The symposium is the second edition of the "Symposium of
Planar Vector Fields" which was held at Lleida in November of 1996. The
main interest of the Symposium is the qualitative theory of planar vector fields
and related topics, as the bifurcation analysis, limit cycles, periodic limit
sets, singularities, desingularization, algebraic invariant curves, integrability,
center  focus problem, isochronous centers, period maps, finite cyclicity,
foliations, etc. The aim of the symposium is twofold: to survey recent progress
made in this field and to explore new directions.
C. Alonso, C. Alvarez, P. Aranda, J. C. Artés, L. Cairó,
F. Cano, A Cima, M. E. Cobo Cortez, T. Coll, A Delshams, Z. Denkovska, F. Dumortier,
T. Ferragut, J. P. Françoise, E. Freire, I A. Garcia, A. Gasull, L. Gavrilov,
H. Giacomini, M. Grau, C. Gutiérrez, X. Jarque, J. Mallol, V. Manyosa,
F. Manyosas, P. Mardesic, J. Moulin Ollagnier, M. Ndiaye, M. Nicolau, R. Ortega,
C. Pantazi, J. S. Pérez del Rio, E. Ponce, R. Prohens, R. Ramirez, M.
Reyes, J. J. Risler, G. Rodriguez, J. A. Rodriguez, R. Roussarie, C. Rousseau,
M. Sabatini, N. Sadovskaia, N. Salih, F. Sanz, D. Schlomiuk, D. S. Shafer, J.
Sorolla, J. Sotomayor, J. M. Strelcyn, E. Strozyna, I. Szantó, M. A.
Teixeira, A. Teruel, J. Torregrosa, J. Villadelprat, Xiang Zhang, H. Zoladek.
There you also can find a time table for the trains from Barcelona
to Lleida and vice versa in
Apartaments Universitaris Campus de Lleida from 18:00 to 21:00
the Saturday, December 16; or from 8:00 to 9:00 the Sunday, December 17.
If you arrive at the airport of Barcelona, you must pick up the
train from the airport to Sants Station at Barcelona. There is a train every
30 minutes, the price of the ticket is aproximately 350 pesetas. In Sants Station
you can take the train to Lleida.
Papers presented to the proceedings of the conference will follow
the standard refereeing process of the journal "Qualitative Theory of Dynamical
Systems". Papers must be written in english and in the macros of the journal
that can be found in http://www.udl.es/dept/matematica/ssd/qtds and must be
sent by email to jllibre@mat.uab.es before January 31, 2001.
Please confirm us the date and approximate time of arrival and
departure.
All the participants must pay 10000 pesetas (60 euros) for participating
into the banket of the symposium and for receiving one copy of the proceedings
of the meeting. This must be payed at the registration of the meeting in Lleida.
If you talk in the meeting and you have not sent the title or
the abstract, please send it to us before November 15.
If you did not send us your institutional adress please send it
to us.

SUNDAY 17

MONDAY 18

TUESDAY 19

WEDNESDAY 20

9:00  9:35


Freire

Gavrilov

Zholadek

9:40  10:15

Dumortier

MoulinOllagnier

Ortega

J. A. Rodríguez

10:20  10:55

Cobo

Delshams

Nicolau

Ponce

11:00  11.35

Sabatini

Risler

Sanz

Cima

BREAK

12:00  12:35

Villarini

Rousseau

Schlomiuk

Teruel

12:40  13:15

Cairó

Cano

Pérez del Río

Zhang

13.20  13:55

García

Denkowska

Ramírez

Sotomayor

LUNCH

16:00  16:35

Françoise

Roussarie

Gutiérrez


16:40  17:15

Mardesic

Shafer

Gasull


BREAK

17:40  18:15

Teixeira

Strelcyn

Szántó

18:20  18:55

Mañosa

Prohens

Jarqué




BANQUET

ABSTRACTS
Darbouxian First Integrals and
Invariants for Real Quadratic Systems
Having an Invariant Conic
Laurent Cairó
Université d'Orléans, France
Jaume Llibre
Universitat Autònoma de Barcelona, Spain
We apply the Darboux theory to study the integrability of real
quadratic differential systems having an invariant conic.The fact that two intersecting
straight lines or two parallel straight lines are particular cases of conics
allows us to study simultaneously the integrability of quadratic systems having
at least two invariant straight lines real or complex.
Topological Equivalence of Vector
Fields in Higher Dimension
Felipe Cano
Universidad de Valladolid, Spain
Not notified.
Period Function for a Class of
Hamiltonian Systems
Anna Cima
Armengol Gasull
Francesc Mañosas
Universitat Autònoma de Barcelona, Spain
In this talk we deal with the period function of the class of
Hamiltonian systems $\dot x=H_y, \dot y=H_x$ where $H(x,y)$ has the special
form $H(x,y)=F(x)+G(y)$ and the origin is a non degenerated center. More concretely,
if $T(h)$ denotes the period of the periodic orbit contained in $H(x,y)=h$ we
solve the inverse problem of characterizing all systems with a given function
$T(h)$. We also characterize the limiting behaviour of $T$ at infinity when
the origin is a global center and apply this result to prove, among other results,
that there are no polynomial isochronous centers in this family.
Interval Exchange and Piecewise
Linear Maps
Milton Cobo
Universidad de Campinas, Brazil
We will study the differenciablity of conjugations between interval exchange
maps and bijective piecewise linear maps of the interval, which we will call
affine interval exchanges. If T is a (uniquely ergodic) interval exchange and
F is an affine interval exchange which is conjugated to T, then we will show
that the differenciability of the conjugation depend only on the vector of derivatives
of the map F, and that this conjugation is not absolutely continuous in most
of the cases, implying that F has an invariant measure which is singular with
respect to Lebesgue.
An approach to the CenterFocus
Problem via Pseudo Normal Forms
Amadeu Delshams
Antoni Guillamon
J. Tomás Lázaro
Universitat Politècnica de Catalunya, Spain
The existence of a convergent transformation $z=\Phi(\zeta,\theta)$
leading a general system $\dot{z}=F(z,\theta)$, $z\in {\bf R}^2$, $\theta\in
{\bf T}$, in a neighborhood of a hyperbolic fixed point, into its Birkhoff Normal
Form, is a classical result due to J. Moser (1956). It is also known its convergence
for autonomous systems of the plane around elliptic fixed points. One of the
applications of these results is to the problem of determining whether a nonlinear
planar vector field with centertype linear part has a center or not ({\it CenterFocus
Problem}). Our approach, which is also due to D. DeLatte and J. Moser, is based
on looking for a close to the identity vector field $\Phi$ and two vector fields
$B$ and $N$ (of an special form) satisfying the relation $D\Phi \cdot N + B
= F\circ \Phi$. It is proved their convergence for any kind of vector fields
on the plane (autonomous or not) in the {\it hyperbolic} case and for autonomous
ones in the {\it elliptic} case. Note that, in particular, if $B\equiv 0$, the
vector field $\Phi$ constitutes a change of variables and, therefore, we get
a Birkhoff Normal Form. That is, our system is integrable. In the elliptic case,
this implies that the equilibrium is a center. The coefficients of $B$ can be
thought as obstructions to have centers and so play a similar role to the {\it
Lyapunov constants}, as well as the coefficients of $N$ contain the {\it constants
of period}. Finally, we want to stress that all this process is carried out
constructively and has been implemented on a computer.
About the Convergence of Kuratowski
in Differential Equations
Zofia Denkowska
Université d'Angers, France
This is a convergence of closed sets, extensively used in PDE,
close to the De Giorgi convergence, useful to obtain some results in generalized
PDE (cf. my common work with my husband, Z. Denkowski, Birkahauser). I have
just come across some applications of this convergence in analytic geometry
(convergence of intersections of analytic sets, complex case) and am trying
to make it out for the first return map. The link seems unknown to me.
Perturbations from an Elliptic
Hamiltonian of Degree Four
Freddy Dumortier
Limburgs Universitair Centrum, Belgium
The talk deals with Li\'enard equations of the form $x'= y, y'=
P(x) + y Q(x)$, with $P$ and $Q$ polynomials of degree respectively 3 and 2.
We present a number of results obtained in cooperation with Li Chengzhi. Attention
goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian
of degree four, treating the different cases: saddle loop, twosaddle cycle,
cuspidal loop, global center and figureeight loop. The study permits to prove
the existence of Li\'enard equations of type (3,2) with a quadruple limit cycle
occuring in a complete swallowtail bifurcation of limit cycles. It also permits
to prove the occurence of Li\'enard equations of type (3,2) having five limit
cycles.
Global Estimates on the Number
of Limit Cycles for Li\'enard's Equations
Jean Pierre Françoise
Université P.M. Curie, Paris VI, France
New complex analytic methods yield explicit estimates for the
number of limit cycles of a Li\'enard equation.
First Derivative of the Period
Function of a Centre
Emilio Freire
Universidad de Sevilla, Spain
Given a centre of a planar differential system, we extend the
use of the Lie bracket to the determination of the monotonicity character of
the period function. As far as we know, there are no general methods to study
this function, and the use of commutators and Lie brackets has been restricted
to prove isochronicity. A new criterion to find isochronous centres, without
looking either for commutators or for linearizations, is given.
On Limit Cycles for Quadratic
Systems with Invariant Algebraic Curves
Javier Chavarriga
Isaac A. García
Universitat de Lleida, Spain
In this paper we consider real quadratic systems. We present new
criteria for the existence and uniqueness of limit cycles for such systems by
using Darbouxian particular solutions. Some results are based on the study of
such systems in ${\C\P^2}$. We also generalize the wellknow result of Bautin
on the nonexistence of limit cycles for quadratic LotkaVolterra systems.
Upper Bounds for the Number of
Limit Cycles of Some Liénard Differential Equations
Armengol Gasull
Universitat Autònoma de Barcelona, Spain
Hector Giacomini
Université de Tours, France
It is well known that the Van der Pol differential equation has
at most one limit cycle. There is a recent proof of Cherkas of this fact by
using a clever Dulac function. In this talk we extend this idea to give the
exact number of limit cycles of other examples of Li\'enard differential equations.
This talk is based on a joint work with Hector Giacomini.
The Infinitesimal 16th Hilbert
Problem in the Quadratic Case
Lubomir Gavrilov
Université Paul Sabatier, France
Let $H(x,y)$ be a real cubic polynomial with four {\em distinct}
critical values (in a complex domain) and let $ X_H= H_y \frac{\partial}{\partial
x}  H_x \frac{\partial}{\partial y} $ be the corresponding Hamiltonian vector
field. We show that there is a neighborhood ${\cal U}$ of $X_H$ in the space
of all quadratic plane vector fields, such that any $X\in {\cal U}$ has at most
two limit cycles.
Injectivity of $\cal{C}^1$ Maps
$\R^2 \to \R^2$ at Infinity and Planar Vector Fields
Carlos Gutiérrez
Universidade de Sao Paulo, Brazil
Let $X:{\R}^2\setminus B_r \to {\R}^2$ be a $\cal{C}^1$ map, where
$r>0$ and $B_r = \{ p\in {\R}^2 : \vert\vert p \vert\vert \le r\}.$ (a) If for
some $\epsilon >0$ and for all $p\in {\R}^2\setminus B_r,$ no eigenvalue of
$DX(p)$ belongs to $(\epsilon, \infty),$ then there exists $s\ge r$, such that
$X,$ restricted to ${\R}^2\setminus B_r$ is injective; (b) If for some $\epsilon
>0$ and for all $p\in {\R}^2\setminus B_r,$ no eigenvalue of $DX(p)$ belongs
to $(\epsilon, 0)\cup \{ z\in{\C}: {\R}e(z)\ge 0 \},$ then the point $\infty$,
of the Riemann sphere ${\R}^2 \cup \{\infty\}, $is either an attractor or a
repellor of $x'= X(x)$.
Isochronicity in a Family of
Planar Polynomial Hamiltonian Systems
Xavier Jarqué
Jordi Villadelprat
Universitat Autònoma de Barcelona, Spain
The talk deals with centers of polynomial Hamiltonian systems
and we are interested in the isochronous ones. We prove that every center of
a polynomial Hamiltonian system of degree four (that is, with its homogeneous
part of degree four not identically zero) is nonisochronous. The proof uses
the geometric properties of the period annulus and it requires the study of
the Hamiltonian systems associated to a Hamiltonian of the form $H(x,y)=A(x)+B(x)y+C(x)y^2+D(x)y^3$.
Monodromy, Stability, and Bifurcation
of a Limit Cycle from
Degenerate Singular Points of
Certain Planar Vector Fields
Víctor Mañosa
Universitat Politècnica de Catalunya, Spain
We solve the monodromy and stability problems (except when the
principal term of the Dulac development of the return map is the identity) for
degenerate singular points of a generic family of vector fields. It is known,
that for the most cases, the stability of degenerate monodromic points can be
determined integrating the first order variational equations, associated to
the edges of the the first polar blowup polycycle. We are motivated by the
fact that this family contains cases, such that the contribution to the stability
of the point, "hidden" in the singular points of the first polar blowup polycycle,
compensates the contribution given by the first order variational equations
associated to the edges of the polycycle. This property is used to generate
a bifurcation of a limit cycle.
Linearizability and Complex Isochronous
Saddles
Colin Christopher
Plymouth University, England
Pavao Mardesic
Université de Bourgogne, France
Christiane Rousseau
Université de Montreal, Canada
This is a part of a joint work with C. Christopher and C. Rousseau.
We define the notion of isochronicity for integrable saddles and prove that
an integrable saddle is isochronous if and only if it is linearizable. We also
generalize this result to normalizable systems.
Some Remarks about the Integration
of Polynomial Planar Vector Fields
Jean MoulinOllagnier
Laboratoire GAGE. École Polytechnique, France
The aim of this talk is to present some tools, especially but
not only from enumerative and algebraic geometry, that are involved in the search
of first integrals of polynomial planar vector fields. As an illustration, we
describe how these tools have been useful in our recent characterization of
all cases of Liouvillian integration of the homogeneous threedimensional LotkaVolterra
system.
On the Hypersurface Solutions
to a Pfaff Equation
Marcel Nicolau
Universitat Autònoma de Barcelona, Spain
We discuss some generalizations of Jouanoulou's theorem on the
finiteness of hypersurface solutions to a Pfaff equation without first integral.
Degeneracy in Periodic Equations
Rafael Ortega
Universidad de Granada, Spain
Consider the equation $$x''+g(t,x)=s$$ or $$x'+g(t,x)=s,$$ where
$g$ is periodic in time and $s$ is a real parameter. This equation will be called
degenerate if it has no periodic solution for any $s\neq 0$. In the theory of
forced oscillations it is useful to characterize the class of degenerate equations.\par
This problem could have some similaritiy with the problem of isochronous centers
of autonomous equations.
Integrable Quadratic Vector Fields
with Generic Algebraic Solutions
Jaume Llibre
Universitat Autònoma de Barcelona
Jesús S. Pérez del Río
J. Ángel Rodríguez
Universidad de Oviedo, Spain
Darboux in 1878 was the first in showing the fascinating relationships
between the integrability (a topological phenomena) and the existence of invariant
algebraic curves. Several authors have obtained improvements to Darboux theory
of integrability. Recently, Christopher and Zholadek have proved the next result:
If X is a polynomial vector field of degree n that has algebraic solutions such
that the sum of their degrees is n+1 and they satisfy some conditions of genericity,
then X is integrable. Moreover, X can be expressed through these solutions and
their associated Hamiltonian vector fields. In this work we consider n=2 and
we study the vector fields that has an invariant curve of degree 3 such that
their irreducible factors verify the assumptions of Christopher and Zholadek.
We do their topological classification in the Poincar\'e's compactificación
and we obtain 46 equivalence classes.
On the DoubleZero Unfolding
in Symmetric Piecewise Linear Systems
Enrique Ponce
Universidad de Sevilla, Spain
We are interested in the dynamic bifurcations, giving rise to
limit cycles, that appear organized by the doublezero bifurcation point in
symmetric piecewise linear systems. One of the lines emanating from this point
corresponds with vertical Hopf points, and the information about limit cycle
amplitude and period evolution is well known. However, similar results concerning
a degenerate pitchfork bifurcation which also involves the birth of a symmetric
pair of limit cycles are not available. Some results concerning this other dynamic
bifurcation will be shown.
Bifurcation of Limit Cycles from
Hamiltonian Systems
Tomeu Coll
Universitat de les Illes Balears, Spain
Armengol Gasull
Universitat Autònoma de Barcelona, Spain
Rafael Prohens
Universitat de les Illes Balears, Spain
We study the number of limit cycles that bifurcate from the periodic
orbits of a center in two cases of planar Hamiltonian systems. We do this by
perturbing the Hamiltonian systems inside either the class of all polynomial
systems or in a family of rational functions, and studying the number of isolated
zeroes of the Abelian integral. As a consequenc, we provide lower bounds for
the Hilbert numbers in terms of the degree of the system.
On the Polynomial Vector Field
of Degree n with n1 Algebraic Limit Cycles
Rafael Ramírez
Universitat Rovira i Virgili, Spain
Natalia Sadovskaia
Universitat Politècnica de Catalunya,
Spain
We construct a polynomial vector field of degree $n$ with $n1$
invariant circumferences. The 16th Hilbert problem for algebraic limit cycles
is study.
Invariant Curves and Topological
Invariants for Real Plane Analytic Vector Fields
Jean Jacques Risler
Université Pierre et Marie Curie, Paris VI, France
Let $Z$ be a germ of a singular real analytic vector field at
$O \in {\bf R}^2$. We give conditions on the multiplicity and the Milnor number
of $Z$ which imply that the foliation defined by $Z$ has a characteristic orbit,
or an analytic invariant curve with the hypothesis that $Z$ is a ``Real Generalized
Curve''. Then it is proved that for a nondicritical ``Real Generalized Curve'',
the multiplicity mod 2 is invariant under bilipshitz homeomorphisms preserving
foliations.
Not Notified
J. Ángel Rodríguez
Universidad de Oviedo, Spain
Reduction of Several to one Parameter
in Analytic Families of Planar Vector Fields
Robert Roussarie
Université de Bourgogne, France
The computation of the number of limit cycles which appear in
an analytic unfolding of planar vector fields is related to the decomposition
of the displacement function of this unfolding in an ideal of functions in the
parameter space, called the Ideal of Bautin. On the other hand, the asymptotic
of the displacement function, for 1parameter unfoldings of hamiltonian vector
fields is given by Melnikov functions which are defined as the coefficients
of Taylor expansion in the parameter. It is interesting to compare these two
notions and to study if the general estimations of the number of limit cycles
in terms of the Bautin ideal could be reduced to the computations of Melnikov
functions for some 1parameter subfamilies.
Analytic Normal Form for SaddleNodes
and Finite Cyclicity of Graphics
Christiane Rousseau
Université de Montreal, Canada
The lecture will present a refinement of the transformation to
smooth normal form for an analytic family of vector fields in the neighborhood
of a saddlenode. This allows to prove the finite cyclicity of families of graphics
("ensembles") occuring inside analytic families of vector fields, for instance
the "lips" in generic conditions. It is essential for the proof of the finite
cyclicity of an hpgraphic through a nilpotent singular point of elliptic type.
It allows substantial improvement in the program to prove the finiteness part
of Hilbert's 16th problem for quadratic vector fields (by showing the finite
cyclicity of 121 graphics).
On the Number of Large Amplitude
Limit Cycles of Second Order Polynomial ODE's
Marco Sabatini
Universitá de Trento, Italy
In this paper we study the number of limit cycles of polynomial
Li\'enard equation $$ x" + f(x)x' + g(x) = 0 $$ that bifurcate from infinity
when a suitable perturbation is introduced. We show that for every positive
integer n there exists a polynomial Li\'enard equation of degree $2n+1$ having
$n$ limit cycles of large amplitude. Similar results are proved for other classes
of {\rm II} order ODE's.
Balanced Coordinates for Spiraling
Vector Fields
Fernando Sanz
Universidad de Valladolid, Spain
Let $X$ be an analytic vector field in $(\R^3,0)$ with a {\em
twister axis\/} $\Gamma$ in the sense of [CMS]. That is, $\Gamma$ is an invariant
analytic curve for which there is a neighbourhood composed of integral curves
$\gamma$ of $X$ that {\em spiral\/} asymptotically around the axis $\Gamma$
and have flat contact with it. Suppose that $\Gamma$ is a smooth curve and take
some coordinates $(x,y,z)$ such that $\Gamma$ is the $z$axis. The spiraling
behaviour of such an integral curve $t\mapsto \gamma(t)=(x(t),y(t),z(t))$ is
given by the fact that its angle $\theta(t)=\arctan{y(t)/x(t)}$ is divergent
while the distance $r(t)=\sqrt{x^2(t)+y^2(t)}$ from the axis goes to zero. We
say that the coordinates $(x,y,z)$ are {\em balanced\/} if the vector field
$X$ is transversal to the foliations $\{y/x=constant\}$ and $\{x^2+y^2=constant\}$.
This means that the angle and the distance from $\Gamma$ is monotone and have
a {\em uniform\/} behaviour for any integral curve $\gamma$. We show with some
examples that this uniformity depends on the coordinates. We prove that there
always exist some balanced coordinates in the case of {\em elementary singularity\/}
(non nilpotent linear part) and $\Gamma$ not contained in the singular locus
of $X$. Also, with this last condition and in the case of nilpotent singularity,
we show the existence of coordinates for which the angle is monotone. These
are obtained as coordinates giving {\em maximal contact\/} for the reduction
of the singular point. The results can be viewed as generalizations of the same
results in dimension two which we study first as a motivation.\par \vspace{5pt}
[CMS] Cano, F.; Moussu, R.; Sanz, F.: ``Oscillation, Spiralement, Tourbillonnement''.
Comm. Math. Helv., 75 (2000), 284318.
Concepts for the Classification
Problems of Planar Quadratic Systems
Dana Schlomiuk
Université de Montreal, Canada
In this lecture we first take a look at the literature on quadratic
systems in the direction of the classification problems. We next proceed by
introducing several geometric concepts helpful for encoding and organizing the
massive information encountered in classification problems. We show how these
concepts link the geometrical study of the vector fields to the algebraic invariant
theory of the systems and clear the way for specific analytic studies, theoretical
and numerical. We illustrate this by specific examples and results.
Geometry of Quadratic Cycles
Douglas S. Shafer
University of North Carolina at Charlotte, USA
We investigate the geometry and affine geometry of periodic trajectories
that occur in the phase portraits of quadratic systems of differential equations.
Umbilic and Tangential Singularities
on Lines od Curvature Configurations
in Surfaces with Boundary
Jorge Sotomayor
Universidade de Sao Paulo, Brazil
A relationship between the umbilic points that appear near the
border of a surface which is approximated by the border of a tubular neighbourhood
is studied. It is stablished that umbilics bifurcate from points of tangency
to the border.
Mathematical Works of M.N. Lagutinskii
(18711915)
Jean Marie Strelcyn
Université de Rouen and LAGA, Institut Galiée,
Université Paris 13, France
The forgotten Russian mathematician M.N. Lagutinskii (18711915)
(see [1] for his biography) was a successor of G. Darboux in what concern the
algebraic Darboux method of search of first integrals in finite terms of polynomial
systems of ODE. The aim of my talk is to present some mathematics results obtained
by M.N. Lagutinskii in this area.
REFERENCES [1] V.A. Dobrovol'skii, N.V. Lokot', J.M. Strelcyn
 Mikhail Nikolaevich Lagutinskii (18711915): Un math\'ematicie m\'ecconu,
Historia Mathematica, Vol.25(1988), 245264.
Isochronous Centers and Real
Invariant Straight Lines in Cubic Planar Systems
Javier Chavarriga
Universitat de Lleida, Spain
Eduardo Sáez
Iván Szántó
Universidad de Santa María, Chile
We prove the existence of oneparameters families of cubic isochronous
centers with three real invariant stright lines.
Reversible Unfolding of Planar
Degenerate Cusps
Ronaldo García
Universidade Federal de Goiaz, Brazil
Marco Antonio Teixeira
Universidad de Campinas, Brazil
In this talk we discuss the bifurcation diagram of the singularity
of the vector field $X(x,y)=(y , 2x^5 + 2x^3 y))$ in the class of reversible
vector fields. The three parameter unfolding of such system is also established.
Limit Cycles, Poincaré
Maps and First Integrals for Piecewise Linear Systems
Jaume Llibre
E. Núñez
Antonio E. Teruel
Universitat Autònoma de Barcelona, Spain
In the qualitative theory of differential equations, research
on limit cycles is an interesting and difficult part. In order to determine
whether there existed a limit cycle for a given differential equation and to
study the properties of limit cycles, Poincar\'e introduced the successor function,
what is actually know as Poincar\'e map. Using Poincar\'e maps we present some
relevant results about the number of limit cycle and their properties for the
family of fundamental systems. Fundamental systems are a particular case of
piecewise linear system. They are three pieces continuous linear systems, and
symmetric with respect to the origin. The same results as before can be achieved
using first integrals of fundamental systems.
Existence of Periodic Orbits
via Fuller Index and Averaging Method
Massimo Villarini
Universitá di Modena e Reggio Emilia, Italy
The problem of existence of periodic orbits for perturbations
of vector fields generating a fibration by circles on a manifold is considered:
relevant examples are coupled oscillators having the same frequencies, or the
geodesic flow on spheres.An existence theorem of Seifert and Fuller is generalized
via a connection with the averaging method of one frequency systems.Some examples
will be discussed.
Local First Integrals of Differential
Systems
Weigu Li
Peking University, China
Jaume Llibre
Universitat Autònoma de Barcelona, Spain
Xiang Zhang
Nanjing Normal University, China
In this paper using theory of linear operators and normal forms
we generalize a result of Poincar\'e about the nonexistence of local first
integrals for systems of differential equations in a neighbourhood of a singular
point. As an application of the generalized result, and under more weak conditions
we obtain a result of Furta about local first integrals of semiquasihomogeneous
systems. Moreover, for periodic differential systems we give definitions of
their first integrals, and generalize the previous results about systems of
differential equations to periodic differential systems in a neighbourhood of
a constant solution.
Planar Periodic Systems Without
Periodic Solutions
Henryk Zoladek
University of Warsaw, Poland
We construct examples of planar systems of the form $dz/dt= z^n+p_{n1}(e^{it})z^{n1}+\ldots
+ p_0(e^{it})$ (where $p_j$ are polynomials) which do not have periodic solutions.
For the Riccati system $dz/dt=z^2+re^{it}$ we express the values $r_j$ of the
parameter for which there is no periodic solution by means of zeroes of the
Bessel function $J_0$.