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Limit Cycles for a Class of Continuous and Discontinuous Cubic Polynomial Differential Systems

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Abstract

We study the maximum number of limit cycles that bifurcate from the periodic solutions of the family of isochronous cubic polynomial centers

$$\begin{aligned} \dot{x}=y(-1+2\alpha x+2\beta x^2),\quad \dot{y}=x+\alpha (y^2-x^2)+2\beta xy^2, \quad \alpha \in \mathbb {R},\,\beta <0, \end{aligned}$$

when it is perturbed inside the classes of all continuous and discontinuous cubic polynomial differential systems with two zones of discontinuity separated by a straight line. We obtain that this number is 3 for the perturbed continuous systems and at least 12 for the discontinuous ones using the averaging method of first order.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193, Barcelona, Catalonia, Spain

    Jaume Llibre & Bruno D. Lopes

  2. Departamento de Matemática, IBILCE-UNESP, Rua C. Colombo, 2265, S. J. Rio Preto, São Paulo, CEP 15054-000, Brazil

    Jaime R. De Moraes

Authors
  1. Jaume Llibre
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  2. Bruno D. Lopes
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  3. Jaime R. De Moraes
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Correspondence to Jaume Llibre.

Additional information

The first author is partially supported by a MINECO/FEDER grant number MTM2009-03437, by an AGAUR grant number 2009SGR-410, by an ICREA Academia, two FP7+PEOPLE+2012+IRSES numbers 316338 and 318999, and FEDER-UNAB10-4E-378. The first and second author are supported by CAPES-MECD grant PHB-2009-0025-PC. The third author is supported by FAPESP-2010/17956-1.

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Llibre, J., Lopes, B.D. & De Moraes, J.R. Limit Cycles for a Class of Continuous and Discontinuous Cubic Polynomial Differential Systems. Qual. Theory Dyn. Syst. 13, 129–148 (2014). https://doi.org/10.1007/s12346-014-0109-9

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Mathematics Subject Classification (1991)

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