Skip to main content
SpringerLink
Log in

Box Dimension and Cyclicity of Canard Cycles

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

It is well known that the slow divergence integral is a useful tool for obtaining a bound on the cyclicity of canard cycles in planar slow–fast systems. In this paper a new approach is introduced to determine upper bounds on the number of relaxation oscillations Hausdorff-close to a balanced canard cycle in planar slow–fast systems, by computing the box dimension of one orbit of a discrete one-dimensional dynamical system (so-called slow relation function) assigned to the canard cycle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Benoît, É.: Chasse au canard. II. Tunnels–entonnoirs–peignes. Collect. Math. 32(2), 77–97 (1981)

    MathSciNet  Google Scholar 

  2. Benoit, É.: Équations différentielles: relation entrée-sortie. C. R. Acad. Sci. Paris Sér. I Math. 293(5), 293–296 (1981)

    MathSciNet  MATH  Google Scholar 

  3. De Maesschalck, P., Dumortier, F.: Time analysis and entry–exit relation near planar turning points. J. Differ. Equ. 215(2), 225–267 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. De Maesschalck, P., Dumortier, F.: Canard solutions at non-generic turning points. Trans. Am. Math. Soc. 358(5), 2291–2334 (2006). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  5. De Maesschalck, P., Dumortier, F.: Canard cycles in the presence of slow dynamics with singularities. Proc. R. Soc. Edinburgh Sect. A 138(2), 265–299 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. De Maesschalck, P., Dumortier, F.: Singular perturbations and vanishing passage through a turning point. J. Differ. Equ. 248(9), 2294–2328 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. De Maesschalck, P., Dumortier, F.: Detectable canard cycles with singular slow dynamics of any order at the turning point. Discrete Contin. Dyn. Syst. 29(1), 109–140 (2011)

    MathSciNet  MATH  Google Scholar 

  8. De Maesschalck, P., Dumortier, F.: Slow–fast Bogdanov–Takens bifurcations. J. Differ. Equ. 250(2), 1000–1025 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. De Maesschalck, P., Dumortier, F., Roussarie, R.: Cyclicity of common slow–fast cycles. Indag. Math. (N.S.) 22(3–4), 165–206 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121(577), x+100 (1996). (With an appendix by Li Chengzhi)

    MathSciNet  MATH  Google Scholar 

  11. Dumortier, F., Roussarie, R.: Multiple canard cycles in generalized Liénard equations. J. Differ. Equ. 174(1), 1–29 (2001)

    Article  MATH  Google Scholar 

  12. Dumortier, F., Roussarie, R.: Canard cycles with two breaking parameters. Discrete Contin. Dyn. Syst. 17(4), 787–806 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dumortier, F., Roussarie, R.: Birth of canard cycles. Discrete Contin. Dyn. Syst. Ser. S 2(4), 723–781 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dumortier, F.: Slow divergence integral and balanced canard solutions. Qual. Theory Dyn. Syst. 10(1), 65–85 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Elezović, N., Županović, V., Žubrinić, D.: Box dimension of trajectories of some discrete dynamical systems. Chaos Solitons Fractals 34(2), 244–252 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Falconer, K.: Fractal Geometry. Mathematical foundations and applications. Wiley, Chichester (1990)

    MATH  Google Scholar 

  17. Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)

    Google Scholar 

  18. 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)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd edn. Elsevier/Academic Press, Amsterdam (2013)

    MATH  Google Scholar 

  21. Huzak, R.: Cyclicity of the origin in slow–fast codimension 3 saddle and elliptic bifurcations. Discrete Contin. Dyn. Syst. 36(1), 171–215 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huzak, R.: Regular and slow–fast codimension 4 saddle-node bifurcations. J. Differ. Equ. 262(2), 1119–1154 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Krantz, S.G., Parks, H.R.: The Geometry of Domains in Space. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Boston, Inc., Boston (1999)

    Google Scholar 

  24. Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  26. Mishchenko, E.F., Kolesov, Y.S., Kolesov, A.Y., Rozov, N.K.: Asymptotic Methods in Singularly Perturbed Systems. Monographs in Contemporary Mathematics. Consultants Bureau, New York (1994). (Translated from the Russian by Irene Aleksanova)

    Book  MATH  Google Scholar 

  27. Mardešić, P., Resman, M., Županović, V.: Multiplicity of fixed points and growth of \(\varepsilon \)-neighborhoods of orbits. J. Differ. Equ. 253(8), 2493–2514 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Tricot, C.: Curves and Fractal Dimension. Springer, New York (1995). (With a foreword by Michel Mendès France, Translated from the 1993 French original)

    Book  MATH  Google Scholar 

  29. Žubrinić, D., Županović, V.: Poincaré map in fractal analysis of spiral trajectories of planar vector fields. Bull. Belg. Math. Soc. Simon Stevin 15(5, Dynamics in perturbations), 947–960 (2008)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank Maja Resman for a number of useful comments.

Author information

Authors and Affiliations

  1. Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, 3590, Diepenbeek, Belgium

    Renato Huzak

Authors
  1. Renato Huzak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Renato Huzak.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huzak, R. Box Dimension and Cyclicity of Canard Cycles. Qual. Theory Dyn. Syst. 17, 475–493 (2018). https://doi.org/10.1007/s12346-017-0248-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-017-0248-x

Search

Navigation