---

Dynamical Systems, Celestial Mechanics / Systèmes dynamiques, Méchanique céleste
Org: Florin Diacu (Victoria)

---

LARRY BATES, Department of Mathematics, University of Calgary
Complete integrabilty beyond Liouville-Arnol'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 point-vortices 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 Rⁿ. 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 non-linear 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 t-l  |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 t-l  |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 quasi-periodiche, o limitate, di sistemi differenziali non lineari quasi-periodici, o limiteti. Ann. Mat. Pura Appl. 39(1955) 97-119.

[2]

S. Bochner, A new approch to almost periodicity. Proc. Nat. Acad. Sci. U.S.A. 48(1962), 2039-2043.

[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, 43-57.

[4]

J. Favard, Lecons sur les functions presque-periodiques. Gauthier-Villars, Paris, 1933

[5]

A. M. Fink, Almost periodic differential equations. Lecture Notes in Math. 377, Springer-Verlag, New York, 1974.

[6]

J. K. Hale, Theory of Functional Differential equations. Springer-Verlag, New York, 1977.

[7]

A. Haraux, Nonlinear Evolution Equation-Global Behavior of Solutions. Lecture Notes in Math. 841, Springer-Verlag, 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), 66-72.

[10]

N. Levinson, Transformation theory of nonlinear differential equations of second order. Ann. of Math. 45(1944), 724-737.

[11]

T. Yoshizawa, Stability Theory and the existenxe of periodic solutions and almost periodic solutions. Springer-Verlag, 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 non-local 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 discrete-time 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 Hamiltonian-Hopf 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 near-Hamiltonian 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 N-body 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 energy-momentum 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.

JUAN-PABLO ORTEGA, CNRS/Université de Franche-Comté, 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 non-generic momentum and move on the plane as particles with a mass calculable through certain normal forms.

ERNESTO PEREZ-CHAVELA, Universidad Autonoma Metropolitana-Iztapalapa, 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 3-point 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' N-centre 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 saddle-node

In this talk we consider generic families of 2-dimensional analytic vector fields unfolding a generic (codimension 1) saddle-node at the origin. We show that a complete modulus of orbital analytic classification for the family is given by an unfolding of the Martinet-Ramis modulus of the saddle-node. The Martinet-Ramis 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" (x²-e)[(¶)/(¶x)] +y (1+a(e)x) [(¶)/(¶y)]. The nontriviality of the Martinet-Ramis 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  92697-3875, USA
Qualitative properties of the anisotropic Manev problem

The anisotropic Manev problem is a two-body 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 blow-up of singularities, some of the collision solutions are transformed into orbits homoclinic to a non-hyperbolic periodic orbit. A suitable generalization of the Poincaré-Melnikov 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 blow-up 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 cotangent-lifted 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 cotangent-lifted actions. This result has two advantages over the earlier more general one: the proof is constructive, and it involves a cotangent-bundle-specific 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 Lyapunov-Schmidt type for determining the bifurcations of q-periodic 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 q-th root of unity q ³ 3), two branches of symmetric q-subharmonics 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 3-body 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 Maria-Cristina Ciocci (Gent), Francesco-Javier Muñoz-Almaraz, 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) 190-217.

[2]

J. Henrard, The web of periodic orbits at L4. Celestial Mech. Dynam. Astronom. 83(2000) 291-302.