Geometry of quadratic di erential systems in the neighborhood of infinity

In this article we give a complete global classification of the class QSess of planar, essentially quadratic differential systems (i.e. defined by relatively prime polynomials and whose points at infinity are not all singular), according to their topological behavior in the vicinity of infinity. This class depends on 12 parameters but due to the action of the affine group and re-scaling of time, the family actually depends on five parameters. Our classification theorem (Theorem 7.1) gives us a complete dictionary connecting very simple integer-valued invariants which encode the geometry of the systems in the vicinity of infinity, with algebraic invariants and comitants which are a powerful tool for computer algebra computations helpful in the route to obtain the full topological classification of the class QS of all quadratic differential systems.

[1]  Cai Suilin,et al.  Quadratic systems with a weak focus , 1991, Bulletin of the Australian Mathematical Society.

[2]  Jaume Llibre,et al.  Chordal quadratic systems , 1986 .

[3]  E. Velasco,et al.  Generic properties of polynomial vector fields at infinity , 1969 .

[4]  Dana Schlomiuk,et al.  Summing up the Dynamics of Quadratic Hamiltonian Systems With a Center , 1997, Canadian Journal of Mathematics.

[5]  H. Poincaré,et al.  Sur les courbes définies par les équations différentielles(III) , 1885 .

[6]  W. Fulton.  Algebraic curves , 1969 .

[7]  Joan C. Artés,et al.  Quadratic vector fields with a weak focus of third order , 1997 .

[8]  Dana Schlomiuk,et al.  Aspects of planar polynomial vector fields: global versus local, real versus complex, analytic versus algebraic and geometric , 2004 .

[9]  R. Kooij,et al.  Infinite singular points of quadratic systems in the plane , 1995 .

[10]  Liam O'Carroll,et al.  CLASSICAL INVARIANT THEORY (London Mathematical Society Student Texts 44) By PETER J. OLVER: 280 pp., £15.95 (LMS members' price £11.96), ISBN 0-521-55821-2 (Cambridge University Press, 1999). , 2001 .

[11]  T. A. Springer.  The Algebra of Invariants , 2007 .

[12]  D. Schlomiuk.  Basic algebro-geometric concepts in the study of planar polynomial vector fields , 1997 .

[13]  L. Markus.  Global structure of ordinary differential equations in the plane , 1954 .

[14]  Konstantin Sergeevich Sibirsky.  Introduction to the Algebraic Theory of Invariants of Differential Equations , 1989 .

[15]  P. Olver.  Classical Invariant Theory , 1999 .

[16]  W. Fulton,et al.  Algebraic Curves: An Introduction to Algebraic Geometry , 1969 .

[17]  I. Nikolaev,et al.  Topological Classification of Quadratic Systems at Infinity , 1997 .

[18]  Jayanta Kumar Pal,et al.  On the geometry in the neighborhood of infinity of quadratic differential systems with a weak focus , 2001 .

[19]  David Pager,et al.  The Categorization of Tag Systems in Terms of Decidability , 1970 .

[20]  Jaume Llibre,et al.  Quadratic Hamiltonian Vector Fields , 1994 .