Algebraic properties of isochronous centers of polynomial quadratic-like Hamiltonian systems

Projecte de recerca elaborat a partir d’una estada a la School of Mathematics and Statistics de la University of Plymouth, United Kingdom, entre abril juliol del 2007.Aquesta investigació és encara oberta i la memòria que presento constitueix un informe de la recerca que estem duent a terme actualment. En aquesta nota estudiem els centres isòcrons dels sistemes Hamiltonians analítics, parant especial atenció en el cas polinomial. Ens centrem en els anomenats quadratic-like Hamiltonian systems. Diverses propietats dels centres isòcrons d'aquest tipus de sistemes van ser donades a [A. Cima, F. Mañosas and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems, J. Di®erential Equations 157 (1999) 373{413]. Aquell article estava centrat principalment en el cas en que A; B i C fossin funcions analítiques. El nostre objectiu amb l'estudi que estem duent a terme és investigar el cas en el que aquestes funcions són polinomis. En aquesta nota formulem una conjectura concreta sobre les propietats algebraiques que venen forçades per la isocronia del centre i provem alguns resultats parcials.

Report for the scientific sojourn carried out at the School of Mathematics and Statistics of the University of Plymouth from April to July the 2007. The research is still open and the report constitutes a review of this work in progress. In this note we study isochronous centers of analytic Hamiltonian systems giving a special attention to the polynomial case. We focus on the so-called quadratic-like Hamiltonian systems. Several properties of the isochronous centers of this type of systems were given in [A. Cima, F. Mañosas and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems, J. Di®erential Equations 157 (1999) 373{413]. That paper focus mainly on the case that A; B and C are analytic functions. Our aim with the present study is to investigate the case in which these functions are polynomial. In this note we pose a concrete conjecture about some strong algebraic properties that are forced by the isochronicity of the center and some partial results are proved.