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Canard Explosion Near Non-Liénard Type Slow–Fast Hopf Point

  • Renato HuzakEmail author
Article
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Abstract

In this paper we study birth of canards near a smooth slow–fast Hopf point of non-Liénard center type which plays an important role in slow–fast codimension 3 saddle and elliptic bifurcations. We show that the number of limit cycles created in the birth of canards in such a slow–fast non-Liénard case is finite. Our paper is also a natural continuation of Dumortier and Roussarie (Discrete Contin Dyn Syst Ser S 2(4):723–781, 2009) where slow–fast Hopf points of Liénard type have been studied. We use geometric singular perturbation theory and the family blow-up.

Keywords

Family blow-up Normal forms Singular perturbation theory Slow–fast Hopf point 

Notes

Acknowledgements

I would like to thank the referee for a number of useful comments.

References

  1. 1.
    Benoît, É.: Chasse au canard. II. Tunnels–entonnoirs–peignes. Collect. Math. 32(2), 77–97 (1981)MathSciNetGoogle Scholar
  2. 2.
    De Maesschalck, P., Dumortier, F., Roussarie, R.: Cyclicity of common slow–fast cycles. Indag. Math. (N.S.) 22(3–4), 165–206 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121(577), x+100 (1996). (With an appendix by Cheng Zhi Li)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dumortier, F., Roussarie, R.: Birth of canard cycles. Discrete Contin. Dyn. Syst. Ser. S 2(4), 723–781 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dumortier, F., Roussarie, R.: Smooth normal linearization of vector fields near lines of singularities. Qual. Theory Dyn. Syst. 9(1–2), 39–87 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Huzak, R., De Maesschalck, P., Dumortier, F.: Limit cycles in slow–fast codimension 3 saddle and elliptic bifurcations. J. Differ. Equ. 255(11), 4012–4051 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Huzak, R., De Maesschalck, P., Dumortier, F.: Primary birth of canard cycles in slow–fast codimension 3 elliptic bifurcations. Commun. Pure Appl. Anal. 13(6), 2641–2673 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Huzak, R.: Limit cycles in slow–fast codimension 3 bifurcations. PhD thesis, Hasselt University, Belgium (2013)Google Scholar
  9. 9.
    Huzak, R.: Cyclicity of the origin in slow–fast codimension 3 saddle and elliptic bifurcations. Discrete Contin. Dyn. Syst. 36(1), 171–215 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huzak, R.: Normal forms of Liénard type for analytic unfoldings of nilpotent singularities. Proc. Am. Math. Soc. 145(10), 4325–4336 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Hasselt UniversityDiepenbeekBelgium

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