# Winter School of Dynamical Systems

- Dates:
- 11.12.97 - 12.12.97
- Place:
- Lleida, Spain
- Organizers:
- Javier Chavarriga and Jaume Llibre

The contents of these courses have been published in the first number of Qualitative Theory of Dynamical Systems

#### Speakers

Javier Chavarriga, Colin Christopher, Freddy Dumortier, Chris Herssens, Jaume Llibre, and Marco Sabatini#### Lessons

- A Survey of Isochronous Centers.

Javier Chavarriga (Universitat de Lleida) and Marco Sabatini (Università degli Estudi di Trento) - Algebraic Aspects of Integrability for Polynomial Systems.

Colin Christopher (University of Plymooth) and Jaume Llibre (UAB) - Tracing Phase Portraits of Planar Polynomial Vector Fields with Detailed Analysis of the Singularites.

Freddy Dumortier (Limburgs Universitair Centrum) and Chris Herssens (Limburgs Universitair Centrum)

#### Contents

*A Survey of Isochronous Centers*

- Introduction.
- The center problem and the isochronicity problem.
- Linearizations.
- Computation of isochronous constants.
- Commuting systems.
- Complex systems.
- Hamiltonian systems.
- Uniformly isochronous centers.
- Non-hamiltonian second order O.D.E's.
- Quadratic systems.
- Cubic systems with homogeneous nonlinearities.
- Cubic reversible systems.
- Polynomial systems with homogeneous nonlinearities.
- Isochronous centers of a cubic system with degenerate infinity.
- Kukles' system.

In this survey we give an overview of the results obtained in the study of isochronous centers of vector fields in the plane. This paper consists of two parts. In the first one (Sections 2-8), we review some general techniques that proved to be useful in the study of isochronicity. In the second one (Sections 9-16), we try to give a picture of the state of the art at the moment this review was written.

In Section 2, we give some basic definitions about centers, isochronous centers, first integrals, integrating factors, particular algebraic solutions, and other related concepts. In this sections we also give some general theorems about centers and isochronous centers, and we give a brief account of the evolution of the researches in this field.

In the successive sections we focus on various methods that have been used in attacking the isochonicity problem. We start with linearizations in Section 3, stating Poincaré's classical theorem and some of its consequences. Section 4 is devoted to describe the procedure that leads to define and compute isochronous constants. In Section 5, commutators are introduced, and basic facts about couples of commuting systems are described. Classical theorems about systems obtained from complex ordinary differential equations are collected in Section 6. Hamiltonian systems are considered in Section 7, where their connection to the study of the Jacobian Conjecture is showed, too. Section 8 is concerned with systems having constant angular speed with respect to some coordinate system.

The second part starts with Section 9, that is devoted to recent results about second order differential equations not immediately reducible to hamiltonian systems. This section also contains the characterization of isochronous centers of reversible Liénard systems. In Section 10 we list all fundamental results about isochronous centers of quadratic systems. Next section contains results about cubic systems with homogeneous nonlinearities. Sections 12 is devoted to cubic reversible systems. In Section 13 we collect results about quartic and quintic systems with homogeneous nonlinearities. A class of particular cubic systems, with degenerate infinity is considered in Section 14. Finally, Section 15 is devoted to Kukles system.

All the sections of the second part, and some of the first part, contain tables, where the main features of the considered systems are collected. When possible, for every class of systems we have written the system in rectangular and polar coordinates, and we have reported a first integral, a commutator, a linearization and a reciprocal integrating factor.

The bibliography contains references both to papers devoted to the study of isochronicity and to papers concerned with integrability of plane systems and the study of the period function of centers. We have tried to make the bibliography so complete as possible for what is concerned with isochronicity. We have made no effort to make it complete for papers about integrability and the study of the period function. We address the reader interested in integrability problems to the forthcoming review paper by Conti.

Algebraic Aspects of Integrability for Polynomial Systems

- Introduction.
- Algebraic curves and Darboux integrability.
- The Darboux method and centres.
- Furthers results.
- Non-existence of limit cycles.
- The inverse problem.
- More non-existence results.
- Elementary and Liouvillian first integrals.
- The centre problem.

The main part of these notes is devoted to explaining the fascinating connection between the local integrability of polynomial differential equations (a topological phenomena) and the existence of exact solutions for these equations (an algebraic one). However, having built up the appropriate methods, it seemed a good idea to digress into a few closely related areas.

There are now several expositions of the Darboux method of integration [9, 20, 1, 6], so our aim here is not at completeness, but at obtaining a general idea of the methods involved. Once the geometric ideas are grasped (and these are not difficult) the more technical theorems are mostly routine. Some of these have been left as exercises in the text.

Unfortunately this is not a subject that lends itself nicely to hand calculations except in the simplest cases. However, we have tried to include a few of the simpler computations to give a of the algebra involved. We have also mentioned a few of the more interesting research projects which suggested themselves as we went.

Tracing Phase Portraits of Planar Polynomial Vector Fields with Detailed Analysis of the Singularites.

- Introduction.
- Study near the singular points, the elementary case.
- Blowing-up non-elementary singularities.
- Poincaré and Poincaré-Lyapunov compactification.
- The program
**P4**. - Treatment of examples.

Our aim is to study ordinary differential equations in two real variables

x'=P(x,y) |
(1) |

y'=Q(x,y) |

with *P* and *Q* both polynomial.

We will also call this a (polynomial) vector field on *R ^{2}*, emphasizing that the object under study can be defined in a coordinate-free way. Another way to express the vector field is by writing it as

*X=P(x,y)d/dx+Q(x,y)d/dy*(2)

Both expressions (1) and (2) represent the vector field in the standard coordinates on *R ^{2}*, but during the analysis we will often use other coordinates, as well linear as non-linear ones, even not always globally defined. In fact our goal is surely not to look for an analytic expression of the global solution of (1). Not only would it be an impossible task for most equations but moreover even in the cases where a precise expression can be found it is not always clear what it really represents. Numerical analysis of (1) together with graphical representation, will be an essential ingredient in the analysis. We will however not limit our study to mere numerical integration. In fact in trying to do this one often encounters serious problems; calculations can take an enormous amount of time or even lead to erroneous results. Based however on a priori knowledge of some essential features of (1) these problems can often be avoided. Qualitative techniques are very appropriate to get such an overall understanding of the equation (1). A clear picture is achieved by drawing a phase portrait in which the relevant qualitative features are represented. Of course, for practical reasons, the representation may not be too far from reality and has to respect some numerical accuracy. These are, in a nutshell, the main ingredients in our approach. In Section 5 we present a computer program based on them. The program is an extension of previous work due to J. C. Artés and J. Llibre. We have called it "Polynomial Planar Phase Portraits", which we abbreviate as

**P4**.

We first start by studying the vector field near the singular points. Section 2 deals with the elementary singularities and Section 3 with the non-elementary ones. In Section 4 we introduce Poincaré and Poincaré- Lyapunov compactification in order to be able to study the vector fields near infinity. In Section 5 we present the program **P4**, while in Section 6 we treat some examples.