some transversality
conditions hold). The orbital stability of these solutions is
established as well.
 FREDDY DUMORTIER, Limburgs Universitair Centrum, Belgium
Bifurcation of relaxation oscillations
[PDF] 
The talk deals with bifurcation of relaxation oscillations in
twodimensional systems, with emphasis on Liénard equations.
Attention goes to the investigation of the transient canard
oscillations during the bifurcation as well as to the techniques used
in proving the results.
The talk relies on recent joint work with Robert Roussarie.
 MARTIN GOLUBITSKY, University of Houston
Nilpotent Hopf Bifurcations in Coupled Cell Systems
[PDF] 
A coupled cell system is a collection of interacting dynamical
systems. Coupled cell models assume that the output from each cell is
important and that signals from two or more cells can be compared so
that patterns of synchrony can emerge. We ask: How much of the
qualitative dynamics observed in coupled cells is the product of
network architecture and how much depends on the specific equations?
For example, network architecture can lead to codimension one
bifurcations from a synchronous equilibrium whose linearization has a
pair of purely imaginary eigenvalues with algebraic multiplicity two
and geometric multiplicity one. In addition, network architecture can
change the nonlinear terms in the normal forms of such bifurcations
and lead to multiple branches of periodic solutions and amplitude
growth rates that differ from that of standard Hopf bifurcation. This
is joint research with Toby Elmhirst.
 HERB KUNZE, University of Guelph, Guelph, ON N1G 2W1
Monotonicity Properties of ReactionDiffusion Systems
[PDF] 
A framework exists for analyzing monotonicity with respect to initial
conditions of solutions to systems of ordinary differential equations.
Monotonicity properties can offer support for or invalidate proposed
mathematical models. I have recently been interested in extending
these ideas to reactiondiffusion systems and will present some
results in this talk. In particular, we will see that some known
results for certain reaction systems carry over to their
reactiondiffusion analogs.
 BILL LANGFORD, Dept. of Math. and Stat., University of Guelph, Guelph, ON
N1G 2W1
NearReversible 1:1 Resonance
[PDF] 
This talk explores the interface between reversible and nonreversible
dynamical systems, near 1:1 resonance; that is, the nearreversible
limit. The linearization of these systems has double, nonsemisimple
purely imaginary eigenvalues. The unfolding of the codimension3
nonreversible case leads to a Whitney umbrella of classical Hopf
bifurcations. The codimension1 reversible case corresponds to the
"handle" of this umbrella. Families of periodic solutions exist as
centers at points along this handle, in the reversible case. We show
that, along rays emanating from this handle, there is a onetoone
correspondence between hyperbolic periodic orbits arising by Hopf
bifurcation in the nonreversible case and particular orbits in the
centers of the reversible case. We also explore the relationship
between quasiperiodic families in the reversible case and invariant
tori in the nonreversible case. This work provides some justification
for using reversible mathematical models for physical systems which
are only close to being reversible.
This is joint work with G. Iooss, INLNCNRS, France.
 VICTOR G. LEBLANC, University of Ottawa
Toroidal normal forms for delaydifferential equations
[PDF] 
We will present some recent results on the realisability and
restrictions of normal forms for parametrized scalar
delaydifferential equations undergoing either nonresonant multiple
Hopf bifurcation, or steadystate/nonresonant multiple Hopf
interaction. Our results fully exploit the toroidal equivariance of
the normal forms for these bifurcations. This allows for the
development of a framework in which generic cases, degenerate cases,
and unfoldings can be treated systematically.
This is joint work with Youn Sun Choi.
 MICHAEL Y. LI, University of Alberta
Global Hopf Bifurcation in a Delayed Nicholson's Blowfly
Equation
[PDF] 
The dynamics of a Nicholson's Blowfly equation with a finite delay are
investigated. We prove that a sequence of Hopf bifurcations occur at
the positive equilibrium as the delay increases. Global extensions of
local Hopf branches for large delays are proved using a
degreetheoretic argument and a higher dimensional Bendixson criterion
for ordinary differential equations.
 LIPING LIU, University of Texas Pan American, 1201 W. University Drive,
Edinburg, TX 78539, USA
The Analysis of The Numerical Harmonic Balance Method: on
Duffing's Oscillator
[PDF] 
This study focuses on a novel harmonic balance formulation, which is
much easier to implement than the standard/classical harmonic balance
method for complex nonlinear mathematical models and algorithms. Both
harmonic balance approaches are applied to Duffing's oscillator to
demonstrate the advantages and disadvantages of the two approaches. A
fundamental understanding of the difference between these two methods
is achieved, and the properties of each method are analyzed in
detail.
 PIERRE MAGAL, Université du Havre, 25 rue Philippe Lebon, 76600 Le Havre,
France
Asymptotic Behavior in Nosocomial Epidemic Models with
Antibiotic Resistance
[PDF] 
We analyze a model of an epidemic in a hospital setting that
incorporates antibiotic nonresistant and resistant strains of
bacterial infection. The model connects two population
levelsbacteria and patients. The bacteria population is divided
into nonresistant and resistant strains. The bacterial strains
satisfy ordinary differential equations describing the recombination
and reversion processes producing the two strains within each infected
individual. The patient population is divided into susceptibles,
infectives infected with the nonresistant bacterial strain, and
infectives infected with the resistant bacterial stain. The infective
classes satisfy partial differential equations for the infection age
densities of the two classes. We investigate the asymptotic behavior
of the solutions of the model with respect to three possible
equilibria:
(1) extinction of both infective classes,
(2) extinction of the resistant infectives and endemicity of
the nonresistant infectives, and
(3) endemicity of both infective classes.
Based on joint work with Erika D'Agata, Shigui Ruan, and Glenn Webb.
 CONNELL McCLUSKEY, McMaster
Hidden Structure in Lyapunov Functions for Epidemic Models
[PDF] 
Recent work by Korobeinikov and Maini has shown that it is feasible to
try to find Lyapunov functions of a certain type which can be used to
demonstrate the global stability of a positive equilibrium of epidemic
models.
In constructing similar Lyapunov functions for two tuberculosis
models, we came across systems of inequalities that must be
simultaneously satisfied by a set of parameters. In each case, there
are more inequalities than there are free parameters and yet a unique
solution exists, implying an underlying structure to the method.
 JAMES MULDOWNEY, University of Alberta, Dept. Math. & Stat. Sci.,
Edmonton, AB T6G 2G1
Evolution of exterior products in dynamics
[PDF] 
The evolution of differential kforms has proved to be a versatile
tool in dynamical systems. It facilitates the investigation of local
and global properties as well as steady state behaviour; topics such
as existence and stability of periodic orbits and questions of
dimension of invariant sets such as global attractors. This approach
is taken by Temam in the treatment of evolutionary partial
differential equations in a Hilbert space setting. The main tools in
finite dimensional dynamics have been multiplicative and additive
kcompound matrices whose algebraic, metric and spectral properties
have provided important insights. This presentation will report on
joint work with Qian Wang on the extension of the concept of these
compound operators to an infinite dimensional setting.
 CHUNHUA OU, York University
Modelling spatiotemporal patterns in epidemiology
[PDF] 
In this talk we present a case study on the spread of rabies in
Europe. We will explain how to apply the nonlocal parabolic model to
study the infectious disease in a concrete example. A new
mathematical model with nonlocal response on the spread of "rabies"
in Europe will be presented. First we consider our model in a finite
domain with Neumann boundary condition. Asymptotic stability of the
equilibriums is given. When the spatial domain becomes the whole real
line, traveling wave fronts are investigated. The spreading speed is
determined by the standard stability analysis and the minimal speed is
confirmed by numerical computation. Rigorous proof of the existence
of traveling fronts is presented when the wave speed is large.
 CHRISTIANE ROUSSEAU, Université de Montréal
The problem of the equivalence of two curvilinear angles in
conformal geometry
[PDF] 
We will first state the problem and show that it is closely related to
the problem of analytic conjugacy of germs of diffeomorphisms of
(C,0) with a fixed point at the origin whose multiplier is on the
unit circle and satisfying a symmetry condition. The lecture will
describe the formal and analytic invariants. In the particular case
of two tangent arcs (a horn) we will present the necessary and
sufficient conditions for the bisection of the angle, and more
generally for the section of the angle into N equal angles. We will
explain the geometric significance of all these conditions by
unfolding the situation.
 LIN WANG, McMaster University, Hamilton, Canada
Convergence of DiscreteTime Neural Networks with Delays
[PDF] 
An LMI (Linear Matrix Inequality) approach and an embedding technique
are employed to derive some sufficient conditions for the global
exponential stability of discretetime neural networks with
timedependent delays. For networks with timedependent parameters
with constant delays, by using the property of internally chain
transitive sets, it is shown that these conditions are also sufficient
for the convergence.
 GAIL WOLKOWICZ, McMaster University, Department of Mathematics and
Statistics, Hamilton, Ontario L8S 4K1
Competition in a chemostat: The effect of delayed response in
growth
[PDF] 
The chemostat, a laboratory apparatus used for the continuous culture
of microorganisms will be described. After first considering the
basic mathematical model for n species of microorganisms competing
exploitatively for a nutrient that is limiting at both low and high
concentrations, under the simplifying assumption that growth based on
consumption occurs instantaneously, we explore the effect of delayed
response to growth. This represents joint work with Huaxing Xia and
Lin Wang.
 WEIGUANG YAO, University of Western Ontario, London, Ontario
Estimate of the population of memory cells and application in
drug treatment
[PDF] 
The effector stage of a primary immune response is a transient
process. We propose to estimate the maximal value of the population
of activated immune cells during the transient process. The value is
expressed in system parameters allowing an analysis on the
contribution of these parameters including drug when a treatment is
performed. Since the population of memory cells is proportional to
the maximal value and the memory cells play an essential role in the
memory stage on protecting the host from secondary challenges, we
apply the estimate to designing optimal drug treatment.
This is a joint work with Lindi Wahl.
 PEI YU, The University of Western Ontario, London, ON
Bifurcation of Limit Cycles from Perturbed Hamiltonian
Systems and Hilbert's 16th Problem
[PDF] 
Some new results will be presented on the study of the second part of
Hilbert's 16th problem. After a brief review of the problem, the main
attention of this talk is focused on the weakened Hilbert's 16th
problem for higherorder Hamiltonian systems. First, a summary will
be given for the results of odd order systems: H(5) ³ 5^{2}1, H(7) ³ 7^{2}, H(9) ³ 9^{2}1, and H(11) ³ 11^{2}, then the particular
attention is given to rarelyconsidered even order systems. With the
aid of the detection function method and normal form theory, both
global and local bifurcation analyses are employed to show that a
quintic Hamiltonian system under a 6thorder perturbation can generate
at least 35 limit cycles, i.e., H(6) ³ 6^{2}1.
Combining this result with other existing results, H(2) ³ 2^{2},
H(4) ³ 4^{2}1, and that for odd order systems, a conjecture is
posed for Hilbert's 16th problem: H(n) ³ n^{2} or n^{2}1.
 YUAN YUAN, Memorial University of Newfoundland, St. John's, Canada
Multiple Bifurcation Analysis of Synchronized Cells with Delays
[PDF] 
The synchronized cells in a neural network model with two discrete
time delays is considered. The local stability of the zero solution
of this system is investigated by studying the distributions of the
eigenvalues of the system. Several groups of conditions have been
given to guarantee the model having multiple synchronized periodic
solutions when the transfer coefficient or time delay is sufficiently
large. A complete bifurcation analysis is given by employing the
center manifold theorem, normal form method and bifurcation theory.
It is shown that the equilibrium point may lose stability via a
transcritical/pitchfork bifurcation, Hopf bifurcation or
BogdanovTakens bifurcation. Some numerical simulation examples are
given to justify the theoretical results.

