Box Dimension and Cyclicity of Canard Cycles

• Huzak, Renato ^[1]
  1. [1] University of Hasselt

     University of Hasselt

     Arrondissement Hasselt, Bélgica

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 17, Nº 2, 2018, págs. 475-493
• Idioma: inglés
• DOI: 10.1007/s12346-017-0248-x
• Enlaces
  • Texto completo
• Resumen

  • 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.

• Referencias bibliográficas
  • 1. Benoît, É.: Chasse au canard. II. Tunnels–entonnoirs–peignes. Collect. Math. 32(2), 77–97 (1981)
  • 2. Benoit, É.: Équations différentielles: relation entrée-sortie. C. R. Acad. Sci. Paris Sér. I Math. 293(5), 293–296 (1981)
  • 3. De Maesschalck, P., Dumortier, F.: Time analysis and entry–exit relation near planar turning points. J. Differ. Equ. 215(2), 225–267 (2005)
  • 4. De Maesschalck, P., Dumortier, F.: Canard solutions at non-generic turning points. Trans. Am. Math. Soc. 358(5), 2291–2334 (2006). (electronic)
  • 5. De Maesschalck, P., Dumortier, F.: Canard cycles in the presence of slow dynamics with singularities. Proc. R. Soc. Edinburgh Sect. A 138(2),...
  • 6. De Maesschalck, P., Dumortier, F.: Singular perturbations and vanishing passage through a turning point. J. Differ. Equ. 248(9), 2294–2328...
  • 7. De Maesschalck, P., Dumortier, F.: Detectable canard cycles with singular slow dynamics of any order at the turning point. Discrete Contin....
  • 8. De Maesschalck, P., Dumortier, F.: Slow–fast Bogdanov–Takens bifurcations. J. Differ. Equ. 250(2), 1000–1025 (2011)
  • 9. De Maesschalck, P., Dumortier, F., Roussarie, R.: Cyclicity of common slow–fast cycles. Indag. Math. (N.S.) 22(3–4), 165–206 (2011)
  • 10. Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121(577), x+100 (1996). (With an appendix by...
  • 11. Dumortier, F., Roussarie, R.: Multiple canard cycles in generalized Liénard equations. J. Differ. Equ. 174(1), 1–29 (2001)
  • 12. Dumortier, F., Roussarie, R.: Canard cycles with two breaking parameters. Discrete Contin. Dyn. Syst. 17(4), 787–806 (2007)
  • 13. Dumortier, F., Roussarie, R.: Birth of canard cycles. Discrete Contin. Dyn. Syst. Ser. S 2(4), 723–781 (2009)
  • 14. Dumortier, F.: Slow divergence integral and balanced canard solutions. Qual. Theory Dyn. Syst. 10(1), 65–85 (2011)
  • 15. Elezovi´c, N., Županovi´c, V., Žubrini´c, D.: Box dimension of trajectories of some discrete dynamical systems. Chaos Solitons Fractals...
  • 16. Falconer, K.: Fractal Geometry. Mathematical foundations and applications. Wiley, Chichester (1990)
  • 17. Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)
  • 18. Huzak, R., De Maesschalck, P., Dumortier, F.: Limit cycles in slow–fast codimension 3 saddle and elliptic bifurcations. J. Differ. Equ....
  • 19. Huzak, R., De Maesschalck, P., Dumortier, F.: Primary birth of canard cycles in slow–fast codimension 3 elliptic bifurcations. Commun....
  • 20. Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd edn. Elsevier/Academic...
  • 21. Huzak, R.: Cyclicity of the origin in slow–fast codimension 3 saddle and elliptic bifurcations. Discrete Contin. Dyn. Syst. 36(1), 171–215...
  • 22. Huzak, R.: Regular and slow–fast codimension 4 saddle-node bifurcations. J. Differ. Equ. 262(2), 1119–1154 (2017)
  • 23. Krantz, S.G., Parks, H.R.: The Geometry of Domains in Space. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts:...
  • 24. Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001)
  • 25. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics,...
  • 26. Mishchenko, E.F., Kolesov, Y.S., Kolesov, A.Y., Rozov, N.K.: Asymptotic Methods in Singularly Perturbed Systems. Monographs in Contemporary...
  • 27. Mardeši´c, P., Resman, M., Županovi´c, V.: Multiplicity of fixed points and growth of ε-neighborhoods of orbits. J. Differ. Equ. 253(8),...
  • 28. Tricot, C.: Curves and Fractal Dimension. Springer, New York (1995). (With a foreword by Michel Mendès France, Translated from the 1993...
  • 29. Žubrini´c, D., Županovi´c, V.: Poincaré map in fractal analysis of spiral trajectories of planar vector fields. Bull. Belg. Math. Soc....