




Dynamics and Symmetry / Dynamique et symétrie (Bill Langford and Jianhong Wu, Organizers)
 JACQUES BÉLAIR, Départment de mathématiques et Centre de recherches
mathématiques, Université de Montréal
Oscillations in drug administration: modeling pharmacokinetics by
incorporating spatial effects

Regular and irregular oscillations are ubiquitous in physiological
systems, both in normal and pathological conditions. It has been
realized recently that schemes for drug administration must be designed
to represent as closely as possible the physiological regimes,
incorporating fluctuations if need be. We present a model to describe
the time course of plasma concentration of neuromuscular blocking
agents used as anaesthetics during surgery. The model overcomes the
limitations of the classical compartmental models commonly used in
pharmacokinetics by incorporating spatial effects due to heterogeneity
in the circulation, and takes the form of a diffusion equation on a
circular domain, with a timedependent leakage term. This term is
fitted to the functional form desired once firststage transients have
died out. Comparisons are made with clinical data by adjusting three
parameters.
[Joint work with Patrick Lafrance, Vincent Lemaire, Fahima Nekka,
France Varin and François DonatiMathématiques, Physique,
Pharmacie, Anesthésiologie and Génie biomédicalUniversité de
Montréal]
 OLEG BOGOYAVLENSKIJ, Department of Mathematics, Queen's University, Kingston,
Ontario K7L 3N6
New symmetries of the magnetohydrodynamics equilibrium equations

New continuous symmetries are introduced for the ideal
magnetohydrodynamics (MHD) equilibrium equations. The equations are
widely used in plasma physics and astrophysics. The symmetries have the
following properties that distinquish them from the Backlund transforms
for the soliton equations (such as the Kortewegde Vries equation,
the SineGordon equation, etc.):
a) Unlike the Backlund transforms for the soliton equations, the new
symmetries depend on all three spatial variables x, y, z and are
given by the explicit algebraic formulae;
b) The symmetries break the geometrical symmetries of some MHD
equilibria;
c) The symmetry transformations form infinitedimensional abelian Lie
groups that depend on the topological properties of the equilibrium
solutions.
As an application of the new symmetries, we have constructed a large
family of the global nonsymmetric MHD equilibria that model
astrophysical jets.
 PIETROLUCIANO BUONO, Centre de Recherches Mathematiques, Universite de Montreal, Montreal,
Quebec H3C 3J7
Bifurcations in reversible equivariant dynamical systems

A vector field x¢=f(x) has a reversible symmetry R if for any
solution curve x(t) then Rx(t) is also a solution curve. This
leads to the following commutation relation: f(Rx)=Rf(x). The group
G of reversible symmetries and regular symmetries of a vector field
is the reversing symmetry group and the vector field is said to
be Greversible equivariant. In this talk, I will present a
method for reducing reversible equivariant steadystate bifurcation
problems of ``separable type'' to equivariant bifurcation problems (in
general with parameter symmetries). The proof relies on the formalism
and results of Gtransversality theory. This is joint work with
J. S. W. Lamb (Imperial College) and M. Roberts (University of
Surrey).
 SUE ANN CAMPBELL, University of Waterloo, Waterloo, Ontario N2L 3G1
Rings of oscillators with delayed coupling

We consider a ring of oscillators linked with nonlinear, time delayed
coupling. When the oscillators are identical we show that mode
interaction can lead to the coexistence of different stable oscillation
patterns, or of an oscillation and a stable nontrivial equilibria. We
show that this behaviour persists for nonindentical oscillators with
appropriate parameter values.
 YUMING CHEN, Department of Mathematics, Wilfred Laurier University, Waterloo,
Ontario N2L 3C5
Discrete Lyapunov functionals and symmetry

We consider a system of delay differential equations with D3symmetry,
which describes the dynamics of a network of three bidirectionally
connected neurons. Basing on the theory of discrete Lyapunov
functionals for cyclic nearest neighbor systems of differential delay
equations developed by MalletParet and Sell (J. Differential
Equations, 125, pp. 385440, pp. 440489), we define another
discrete Lyapunov functional for the considered system. A Morse
decomposition of the global attractor is obtained by using this
discrete Lyapunov functional.
 BENOIT DIONNE, University of Ottawa, Ottawa, Ontario
Heteroclinic cycles and wreath product symmetries

We consider the existence and stability of heteroclinic cycles arising
by local bifurcation in dynamical systems with wreath product symmetry
G = Z_{2} \wr G, where Z_{2} acts by
±1 on R and G is a transitive subgroup of
the permutation group S_{N}. The group G acts
absolutely irreducibly on R^{N}. We consider primary
(codimension one) bifurcations from an equilibrium to heteroclinic
cycles as real eigenvalues pass through zero. We relate the
possibility of such cycles to the existence of nongradient equivariant
vector fields of cubic order. Using Hilbert series and the software
package MAGMA we show that apart from the cyclic groups G
(already studied by other authors) only five groups G of
degree less than 7 are candidates for the existence of heteroclinic
cycles. We establish the existence of certain types of heteroclinic
cycle in these cases by making use of the concept of a subcycle.
 VICTOR G. LEBLANC, University of Ottawa, Ottawa, Ontario K1N 6N5
Forced symmetrybreaking for spiral waves

We study the effects on SE(2)equivariant dynamical systems of
forced symmetrybreaking to a proper subgroup S Ì SE(2).
Specifically, we are interested in the perturbed dynamics on
normallyhyperbolic relative equilibria and relative periodic
solutions, in the cases where:
(1) S = SO(2) (translational symmetrybreaking), and
(2) S = Z_{l}+¢R^{2} (rotational symmetrybreaking).
The goal is to obtain a phenomenological explanation of certain dynamic
states which have been observed for spiral waves in excitable media,
and which are inconsistent with full SE(2) symmetry (e.g.
anchoring by inhomogeneities, drifting along boundaries, phaselocking
in anisotropic media, transition to retracting tip).
 GREGORY LEWIS, The Fields Institute
Double Hopf bifurcations in the differentially heated rotating
fluid annulus

Symmetry breaking bifurcations that occur in a mathematical model of
the differentially heated rotating fluid annulus are analyzed. More
specifically, center manifold reduction and normal forms are used to
evaluate the double Hopf bifurcations that occur along the transition
between axisymmetric steady solutions and nonaxisymmetric rotating
waves. A combination of analytical and numerical methods lead to
numerical approximations of the coefficients of the normal form
equations.
The results indicate that, close to the bifurcation points, there are
regions in the space of parameters where multiple stable waves are
possible. Hysteresis of these waves is predicted. The numerical
approximations are validated by comparison with experimental
observations.
 ROBERT MILSON, Dalhousie
Invariant modules and dynamical symmetries of nonlinear
evolution equations

Classically symmetry reduction for PDEs relies on the notions of point
transformation symmetries and invariant solutions. For evolutionary
PDE's it is possible to consider an alternative reduction formalism
based on the notion of invariant function modules. In order to apply
this formalism to nonlinear equations it becomes necessary to
characterize nonlinear partialdifferential operators with
finitedimensional invariant modules. In this talk we will consider
how this can be accomplished using the idea of an affine annihilator of
finitely generated module of analytic functions.
 WAYNE NAGATA, Department of Mathematics, University of British Columbia,
Vancouver, British Columbia V6T 1Z2
Reactiondiffusion models of growing plant tips: bifurcations
on hemispheres

We study two chemical models for pattern formation in growing plant
tips. The models are twomorphogen reactiondiffusion systems on the
surface of a hemispherical shell, with Dirichlet boundary conditions.
Bifurcation analysis shows that both models give possible mechanisms
for dichotomous branching of the growing tips. Symmetries of the
models are used in the analysis.
 GEORGE W. PATRICK, University of Saskatchewan, Saskatchewan
Stability of relative equilibrianoncompact symmetry
and nongeneric momentumm

For Hamiltonian systems with symmetry, stability of a relative
equilibrium follows from confinement by energy on its reduced space if
its momentum is generic or the symmetry group is compact. Such
confinement is insufficient to establish stability of relative
equilibria at nongeneric momenta in the case of a noncompact symmetry
group. We link this to the appearance of nonHausdorff topologies in
the quotient space of the Poisson reduce space by its symplectic
leaves, and, based on this topology, derive sufficient conditions for
stability of such relative equilibria. The theory is applicable to
certain relative equilibria of underwater vehicles.
 SHIGUI RUAN, Department of Mathematics and Statistics, Dalhousie University,
Halifax, Nova Scotia B3H 3J5, and Department of Mathematics, Vanderbilt
University, Nashville, Tennessee 37240, USA
Global dynamics of a ratiodependent predatorprey system

Recently, ratiodependent predatorprey systems have been regarded by
some researchers to be more appropriate for predatorprey interactions
where predation involves serious searching processes. However, such
models have set up a challenging issue regarding their dynamics near
the origin since these models are not welldefined there. In this
paper, the qualitative behavior of a class of ratiodependent
predatorprey system at the origin in the interior of the first
quadrant is studied. It is shown that the origin is indeed a critical
point of higher order. There can exist numerous kinds of topological
structures in a neighborhood of the origin including the parabolic
orbits, the elliptic orbits, the hyperbolic orbits, and any combination
of them. These structures have important implications for the global
behavior of the model. Global qualitative analysis of the model
depending on all parameters is carried out, and conditions of existence
and nonexistence of limit cycles for the model are given. Computer
simulations are presented to illustrate the conclusions.
 JEDRZEJ SNIATYCKI, University of Calgary, Calgary, Alberta
Nonholonomic singular reduction

Hamiltonian dynamics with linear nonholonomic constraints is described
as a distributional Hamiltonian system. We show that the space of
orbits of the proper action of a symmetry group is a differential space
partitioned by smooth manifolds preserved by the reduced evolution. The
reduced dynamics is given by distributional Hamiltonian systems on the
manifolds of the partition.
 XIAOQIANG ZHAO, Department of Mathematics and Statistics, Memorial University,
St. John's, Newfoundland A1C 5S7
Fisher waves in an epidemic model

In this talk, we will report our recent results on the existence of
Fisher type monotone traveling waves and the minimal wave speed for a
reactiondiffusion system modeling manenvironmentman epidemics. Our
approach is via the method of upper and lower solutions as applied to a
reduced second order ordinary differential equation with infinite time
delay. It may be expected naturally that the minimal wave speed is also
the asymptotic speed of propagation in AronsonWeinberger's sense.
This problem will be discussed if time permits.
 XINGFU ZOU, Memorial University of Newfoundland, St. John's,
Newfoundland A1C 5S7
Monotonicity jointly induced by delay and nonlocal
effect in a population model

We show that a certain type of reaction diffusion equations with time
delay and nonlocal effect defines a monotone semiflow whereas the
corresponding nondelay problem and local problem do not. A prototype
of such equations is a population model containing both time delay and
nonlocal effect, which is derived from a age structure of the
population.

