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Quadratic Perturbations of a Quadratic Reversible Lotka–Volterra System

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Abstract

We prove that perturbing the two periodic annuli of the quadratic polynomial reversible Lotka–Volterra differential system ẋ = −y + x 2 − y 2, ẏ = x(1 + 2y), inside the class of all quadratic polynomial differential systems we can obtain the following configurations of limit cycles (0, 0), (1, 0), (2, 0), (1, 1) and (1, 2).

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References

  1. Buică A., Llibre J.: Averaging methods for finding periodic orbits via Brouwer degree. Bull. Sci. Math. 128, 7–22 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chicone C., Jacobs M.: Bifurcation of limit cycles from quadratic isochrones. J. Differ. Equ. 91, 268–326 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Iliev I.D.: Perturbations of quadratic centers. Bull. Sci. Math. 122, 107–161 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Llibre J.: Averaging theory and limit cycles for quadratic systems. Radovi Math. 11, 1–14 (2002)

    MathSciNet  Google Scholar 

  5. Loud W.S.: Behavior of the period of solutions of certain plane autonomous systems near centers. Contrib. Differ. Equ. 3, 21–36 (1964)

    MathSciNet  Google Scholar 

  6. Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. Appl. Math. Sci., vol. 59. Springer, Berlin (1985)

  7. Verhulst F.: Nonlinear Differential Equations and Dynamical Systems. Universitext. Springer, Berlin (1991)

    Google Scholar 

  8. Zhang P.: On the distribution and number of limit cycles for quadratic systems with two foci. Qual. Theory Dyn. Syst. 3, 437–463 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Żoła̧dek H.: Quadratic systems with center and their perturbations. J. Differ. Equ. 109, 223–273 (1994)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Chengzhi Li

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

    Jaume Llibre

Authors
  1. Chengzhi Li
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  2. Jaume Llibre
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Corresponding author

Correspondence to Chengzhi Li.

Additional information

The first author is partially supported by NSFC-10671005 and NFSC-10831003. The second author is partially supported by MCYT grant MTM2008-03437 and by a CIRIT grant number 2009SGR 410.

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Li, C., Llibre, J. Quadratic Perturbations of a Quadratic Reversible Lotka–Volterra System. Qual. Theory Dyn. Syst. 9, 235–249 (2010). https://doi.org/10.1007/s12346-010-0026-5

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  • DOI: https://doi.org/10.1007/s12346-010-0026-5

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