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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 time-dependent leakage term. This term is fitted to the functional form desired once first-stage 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 Donati-Mathématiques, Physique, Pharmacie, Anesthésiologie and Génie biomédical-Université 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 Korteweg-de Vries equation, the Sine-Gordon 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 infinite-dimensional 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 non-symmetric MHD equilibria that model astrophysical jets.

PIETRO-LUCIANO 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 G-reversible equivariant. In this talk, I will present a method for reducing reversible equivariant steady-state bifurcation problems of ``separable type'' to equivariant bifurcation problems (in general with parameter symmetries). The proof relies on the formalism and results of G-transversality 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 D3-symmetry, 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 Mallet-Paret and Sell (J. Differential Equations, 125, pp. 385-440, pp. 440-489), 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 = Z2 \wr G, where Z2 acts by ±1 on R and G is a transitive subgroup of the permutation group SN. The group G acts absolutely irreducibly on RN. 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 non-gradient 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 symmetry-breaking for spiral waves

We study the effects on SE(2)-equivariant dynamical systems of forced symmetry-breaking to a proper subgroup S Ì SE(2). Specifically, we are interested in the perturbed dynamics on normally-hyperbolic relative equilibria and relative periodic solutions, in the cases where:

(1)  S = SO(2) (translational symmetry-breaking), and

(2)  S = Zl+¢R2 (rotational symmetry-breaking).

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, phase-locking 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 non-axisymmetric 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.

Invariant modules and dynamical symmetries of non-linear 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 non-linear equations it becomes necessary to characterize non-linear partial-differential operators with finite-dimensional 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
Reaction-diffusion models of growing plant tips: bifurcations on hemispheres

We study two chemical models for pattern formation in growing plant tips. The models are two-morphogen reaction-diffusion 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 equilibria-noncompact 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 non-Hausdorff 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 ratio-dependent predator-prey system

Recently, ratio-dependent predator-prey systems have been regarded by some researchers to be more appropriate for predator-prey 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 well-defined there. In this paper, the qualitative behavior of a class of ratio-dependent predator-prey 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 non-existence of limit cycles for the model are given. Computer simulations are presented to illustrate the conclusions.

JEDRZEJ SNIATYCKI, University of Calgary, Calgary, Alberta
Non-holonomic singular reduction

Hamiltonian dynamics with linear non-holonomic 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 reaction-diffusion system modeling man-environment-man 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 Aronson-Weinberger'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 non-local effect in a population model

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

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