Center problem and ν-cyclicity of polynomial zero-Hopf singularities with non-singular rotation axis
We consider three-dimensional polynomial families of vector fields parameterized by the admissible coefficients having a fixed zero-Hopf equilibrium and a non-singular rotation axis through it. We are interested in the periodic ν-orbits, that is, those orbits that makes a fixed arbitrary number ν (o...
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:10459.1/71375 |
| Acceso en línea: | https://doi.org/10.1016/j.jde.2021.05.054 http://hdl.handle.net/10459.1/71375 |
| Access Level: | acceso abierto |
| Palabra clave: | Zero-Hopf singularity Periodic orbits Poincaré map V-cyclicity |
| Sumario: | We consider three-dimensional polynomial families of vector fields parameterized by the admissible coefficients having a fixed zero-Hopf equilibrium and a non-singular rotation axis through it. We are interested in the periodic ν-orbits, that is, those orbits that makes a fixed arbitrary number ν (or a divisor of ν) of revolutions about the rotation axis and then returns to the initial point closing the orbit. We develop a Bautin-type method to study the ν-cyclicity of the equilibrium, that is, the maximum number of small-amplitude ν-limit cycles (isolated periodic ν-orbits) that can be made to bifurcate from the equilibrium by moving the parameters of the family restricted to some open semi-algebraic sets. The method uses branching theory based on Newton-Puiseux Theorem to get a finite number of an analytic reduced one-dimensional Poincaré maps with associated Bautin ideal on certain Noetherian ring of rational functions on an extended parameter space. We derive global upper bounds on the number of bifurcated ν-limit cycles even in the infinite codimension case for which the perturbation of a local two-dimensional invariant manifold through the singularity completely foliated by periodic ν-orbits needs to be performed. A cubic normal form serves as an example of this procedure. |
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