AARMS

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Trends in Differential Equations and Dynamical Systems

October 22-24, 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 Content-Addressable Memory
We consider a system of delay differential equations modelling the dynamics of additive short-term 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 content-addressable memory in applications to memory restoring and retrieving. We will report some recent progress on phased-locked 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 Hans-otto 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+ax2+bx3+o(x3) as x 0. We characterize those small values of e > 0 and l R for which there are periodic solutions of periods approximately 2/k with k N of the following system arising from a network of neurons






e .
x
 
(t)
=
-x(t)+f(y(t-1),l),
e .
y
 
(t)
=
-y(t)+f(x(t-1),l).
The periodic solutions are synchronized if k is even and phase-locked 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 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(t-1),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 free-flight or "user-preferred 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 real-time solution procedures for aircraft executing cooperative and non-cooperative 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, 265-280 (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, 469-478, 1997.

[4] Menon, P. K., Sweriduk, G. D. and Sridhar, B., `Optimal strategies for free-flight air traffic conflict resolution', J. Guidance, Control and Dynamics, 33(2), 202-211 (1999).


Speaker Alan A. Coley, Dalhousie University
Time Saturday, 4:30 p.m.
Title Dynamical Systems in Cosmology
Dynamical systems theory is especially well-suited 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 self-similar 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 one-point 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)×S1)-maps. Appearance of the SO(3)-symmetric Hopf bifurcation is related to nontrivially of the so called primary topological obstructions and the steady-state bifurcation is a result of non-zero secondary obstructions. We use the multiplicativity property, which is one of the most interesting properties of the SO(3)×S1-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 Wei-Jiu 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 closed-loop 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 Integro-differential Equations
This talk concerns the solution of systems Volterra integro-differential 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 Floquet-Lyapunov Factorizations
Floquet-Lyapunov Theory gives the general representation of a fundamental matrix solution of a linear, time-periodic 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 Discrete-time Nonlinear Dynamical Systems in the Sciences
Discrete-time 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 discrete-time systems is important,in addition,because differential equations are often solved in discretised form. It is well established that even simple nonlinear discrete-time systems can reveal rich dynamical behaviour including the period-doubling 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 discrete-time systems dating from 1-dimensional examples from May's (1976) paper in Nature,through the 2-dimensional examples of Henon and Mandelbrot,to the recent 3-dimensional "chaotic beetle" system that has caused excitement in ecology. The second part of the talk will examine the behaviour of some specific 1- and 2-dimensional 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, pitch-fork 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 Kolmogorov-type ordinary differential equations. This talk is mainly based on a recent paper jointly with M. W. Hirsch and H. L. Smith.