Upper bounds for the number of zeroes for some Abelian integrals
Consider the vector field x' = -yG(x, y), y' = xG(x, y), where the set of critical points {G(x, y) = 0} is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2012 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:150551 |
| Acceso en línea: | https://ddd.uab.cat/record/150551 https://dx.doi.org/urn:doi:10.1016/j.na.2012.04.033 |
| Access Level: | acceso abierto |
| Palabra clave: | Abelian integrals Weak 16th Hilbert's Problem Limit cycles Chebyshev system Number of zeroes of real functions |
| Sumario: | Consider the vector field x' = -yG(x, y), y' = xG(x, y), where the set of critical points {G(x, y) = 0} is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study which is the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K and n. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and in a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K ≤ 4 we recover or improve some results obtained in several previous works. |
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