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Journal of Dynamics and Differential Equations

, Volume 22, Issue 4, pp 657–675 | Cite as

Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems

  • Jaume LlibreEmail author
  • Clàudia Valls
Article

Abstract

We classify new classes of centers and of isochronous centers for polynomial differential systems in \({\mathbb R^2}\) of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as
$$\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),$$
where j is either 0 or \({1, \lambda \in \mathbb R}\) and \({A,B,C \in \mathbb C }\) . Note that if j = 0 and d = 7 we obtain a special case of seventh polynomial differential systems which can have a center at the origin, and if j = 1 and d = 9 we obtain a special case of ninth polynomial differential systems which can have a center at the origin.

Keywords

Centers Isochronous Polynomial vector fields 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain
  2. 2.Departamento de MatemáticaInstituto Superior TécnicoLisboaPortugal

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