




Dynamical Systems / Systèmes dynamiques (Michael Radin, Organizer)
 BHAGWAN AGGARWALA, University of Calgary, Calgary, Alberta
Application of predator prey equations to development of
HIV/AIDS in Canada

We consider the ratio dependent predatorprey equations of mathematical
biology and watch the very rich behavior of solutions unfold as the
conversion factor changes. For large values of the conversion factor,
we encounter repeated bifurcations and stable and unstable limit
cycles. These limit cycles disappear as this parameter is further
increased. We apply these equations to the spread of HIV/AIDS in the
Canadian society and notice that unless the number of healthy people in
the society begins to grow much faster than at present moment, we are
in for hard times!
 WILLIAM BASENER, Rochester Institute of Technology
Disk global cross sections in flows on manifolds

For a flow on a manifold, a global cross section is an imbedded
codimension 1 manifold which intersects the orbit through every point
in positive and negative time. Global cross sections without boundary
are a standard tool which capture global topological properties of a
flow. We show that global cross sections that are a disk capture many
of the same properties of the flow. We will prove that, unlike global
cross sections without boundary, every continuous flow on a manifold of
dimension greater than 2 has a global cross section that is a disk.
 BERNARD BROOKS, Rochester Institute of Technology
A rrequency and density dependent multiallele haploid dynamic

A discrete model of frequency and density dependent haploid evolution
will be presented along with the conditions for the linear stability of
the equilibria of the dynamical system. The organisms in question are
multiallele and simulations of the results will be shown.
 PIETROLUCIANO BUONO, CRM, Universite de Montreal, Montreal, Quebec H3C 3J7
Restrictions on flows in double Hopf bifurcations for
functional differential equations

I am going to discuss the normal form of vector fields generated by
delay differential equations at double Hopf bifurcation points. The
main results presented will be:
1) there are generically no restrictions on the possible flows of
scalar first order equations with two delays in the nonlinearity,
2) There are always restrictions on the possible flows for scalar
first order equations with one delay in the nonlinearity and nth
order delay equations (n ³ 2) with one delay in the feedback.
These results are based on computations of normal form for delay
differential equations via the method of Faria and Magalhaes.
 MONICA GABRIELA COJOCARU, Queen's University
Projected dynamical systems

We present a dynamical system constrained to a closed, convex subset of
a Hilbert space of arbitrary dimension. It is a generalization of a
notion introduced in the Euclidean space. A projected dynamical system
is defined by the solutions of a differential equation with a
discontinuous and nonlinear righthand side. We will outline the proof
of existence of such systems and emphasize the difficulties encountered
and the techniques we used.
 ABBA GUMEL, University of Manitoba
Dynamics analysis of the effect of condom use on HIV epidemiology

(joint work with S. M. Moghadas and R. Gordon (University of Manitoba).)
There are two major approaches for controlling the HIV epidemic namely
``prevention'' and therapeutic treatment. Although the new antiHIV
therapies appear promising, prevention remains the most effective
weapon against the epidemic. This paper studies the effect of condom
use in HIV prevention in the absence of any treatment. A new
deterministic model, for monitoring the communitywide effect of condom
use, is designed and analyzed qualitatively. Based on the stability
and numerical analyses of the model, our study shows that
communitywide eradication of HIV is feasible provided the condom
``preventability'' parameter, defined as the product of condom efficacy
and compliance in its use, exceeds a certain threshold.
 HENRIK KALISCH, McMaster University
The effect of attenuation and amplification on dispersionmanaged
solitons

The dynamic properties of pulses in a dispersionmanaged optical fiber
transmission system are investigated using a reduction to a
twodimensional system of ordinary differential equations based on a
Gaussian Approximation. In particular, pulses which are nearly
stationary are investigated in regard to stability and their behavior
with respect to limits of certain system parameters.
 CANDACE KENT, Virginia Commonwealth University
The boundedness nature of positive solutions of the sifference squation
x[n+1] = max{ A[n]/x[n], B[n]/x[n1]} with periodic parameters

We investigate the boundedness nature of positive solutions of the
difference equation
x[n+1] = 
max
 { A[n]/x[n], B[n]/x[n1] }, n=0,1,..., 

where {A[n]} is a periodic sequence of positive numbers with prime
period p and {B[n]} is a periodic sequence of positive numbers
with prime period q. We will give necessary and sufficient conditions
for every positive solution to be unbounded.
 WITOLD KOSMALA, Appalachian State University
On the recursive sequence x(n+1) = (p + x(nk))/x(n)

We investigate the periodic character, boundedness, stability of
solutions, and properties of semicycles of the nonlinear difference
equation
x(n+1) = 
æ è

p + x(nk) 
ö ø

/x(n) 

where the parameter p and the initial conditions are positive real
numbers. Joint work together with Elias Camouzis and Richard DeVault.
 HERBERT KUNZE, University of Guelph
Solving inverse problems in differential equations using Banach's
fixed point theorem

Banach's fixed point theorem plays a role in the (often approximate)
solution of many inverse problems in mathematics. The essential idea is
to express the problem in terms of a contractive map on a complete
metric space in order to use the fixed point theorem and some more
recently developed related results, most notably Barnsley's collage
theorem. For the inverse problem, one is given a solution, possibly an
approximation or an interpolation of experimental data, and then one
seeks a map of a particular form which has this ``target'' solution as
a fixed point. In this talk, we consider the problem: Given a function
x(t), possibly an interpolation of data points, with t Î [0,1],
find an ODE x¢=f(x,t), x(0)=x_{0}, that admits x(t) as a
solution as closely as desired, where f is restricted to a class of
functional forms (affine, quadratic in x, etc.). A
collagelike theorem is established, with the end result being an
algorithm which produces a leastsquares minimization problem. Many of
the parameter estimation techniques used by researchers in differential
equations can be viewed as inverse problems of the outlined type; this
work appears to be the first rigorous mathematical justification of
such ad hoc methods of error minimization.
 WILLIAM LANGFORD, University of Guelph
Nearreversible 1:1 resonant Hopf bifurcation

The standard Hopf bifurcation theorem does not apply to reversible
vector fields, but is replaced by the ``reversible Hopf bifurcation
theorem'' in which two pairs of complex conjugate eigenvalues approach
the imaginary axis symmetrically from the left and right, then merge in
double purely imaginary eigenvalues and separate along the imaginary
axis. The merging is called 1:1 resonance, and has codimension one in
the class of reversible vector fields. In the general case of
nonreversible vector fields, occurrence of 1:1 resonant double
imaginary eigenvalues has codimension three. This talk will present an
investigation of the interface between the reversible and
nonreversible 1:1 resonant Hopf bifurcations, which we call the
nearreversible case. The investigation proceeds in both directions;
first as a restriction of the general case of codimension three 1:1
resonance, and second as a dissipative perturbation of the reversible
case taking advantage of the first integrals that exist there. We show
that interesting bifurcation phenomena known to exist in the general
nonreversible case persist in the nearreversible case, and we reveal
some new phenomena.
This is joint work with G. Iooss, INLNCNRS, France.
 ISRAEL NCUBE, York University
Possible change in criticality of Hopf bifurcation in a
multipledelayed neural system

We consider a network of three multipledelayed identical neurons,
consisting of a single discrete delay and a continuously distributed
delay. The model for such a network is a system of delay differential
equations (DDEs). It is shown that two cases of a single Hopf
bifurcation may occur at the trivial fixed point of the full nonlinear
system of DDEs. These are (i) the simple root Hopf, and (ii) the
double root Hopf. This presentation focusses on (ii), with particular
attention paid to the mechanics (and problems) of deriving the
corresponding system of ordinary differential equations describing the
semiflow on the centre manifold. Our long term objective is to
characterise the criticality of the single Hopf double root case.
 DMITRY PELINOVSKY, McMaster University
Spectral analysis of optical bistability in finite and
semiinfinite nonlinear photonic gratings

We study optical bistability in nonlinear periodic structures of finite
and semiinfinite length. For finitelength structures, the system
exhibits instability mechanisms typical for dissipative dynamical
systems like delaydifferential equations. We show that the
LeraySchauder degree equals the sign of the Evans function in
l = 0. Using the Evans function methods, we rigorously prove that
the stationary solutions with optically bistable transmission
characteristics are indeed spectrally unstable. For semiinfinite
structures, the system exhibits two pulse solutions, when the infinite
Hamiltonian dynamical system has soliton solutions, and one pulse
solution, otherwise. We show that the nonmonotonic pulse solution is
always spectrally unstable. Numerical computations of the linear
stability problem and timedependent coupledmode equations confirm the
analytical results.
In timedependent dynamics, the incident light waves are uniformly
stable and alloptical limited if the photonic grating has zero
netaverage Kerr nonlinearity. When the average nonlinearity is weak
but nonzero, light waves exhibit oscillatory instabilities and
longterm highamplitude oscillations in the outofphase linear
gratings. We show that a twostep transmission map between
lowertransmissive and highertransmissive states has a narrow
stability domain, which limits its applicability for logic and
switching functions. Light waves exhibit cascades of real and complex
instabilities in the multistable gratings with strong netaverage Kerr
nonlinearity. Only the first lowertransmissive stationary state can
be stimulated by the incident light of small intensities. Light waves
of moderate and large intensities are essentially nonstationary in the
multistable gratings and they exhibit periodic generation of Bragg
solitons and blowup.
 MARIO ROY, Concordia University
Is the composition of two expansive maps expansive?

We will investigate the question whether the composition of two
(positively) expansive maps is itself expansive. Two mutually opposed
situations will confront us. On the one hand, recall that the
composition of two maps that are expanding with respect to the same
metric is expanding. On the other hand, the composition of two
expansive homeomorphisms is generally not expansive (for instance,
consider an expansive homeomorphism and its inverse). Which of these
two situations will ``prevail''?
 SHIGUI RUAN, Dalhousie University
Dynamical behavior of an epidemic model with a nonlinear incidence
rate

In this paper we study the global dynamics of an epidemic model with
vital dynamics and nonlinear incidence rate of saturated mass action.
By carrying out global qualitative and bifuraction analyses, it is
shown that either the number of infective individuals tends to zero as
time evolves or there is a region such that the disease will be
persistent if the initial position lies in the region and the disease
will disappear if the initial position lies outside this region. When
such a region exists, it is shown that the model undergoes a
BogdanovTakens bifurcation, i.e., it exhibits a saddlenode
bifurcation, Hopf bifurcations, and a homoclinic bifurcation. Existence
of none, one or two limit cycles is also discussed.
 HASSAN SEDAGHAT, Department of Mathematics, Virginia Commonwealth University
Richmond, Virginia 232842014, USA
On semiconjugates of one dimensional maps

Rsemiconjugates maps are defined as a natural means of relating a
map F of the mdimensional Euclidean space to a mapping g of the
interval via a link map H. Invariants are seen to be special cases of
semiconjugate links where g is the identity. Basic relationships
between the dynamical behaviors of g and F, and conditions under
which a link map H is a Liapunov function are outlined. Examples and
applications involving concepts from stability to chaos are discussed.
 MICHAEL YAMPOLSKY, University of Toronto
Mating of quadratic polynomials

Mating is a heuristic procedure proposed by Douady and Hubbard to
dynamically parameterize quadratic rational maps by pairs of quadratic
polynomials. We will survey the known results, and some conjectures
related to matings, including our own recent joint work with
S. Zakeri.

