The blow-up method applied to monodromic singularities

• Published in: Electronic Journal of Qualitative Theory of Differential Equations
• Main Authors: Brigita Ferčec, Jaume Giné
• Format: Article
• Language: English
• Published: University of Szeged 2024-01-01
• Subjects: monodromic singularity; blow-up; center problem; formal first integral
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=10602
• ISSN: 1417-3875

The blow-up method proves its effectiveness to characterize the integrability of the resonant saddles giving the necessary conditions to have formal integrability and the sufficiency doing the resolution of the associated recurrence differential equation using induction. In this work we apply the blow-up method to monodromic singularities in order to solve the center-focus problem. The case of nondegenerate monodromic singularities is straightforward since any real nondegenerate monodromy singularity can be embedded into a complex system with a resonant saddle. Here we apply the method to nilpotent and degenerate monodromic singularities solving the center problem when the center conditions are algebraic.