Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/24036
Full metadata record
DC FieldValueLanguage
dc.contributor.authorHUZAK, Renato-
dc.date.accessioned2017-08-01T13:26:20Z-
dc.date.available2017-08-01T13:26:20Z-
dc.date.issued2018-
dc.identifier.citationQualitative Theory of Dynamical Systems, 17 (2), p. 475-493-
dc.identifier.issn1575-5460-
dc.identifier.urihttp://hdl.handle.net/1942/24036-
dc.description.abstractIt 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.-
dc.language.isoen-
dc.rights© Springer International Publishing AG 2017-
dc.subject.otherbox dimension; slow-fast systems; slow relation function; slow divergence integral-
dc.titleBox Dimension and Cyclicity of Canard Cycles-
dc.typeJournal Contribution-
dc.identifier.epage493-
dc.identifier.issue2-
dc.identifier.spage475-
dc.identifier.volume17-
local.bibliographicCitation.jcatA1-
dc.description.notesHuzak, R (reprint author), Hasselt Univ, Campus Diepenbeek,Agoralaan Gebouw D, B-3590 Diepenbeek, Belgium. renato.huzak@uhasselt.be-
dc.relation.references1. 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), 265–299 (2008) 6. De Maesschalck, P., Dumortier, F.: Singular perturbations and vanishing passage through a turning point. J. Differ. Equ. 248(9), 2294–2328 (2010) 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) 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 Li Chengzhi) 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. Elezović, N., Županović, V., Žubrinić, D.: Box dimension of trajectories of some discrete dynamical systems. Chaos Solitons Fractals 34(2), 244–252 (2007) 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. 255(11), 4012–4051 (2013) 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) 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) 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) 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: Basel Textbooks]. Birkhäuser Boston, Inc., Boston (1999) 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, vol. 44. Cambridge University Press, Cambridge (1995) 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) 27. Mardešić, P., Resman, M., Županović, V.: Multiplicity of fixed points and growth of ε -neighborhoods of orbits. J. Differ. Equ. 253(8), 2493–2514 (2012) 28. Tricot, C.: Curves and Fractal Dimension. Springer, New York (1995). (With a foreword by Michel Mendès France, Translated from the 1993 French original) 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)-
local.type.refereedRefereed-
local.type.specifiedArticle-
dc.identifier.doi10.1007/s12346-017-0248-x-
dc.identifier.isi000434286700012-
item.fullcitationHUZAK, Renato (2018) Box Dimension and Cyclicity of Canard Cycles. In: Qualitative Theory of Dynamical Systems, 17 (2), p. 475-493.-
item.validationecoom 2019-
item.accessRightsOpen Access-
item.fulltextWith Fulltext-
item.contributorHUZAK, Renato-
crisitem.journal.issn1575-5460-
crisitem.journal.eissn1662-3592-
Appears in Collections:Research publications
Files in This Item:
File Description SizeFormat 
BoxDimSlowFast.pdfPeer-reviewed author version585.01 kBAdobe PDFView/Open
aaaa.pdf
  Restricted Access		Published version805.82 kBAdobe PDFView/Open    Request a copy
Show simple item record

SCOPUSTM   
Citations

1
checked on Sep 5, 2020

WEB OF SCIENCETM
Citations

7
checked on Jul 13, 2024

Page view(s)

144
checked on Sep 7, 2022

Download(s)

280
checked on Sep 7, 2022

Google ScholarTM

Check

Altmetric


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.