Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1295–1310 | Cite as

Centers and Uniform Isochronous Centers of Planar Polynomial Differential Systems

  • Jaume LlibreEmail author
  • Rafael Ramírez
  • Valentín Ramírez
  • Natalia Sadovskaia


For planar polynomial vector fields of the form
$$\begin{aligned} (-y+X(x,y))\dfrac{\partial }{\partial x}+(x+Y(x,y))\dfrac{\partial }{\partial y}, \end{aligned}$$
where X and Y start at least with terms of second order in the variables x and y, we determine necessary and sufficient conditions under which the origin is a center or a uniform isochronous centers.


Center-focus problem Polynomial planar differential system Uniform isochronous centers 

Mathematics Subject Classification




We thank to the reviewer his/her comments which help us to improve the presentation of the results of this paper. The author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR Grant Number 2014SGR-568, and the Grants FP7-PEOPLE-2012-IRSES 318999 and 316338. The second author was partly supported by the Spanish Ministry of Education through Projects TIN2014-57364-C2-1-R, TSI2007-65406-C03-01 “AEGIS” and Consolider CSD2007-00004 “ES”.


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Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBarcelonaSpain
  2. 2.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain
  3. 3.Universitat de BarcelonaBarcelonaSpain
  4. 4.Departament de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain

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