A criticality result for polycycles in a family of quadratic reversible centers
We consider the family of dehomogenized Loud's centers Xµ_=y(x-1)∂ₓ + (x + Dx² + Fy²)_y, where µ=(D,F)єR², and we study the number of critical periodic orbits that emerge or dissapear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the we...
| Autores: | , |
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| Formato: | artículo |
| Fecha de publicación: | 2018 |
| País: | España |
| Recursos: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:199353 |
| Acesso em linha: | https://ddd.uab.cat/record/199353 https://dx.doi.org/urn:doi:10.1016/j.jde.2018.01.042 |
| Access Level: | acceso abierto |
| Palavra-chave: | Bifurcation Center Critical periodic orbit Criticality Ceriod function |
| Resumo: | We consider the family of dehomogenized Loud's centers Xµ_=y(x-1)∂ₓ + (x + Dx² + Fy²)_y, where µ=(D,F)єR², and we study the number of critical periodic orbits that emerge or dissapear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the well-known notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family {Xµ,µ є R²} distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set ΓB of codimension 1 in R². In the present paper we succeed in proving that a subset of ΓB has criticality equal to one. |
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