


Dynamical Systems, Celestial Mechanics / Systèmes dynamiques, Méchanique céleste Org: Florin Diacu (Victoria)
 LARRY BATES, Department of Mathematics, University of Calgary
Complete integrabilty beyond LiouvilleArnol'd

We give some examples of completely integrable systems where the
fibres are not necessarily torii. Some of these systems are the first
known that have finite order monodromy.
 STEFANELLA BOATTO, Fields Institute, Toronto, Ontario M5T 3J1 and Department
of Mathematics and Statistics, McMaster University, Hamilton,
Ontario L8S 4K1
Curvature perturbations and vortex stability

A new geometrical formulation of the hydrodynamics equations is given
for pointvortices on surfaces with costant curvature. More
specifically, we consider regular polygonal configurations and the
transition in their stability properties when passing from spherical to
hyperbolic geometry. The problem has been formulated in a unifying
geometrical setting. (work in collaboration with Tadashi Tokieda)
 MONICA COJOCARU, University of Guelph, Dept. of Mathematics and Statistics, Guelph,
Ontario
Periodic solutions for projected dynamical systems

We present here a part of the work in progress on the topic of
existence of periodic solutions for the class of projected dynamical
systems. This class has been recently generalized to infinite
dimensional Hilbert spaces. However, although the matter of existence
and stability of perturbed equilibria for this class was studied by a
number of authors, the question of periodicity was not posed so far. We
will present our approach in this direction.
 ZUOSHENG HU, Carleton University
Favard's theorem for almost periodic processes on Banach
space

In the paper [8], we obtained an extension of Favard's theorem
for the linear almost periodic differential equations in R^{n}. At the
same time, we got a stronger result for scalar linear almost periodic
differential equations. We mentioned that these results can be extended
to linear almost periodic processes. This paper is to state these
generalized results.
The original work on processes is the fundamental paper of Levinson
[10] dealing with periodic differential equations in the plane.
Over the years, a tremendous literature on this subject has
accumulated. Continuing in the spirit of Levinson for finite
dimensions, many authors discussed processes on a general Banach space
in order to solve more complicated problems. A process can be
considered as being generated by a differential equation. So, the
properties of a process can describe some properties of solutions of
corresponding differential equations. At the same time, an abstract
process can characterize different kinds of differential equations,
such as ordinary differential equations, functional differential
equations, partial differential equations, and etc.
Dafermos [3] established some basic properties of trajectories
for almost periodic processes. We will refer to some of his results in
this paper. H. Ishii [9], A. Haraux [7] considered the
nonlinear contractive almost periodic processes on a Banach space and,
under some assumptions, they both established the existence of almost
periodic orbits in different ways. In particular, Ishii [9]
mentioned that when A(t) is a skew symmetric n×n matrix for
all t Î R, the corresponding result can imply Favard's Theorem.
We will also start within the framework of almost periodic processes on
a Banach space, but instead of the assumption of contractivity, we
assume the convexity and homogeneity of almost periodic
processes. At the same time, our main point is that we can weaken
Favard's condition by requiring instead that

inf
t Î R


æ è


1
2l


ó õ

t+l
tl

x(s) ds 
ö ø

> 0 
 (1) 
where l > 0 is some real number. On the other hand, if we assume some
uniformity of the almost periodic processes, the weaker condition

inf
t Î R


æ è


liminf
l® ¥


1
2l


ó õ

t+l
tl

x(s) ds 
ö ø

> 0 
 (2) 
also guarantees the existence of almost periodic orbits of the
corresponding processes. Finally, we apply our results to linear
retarded functional different equations, that is, we got an extension
of Favard Theorem for linear retarded functional different equations.
References
 [1]
 L. Amerio, Soluzioni quasiperiodiche, o limitate,
di sistemi differenziali non lineari quasiperiodici, o limiteti.
Ann. Mat. Pura Appl. 39(1955) 97119.
 [2]
 S. Bochner, A new approch to almost periodicity.
Proc. Nat. Acad. Sci. U.S.A. 48(1962), 20392043.
 [3]
 C. M. Dafermos, Almost periodic processes and almost
periodic solutions of evolution equations. In: Dynamical Systems,
Proceedings of a University of Florida International Symposium, Academic
Press, New York, 1977, 4357.
 [4]
 J. Favard, Lecons sur les functions
presqueperiodiques. GauthierVillars, Paris, 1933
 [5]
 A. M. Fink, Almost periodic differential equations.
Lecture Notes in Math. 377, SpringerVerlag, New York, 1974.
 [6]
 J. K. Hale, Theory of Functional Differential
equations. SpringerVerlag, New York, 1977.
 [7]
 A. Haraux, Nonlinear Evolution EquationGlobal
Behavior of Solutions. Lecture Notes in Math. 841,
SpringerVerlag, New York, 1981.
 [8]
 Z. Hu and Mingarelli, On a Favard Theorem.
September, (2003), Proc. Amer. Math. Soc., to appear.
 [9]
 H. Ishii, On the existence of almost periodic
trajectories for contractive almost periodic processes.
J. Differential Equations 43(1983), 6672.
 [10]
 N. Levinson, Transformation theory of nonlinear
differential equations of second order. Ann. of Math. 45(1944),
724737.
 [11]
 T. Yoshizawa, Stability Theory and the
existenxe of periodic solutions and almost periodic solutions.
SpringerVerlag, New York, 1975.
 WU JIANHONG, York University
Hyperbolic PDEs with delay: delayed diffusion and
retarded reaction in biological systems

We show how a hyperbolic partial differential equation with delayed
nonlocal nonlinearity arises naturally from the interaction of delayed
diffusion and retarded feedback in biological systems. We discuss
traveling waves and global dynamics of such a hyperbolic PDE.
 TOMASZ KACZYNSKI, Département de mathématiques et d'informatique,
Université de Sherbrooke, Sherbrooke, Québec J1K 2R1
Detection of interesting dynamics via combinatorial index
pairs

In the past decade several generalizations of the Conley Index to
discretetime case were obtained with the aim of applying them to prove
chaotic behavior (or other interesting features) of solutions to
classical differential equations. In this talk, we shall discuss
collective efforts made to provide more efficient and, also, more
elementary construction of index pairs and index maps which are the
base for those applications.
 BILL LANGFORD, University of Guelph, Department of Mathematics and Statistics,
Guelph, Ontario N1G 2W1
Hopf meets Hamilton under Whitney's umbrella

The classical Hopf Bifurcation Theorem describes the generic mechanism
for the birth or death of a periodic solution, near an equilibrium
point of a dynamical system. However, it does not apply to the
important case of Hamiltonian systems, where the generic bifurcation of
periodic solutions is described by what has been called the
HamiltonianHopf bifurcation theorem.
This talk will explore the interface between the classical Hopf
Bifurcation Theorem and the Hamiltonian case; that is, the general
setting of systems that are nearly Hamiltonian (also called weakly
dissipative). This Hamiltonian limit leads to an interesting
singularity known as Whitney's umbrella, which will be described. In
fact, the Hamiltonian case lies on the "handle" of Whitney's
umbrella. There exist limit cycles that persist in the
nearHamiltonian case (just off the handle of the umbrella), that
correspond to the two families of centers that exist in the Hamiltonian
case.
 DAN OFFIN, Queen's University
Lagrangian singularities and stabiltity of periodic orbits

Recent results using variational methods to detect periodic orbits in
the Nbody problem have been widely celebrated. One question which is
unresolved is whether the stability type of the periodic orbits can be
determined from the variational principle. With this goal in mind, we
present general results connecting the stability type of periodic
orbits on three dimensional energymomentum surfaces, with the
measurement of Lagrangian singularities along the orbits. The counting
of the singularities, given by the Maslov index, is shown to be closely
tied to the stability type. We illustrate the general results with
numerical simulations of the Isosceles three body problem, and a
variational principle to determine symmetric hyperbolic periodic orbits
for this system.
 JUANPABLO ORTEGA, CNRS/Université de FrancheComté, Besancon, France
The geometry of Hamiltonian conservation laws

Part of this talk is joint work with Tudor Ratiu.
In his talk we will describe several approaches to the description of
the conservation laws associated to the symmetries of a Hamiltonian
dynamical system. More specifically, we will discuss the role of
singular foliation theory and its interplay with some aspects of the
theory of groupoids in the introduction of the so called optimal
momemtum map. These ideas have an interesting interplay with cylinder
valued and Lie group valued momentum maps and with the notions of
symplectic and Poisson duality that will also be discussed.
 GEORGE PATRICK, University of Saskatchewan
Interactions of relative equilibria of point vortices

The Hamiltonian system of point vortices in the plane admits relative
equilibria consisting of 3 point vortices of strength G in
symmetric orbit about a central vortex of strength 3G. I will
outline the theory and interactions of these these relative equilibria,
which occur at nongeneric momentum and move on the plane as particles
with a mass calculable through certain normal forms.
 ERNESTO PEREZCHAVELA, Universidad Autonoma MetropolitanaIztapalapa, Departamento de
Matematicas, Av Rafael Atlixco 186, Col. Vicentina, Mexico D.F. o9340
Relative equilibria in the charged three body problem

The charged three body problem concerns the study of the dynamic of
3point particles endowed with a positive mass and an electrostatic
charge of any sign, moving under the influence of the corresponding
Newtonian and Coulombian forces. In this work we study some special
kind of periodic orbits, called relative equilibria and their
stability.
 MARK ROBERTS, University of Surrey
Relative periodic solutions of symmetric Lagrangian systems

This talk will describe some recent results on the algebraic topology
of the `relative loop spaces' of configuration spaces of Lagrangian
mechanical system with symmetries. Applications to the existence of
periodic solutions with prescribed symmetries and homotopy classes will
be described. Illustrative examples will include geodesic flows on tori
and the `strong force' Ncentre problem. This is joint work with
Chris McCord, James Montaldi and Luca Sbano.
 CHRISTIANE ROUSSEAU, Université de Montréal
Modulus of orbital analytic classification of a family
unfolding a generic saddlenode

In this talk we consider generic families of 2dimensional analytic
vector fields unfolding a generic (codimension 1) saddlenode at the
origin. We show that a complete modulus of orbital analytic
classification for the family is given by an unfolding of the
MartinetRamis modulus of the saddlenode. The MartinetRamis modulus
is given by a pair of germs of diffeomorphisms, one of which is an
affine map. We show that the unfolding of this diffeomorphism in the
modulus of the family is again an affine map. The point of view taken
is to compare the family with the "model family" (x^{2}e)[(¶)/(¶x)] +y (1+a(e)x) [(¶)/(¶y)]. The nontriviality of the MartinetRamis modulus implies
geometric "pathologies" for the perturbed vector fields, in the sense
that the deformed family does not behave as the standard family.
 MANUELE SANTOPRETE, University of California, Irvine, Department of Mathematics,
Irvine, California 926973875, USA
Qualitative properties of the anisotropic Manev problem

The anisotropic Manev problem is a twobody problem given by the Manev
potential in an anisotropic space. This system is important to help
understand connections between classical mechanics, quantum mechanics,
and relativity. Aside from this physical implications, the anisotropic
Manev problem also exhibits many phenomena of considerable mathematical
interest; in this talk we will exclusively consider this aspect of the
problem.
The first phenomenon that will be illustrated arises from the collision
orbits. It will be shown that, after a McGehee blowup of
singularities, some of the collision solutions are transformed into
orbits homoclinic to a nonhyperbolic periodic orbit. A suitable
generalization of the PoincaréMelnikov method will be applied to
those orbits to show that the negatively and positively invariant sets
of the periodic orbit intersect transversely.
The second interesting behavior, that will be discussed, is related to
the properties of the solutions at infinity. These properties will be
described performing another blowup transformation and analyzing the
flow on the infinity manifold.
 DANA SCHLOMIUK, Universite de Montreal
Recent results on global problems about planar polynomial
vector fields

There are several longstanding famous open problems on the global
behavior of solutions of planar polynomial vector fields. Recent
progress on the global theory of quadratic systems show that there
exist tight connections among these problems. In this lecture we
present new results on planar quadratic differential systems which show
very clearly the interdependence of these problems.
 TANYA SCHMAH, University of Surrey, Guildford, Surrey, United Kingdom
A slice theorem for cotangentlifted symmetries

Slice theorems give local models for manifolds with Lie group
symmetries. The Hamiltonian Slice Theorem of Marle, Guillemin and
Sternberg is a slice theorem for symplectic manifolds. It is a
fundamental tool in the study of Hamiltonian systems with symmetry,
with applications to singular reduction and to questions of stability
and bifurcations.
In this talk, we present an extension of the Hamiltonian slice theorem
that is specific to cotangentlifted actions. This result has two
advantages over the earlier more general one: the proof is
constructive, and it involves a cotangentbundlespecific decomposition
of the symplectic normal space.
 JEDRZEJ SNIATYCKI, University of Calgary, Calgary, Alberta T2N 1N4
Poisson algebras in reduction of symmetries

Reduction of symmetries of Hamiltonian systems consists of two steps:
1. Projection to the orbit space,
2. Localization at the chosen level of the conserved momentum map.
I shall discuss the dependence of the resulting Poisson algebra on the
order of these steps and on the choice of the localization procedure.
 CRISTINA STOICA, Univeristy of Surrey, Guildford, Surrey, United Kingdom
Bifurcations from collinear relative equilibria for triatomic
molecules

Triatomic molecules can be modelled as a generalized three body problem
with a rotational invariant bonding potential. Such systems admit
relative equilibria in a collinear configuration.
Using Hamiltonian slice equations for simple mechanical systems, we
present a study of the bifurcation diagram emanating from a collinear
relative equilibrium.
This work is joint with Mark Roberts and Tanya Schmah.
 ANDRE VANDERBAUWHEDE, University of Ghent, Belgium
Bananas and banana splits: degenerate subharmonic branching
in reversible systems

We present a general reduction method of LyapunovSchmidt type for
determining the bifurcations of qperiodic orbits from a symmetric
fixed point in families of equivariant reversible diffeomorphisms; it
is shown that this problem reduces to a similar problem on a reduced
space, involving an equivariant reversible diffeomorphism with an
additional Z_{q}symmetry, and such that the bifurcating
orbits are also Z_{q}orbits. Next we show how this reduction
can be used to study subharmonic branching in reversible vectorfields.
Typically, when along a branch of symmetric periodic orbits a pair of
complex conjugate multipliers passes (in a transversal way) through a
qth root of unity q ³ 3), two branches of symmetric
qsubharmonics will bifurcate. However, in the degenerate case where
the transversality condition fails two bifurcation scenarios appear
that are reminiscent of the "banana" and "banana split" scenarios
discovered a decade ago by Peckham, Frouzakis and Kevrekidis [1] for
general diffeomorphisms. An example of the banana scenario can be
found in the restricted 3body problem where it plays an important role
in the creation and bifurcations of the web of periodic orbits near
L4 and L5 (see [2]).
This is work in progress in collaboration with MariaCristina Ciocci
(Gent), FrancescoJavier MuñozAlmaraz, Emilio Freire and Jorge
Galán (Sevilla).
References
 [1]
 B. Pecham, C. Frouzakis and I. Kevrekidis, Bananas and
banana splits: a parametric degeneracy in the Hopf bifurcation for maps.
SIAM J. Math. Anal. 26(1995) 190217.
 [2]
 J. Henrard, The web of periodic orbits at L4.
Celestial Mech. Dynam. Astronom. 83(2000) 291302.

