Counting configurations of limit cycles and centers

Armengol Gasull^ab, Antoni Guillamon^cbd, Víctor Mañosa<sup>ec

a</sup> Departament de Matemàtiques, Universitat Autònoma de Barcelona. Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del Vallès, Spain
^b Centre de Recerca Matemàtica. Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del Vallès, Spain
^c Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech), Universitat Politècnica de Catalunya, Barcelona, Spain
^d Departament de Matemàtiques, Universitat Politècnica de Catalunya. EPSEB. Av. Dr. Marañón 44--50, 08028 Barcelona, Spain
^e Departament de Matemàtiques, Universitat Politècnica de Catalunya. ESEIAAT. Colom 11, 08222 Terrassa, Spain

Abstract: We present several results on the determination of the number and distribution of limit cycles or centers for planar systems of differential equations. In most cases, the study of a recurrence is one of the key points of our approach. These results include the counting of the number of configurations of stabilities of nested limit cycles, the study of the number of different configurations of a given number of limit cycles, the proof of some quadratic lower bounds for Hilbert numbers and some questions about the number of centers for planar polynomial vector fields.

Keywords and phrases: limit cycle, configuration, center, phase portrait, recurrence, Fibonacci numbers.

MSC: 34C05, 34C07, 37C27

Received: 10.04.2023

Language: English

DOI: 10.56415/basm.y2023.i1.p78