Planar polynomial differential system, algebraic limit cycles and plane Cremona maps [ Back ]

Date:
04.12.17   
Times:
16:00 to 17:00
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Antoni Ferragut
University:
Universitat Jaume I

Abstract:

We present some tools coming from the Algebraic Geometry, specially the Cremona transformations, that are allowing us to evolve in some specific problems in differential systems.

Concretely, we show how we can transform accurately quadratic systems into new quadratic systems after some kind of birational transformations, the quadratic plane Cremona maps. We apply afterwards these transformations to the families of quadratic differential systems having an algebraic limit cycle. As a consequence, we provide a new family of quadratic systems having an algebraic limit cycle of degree 5. Moreover we show how the known families of quadratic differential systems having an algebraic limit cycle of degree greater than four are obtained using these transformations.
Furthermore, the use of plane Cremona maps allow us to obtain new examples of families having specific and new features, such as 90 (new) cubic differential systems having algebraic limit cycles of degrees from 2 to 10 and a cubic differential system, not Liouville integrable, having an invariant algebraic curve of degree 22.
This talk is the result of joint work (in progress) with Maria Alberich-Carramiñana (UPC) and Jaume Llibre (UAB).