Trends in Differential Equations and Dynamical Systems
October 2224, 1999
Memorial University of Newfoundland
Sponsored by AARMS,
the Atlantic Association for Research in the Mathematical
Sciences
This is a special meeting which will run parallel to the
annual APICS Mathematics/Statistics/Computer Science Conferences.
Contributed papers of 20 minutes duration are encouraged.
Abstracts and titles should be sent by email to
Xingfu Zou no later than
September 30. In your email, please make it very clear that your
talk is intended for the AARMS session.
While there is no registration fee for those attending just this
AARMS session, we hope that many participants will find the time
to enjoy some of the APICS meeting
and choose to register for it (CDN$75) through
online registration.
APICS registration includes a wine and cheese reception on Friday
night and lunch Saturday. Individual tickets for the reception
and lunch will also be available.
Confirmed Invited Papers
Speaker 
Jack Hale, Regents Professor, Center for Dynamical
Systems and Nonlinear Studies, Georgia Institute of Technology 
Time 
Saturday, 9:00 a.m. 
Title 
Convective diffusion equations 
In this talk, we discuss
the dynamics on the compact global attractor
of a scalar parabolic equation in one space dimension
with convective and source terms.
We are particularly interested in the effect of having the
convective terms and in the situation when the diffusion
approaches zero. In the latter case, we have a conservation
law with a source term. The talk will expose results and
ideas rather than proofs.

Speaker 
Bill Langford, Fields Institute/Guelph University

Time 
Friday, 3:00 p.m. 
Title 
Dynamical Disease 
Medical researchers are increasingly concerned with the
temporal behaviour of physiological systems. A large number
of normal and abnormal rhythms have been identified in cardiology,
respirology, neurology and endocrinology. Arrythmias may be
significant causes of illness and even death.
The term "Dynamical Disease" has been coined to describe
ailments caused not by an invasion of bacteria or viruses,
but by a temporal malfunction of a physiological system.
Recently, mathematicians working together with medical
researchers have made significant advances in this field.
Mathematical tools from dynamical systems theory and
control theory are being adapted to study these medical problems.
This talk will describe two examples of recent medical
advances where mathematics has played a crucial role: one in
AIDS research and the other in the human cardiovascular
system. The presentation is intended for a general audience.

Speaker 
Konstantin Mischaikow, Center for Dynamical Systems and
Nonlinear Studies, Georgia Institute of Technology 
Time 
Sunday, 9 a.m. 
Title 
Coarse Numerics for Exploratory and Rigorous Dynamics 
A rapidly growing area of numerical analysis is that
of approximating the dynamics of differential equations, e.g.
the computation of periodic orbits, heteroclinic orbits,
invariant tori, etc. Many of these methods are based on knowing
what object one wants to study and having a reasonable approximation
of that object.
This talk will be about our attempts to develop efficient
numerical methods that can be used to detect the existence
of these types of objects, to give a reasonable approximation
of their location in phase space and even to give rigorous
computer assisted proofs of existences. These methods involve
a nice mixture of computational geometry, dynamics and algebraic
topology.
Since this talk is intended for a general audience,
most of the material will be presented in the context of
differential equations in the plane.)

Speaker 
Jianhong Wu, York University 
Time 
Saturday, 3:00 p.m. 
Title 
Neural Dynamics: Signal Delay and Its Impact on
ContentAddressable Memory 
We consider a system of delay differential equations modelling the dynamics
of additive shortterm memory traces of networks of neurons. We will give a
brief review of the derivation of the model for biological networks of neurons and
for their hardware implementations, and we will discuss the connection
between the global attractor of the model and the contentaddressable memory in
applications to memory restoring and retrieving. We will report some
recent progress on phasedlocked oscillation and synchronization caused by
the temporal delay of the signals among neurons (jointly with Yuming Chen)
and on the invariant stratification of global attractors (jointly with
Tibor Krisztin and Hansotto Walther).

Contributed Papers
Speaker 
Hermann Brunner, Memorial University 
Time 
Saturday, 10:15 a.m. 
Title 
On Collocation Methods for Functional Equations
with Proportional Delays 
The convergence and stability of discretizations of differential and Volterra
integral
equations with proportional delay qt (0 < q < 1) is not yet well
understood. In this survey I shall describe some recent progress
as well as open problems in collocation methods for the numerical solution
of such functional equations in piecewise polynomial spaces.

Speaker 
Yuming Chen, York 
Time 
Saturday, 5:00 p.m. 
Title 
Limiting Profiles of Periodic Solutions of Neural Networks with
Synaptic Delays 
Let f(·,l):R
® R be
given so that f(0,l) = 0 and
f(x,l) = (1+l)x+ax^{2}+bx^{3}+o(x^{3}) as x ® 0. We
characterize those small values of
e > 0 and l Î R for which there are
periodic solutions of periods approximately ^{2}/_{k} with
k Î N of the following system arising from a
network of neurons
The periodic solutions are synchronized if k is even and
phaselocked if k is odd. We show that, as e® 0,
these periodic solutions approach square waves if a = 0 and b < 0,
and pulses if a = 0 and b > 0 or if a ¹ 0. From the fact that
the periodic solutions are synchronized when k is even and the
uniqueness of the periodic orbit, we can deduce same results for
the scalar case (a single neuron)
e 
. x

(t) = x(t)+f(x(t1),l). 


Speaker 
John C. Clements, Dalhousie University 
Time 
Sunday, 10:30 a.m. 
Title 
Some Optimal Control Problems for the Next Generation of
Air Traffic Management Systems 
An important concern in air traffic management is that the worlds airways,
as they are presently constituted, are becoming inefficient and
overcrowded. The approach currently proposed for resolving these problems
both internationally and domestically is evolution to a freeflight or
"userpreferred trajectories" environment. An essential component of such
a system will be the control software required to predict and resolve
conflicting aircraft trajectories. This talk will explore some optimal
control models and realtime solution procedures for aircraft executing
cooperative and noncooperative avoidance maneuvers. It is anicipated that
the results of such studies will form part of an automated or advisory
conflict resolution system which will be incorporated into the Canadian air
traffic management program.
[1] Clements, J.C. and Ingalls, B., `An extended model for pairwise
conflict resolution in air traffic management', Optimal Control Appl.
Methods, to appear.
[2] Clements, J. C., `The optimal control of collision avoidance
trajectories in air traffic management', Transportation Research
22B, 265280 (1999).
[3] Kuchar, K.K. and Yang, C.Y., `Survey of conflict detection and
resolution modeling methods', Proceedings of the AIAA Guidance,
Navigation and Control Conference, New Orleans, LA, 469478, 1997.
[4] Menon, P. K., Sweriduk, G. D. and Sridhar, B., `Optimal strategies for
freeflight air traffic conflict resolution', J. Guidance, Control and
Dynamics, 33(2), 202211 (1999).

Speaker 
Alan A. Coley, Dalhousie University 
Time 
Saturday, 4:30 p.m. 
Title 
Dynamical Systems in Cosmology 
Dynamical systems theory is especially wellsuited for determining the
possible asymptotic states (at both early and late times) of cosmological models,
particularly when the governing equations are a finite system of
autonomous ordinary differential equations.
We discuss cosmological models as dynamical systems, pointing out the
important role of selfsimilar models, and we describe some of the asymptotic
properties of spatially homogeneous models with a perfect fluid source.

Speaker 
Andrew Foster, Memorial University 
Time 
Friday, 4:00 p.m. 
Title 
Numerical ODE Solvers from a dynamical systems perspective 
Numerical methods for solving differential equations generally
involve approximating the (continuous) flow by a (discrete) map.
Dynamical systems theory is a natural choice for studying the
possible qualitative differences between an originating flow and
its numerical approximation. In this talk, the common local
bifurcations of scalar flows are transformed under linearized
onepoint collocation methods. Through normal forms, it is shown
that each such bifurcation gives rise to an exactly corresponding
one in its discretization. However, spurious period doubling
behavior can arise, and a singular set induced by the methods has
significant behavioral consequences.

Speaker 
Wieslaw Krawcewicz, University of Alberta 
Time 
Saturday 11:15 a.m. 
Title 
Steady State/Hopf Bifurcation interaction with SO(3)symmetries 
We will discuss the interaction between steady state
and Hopf bifurcations by applying the equivariant
degree method for (SO(3)×S^{1})maps. Appearance of the
SO(3)symmetric Hopf bifurcation is related to nontrivially of
the so called primary topological obstructions and the
steadystate bifurcation is a result of nonzero secondary
obstructions. We use the multiplicativity property, which is
one of the most interesting properties
of the SO(3)×S^{1}equivariant degree to fully
compute and classify the equivariant degree characterizing
the symmetric properties of bifurcating periodic solutions.
This results were obtained in collaboration with Z. Balanov
and H. Steinlein.

Speaker 
WeiJiu Liu, Dalhousie 
Time 
Friday 4:30 p.m. 
Title 
Adaptive Control of Burgers' Equation with Unknown Viscosity 
In this talk we propose a fortified boundary control law and an
adaptation law for Burgers' equation with unknown viscosity, where no a priori
knowledge of a lower bound on viscosity is needed. This control law is
decentralized, i.e., implementable without the need for central computer
and wiring. Using the Lyapunov method,
we prove that the closedloop system, including the parameter estimator as a dynamic
component, is globally $H^1$ stable and well posed.
Furthermore, we show that the state of the system is
regulated to zero by developing an alternative to
Barbalat's Lemma which cannot be used in the present situation.

Speaker 
Wade Parsons, College of the North Atlantic 
Time 
Sunday, 11:00 a.m. 
Title 
Waveform Relaxation Methods for Volterra Integrodifferential
Equations 
This talk concerns the solution of systems Volterra
integrodifferential equations by the application of waveform
relaxation methods. This is a timely topic since such methods
can often be implemented efficiently on parallel architectures.
I give convergence results for both the
regular kernel and the weakly singular kernel cases, and
although my primary concern is with analytic solutions,
numerical methods are mentioned.

Speaker 
Shigui Ruan, Dalhousie University 
Time 
Saturday, 10:45 a.m. 
Title 
Homoclinic Bifurcations in Dynamical Systems 
Suppose a family of differential equations has a homoclinic orbit
asymptotic to an isolated equilibrium at certain critical parameter value.
Under some technical assumptions a unique stable periodic orbit could
bifurcate from the homoclinic orbit. In this talk, we first briefly
review Sil'nikov's technique in studying homoclinic bifurcations.
We then generalize the ideas of Sil'nikov and Chow and Deng to
infinite dimensional systems generated by a family of partial
functional differential equations. (Based on a join paper with
J. Wei and J. Wu)

Speaker 
Raymond Spiteri, Acadia 
Time 
Friday, 5:00 p.m. 
Title 
On the Existence of Real FloquetLyapunov Factorizations 
FloquetLyapunov Theory gives the general representation of a fundamental
matrix solution of a linear, timeperiodic system in terms of two factors:
a periodic matrix L(t) and a constant matrix F. It is well known that L(t)
and F may be complex even if the original differential system is real.
In practice, the question often arises as to when real factors exist.
In this talk, I will discuss necessary and sufficient conditions
for the existence of real factors such that L(t) has the same primary period
as that of the original system. I will further show that it is always possible
to construct real factors with information derived on one primary period, and
we give a numerically sound algorithm to do so.

Speaker 
Danny Summers, Memorial University 
Time 
Saturday, 4:00 p.m. 
Title 
Discretetime Nonlinear Dynamical Systems in the Sciences 
Discretetime dynamical systems (or iterative maps or difference
equations)arise in many contexts in the sciences including the biological,
economic,and social sciences. Such systems are also of intrinsic
mathematical interest. Understanding the behaviour of discretetime
systems is important,in addition,because differential equations are
often solved in discretised form. It is well established that even
simple nonlinear discretetime systems can reveal rich dynamical
behaviour including the perioddoubling route to chaos,the stretching
and folding of strange attractors,and so on.The first part of this talk
will contain a brief retrospective of some famous nonlinear discretetime
systems dating from 1dimensional examples from May's (1976) paper in
Nature,through the 2dimensional examples of Henon and Mandelbrot,to
the recent 3dimensional "chaotic beetle" system that has caused
excitement in ecology. The second part of the talk will examine the
behaviour of some specific 1 and 2dimensional systems under the action
of periodic forcing. Bifurcation diagrams are constructed with the
forcing amplitude as the bifurcation parameter,and these are observed
to display rich structure including chaotic bands with periodic windows,
pitchfork and tangent bifurcations,and attractor crises.

Speaker 
Xiaoqiang Zhao, Memorial University 
Time 
Sunday, 10:00 a.m. 
Title 
Chain transitivity, attractivity and strong
repellors for semidynamical systems 
The notion of chain recurrence, introduced by C. Conley, is a
way of getting at the recurrence properties of a dynamical
system. It is proving increasingly useful in a variety of
fields such as population dynamics, epidemiology, economics,
game theory, numerical analysis and stochastic approximation
algorithms. In this talk, we will discuss the (internal) chain
transitivity in semidynamical systems(continuous maps and
semiflows) on a metric space and its applications to attractivity,
convergence, strong repellors and stability of uniform persistence.
As an example, the robust permanence is established for the
Kolmogorovtype ordinary differential equations. This talk is
mainly based on a recent paper jointly with M. W. Hirsch and
H. L. Smith.

