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Mosc. Math. J., 2001, Volume 1, Number 1, Pages 27–47 (Mi mmj10)  

This article is cited in 16 scientific papers (total in 16 papers)

The duck and the devil: canards on the staircase

J. Guckenheimera, Yu. S. Ilyashenkobacd

a Cornell University
b Independent University of Moscow
c M. V. Lomonosov Moscow State University
d Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Slow-fast systems on the two-torus $T^2$ provide new effects not observed for systems on the plane. Namely, there exist families without auxiliary parameters that have attracting canard cycles for arbitrary small values of the time scaling parameter $\epsilon$. In order to demonstrate the new effect, we have chosen a particularly simple family, namely $\dot x=a-\cos x-\cos y$, $\dot y=\epsilon$, $a\in(1,2)$ being fixed. There is no doubt that a similar effect may be observed in generic slow-fast systems on $T^2$. The proposed paper is the first step in the proof of this conjecture.

Key words and phrases: Slow-fast systems on the torus, canard solution, devil's staircase, Poincaré map.

DOI: https://doi.org/10.17323/1609-4514-2001-1-1-27-47

Full text: http://www.ams.org/.../abstracts-1-1.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 34A26, 34E15
Received: September 27, 2000; in revised form February 2, 2001
Language:

Citation: J. Guckenheimer, Yu. S. Ilyashenko, “The duck and the devil: canards on the staircase”, Mosc. Math. J., 1:1 (2001), 27–47

Citation in format AMSBIB
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\by J.~Guckenheimer, Yu.~S.~Ilyashenko
\paper The duck and the devil: canards on the staircase
\jour Mosc. Math.~J.
\yr 2001
\vol 1
\issue 1
\pages 27--47
\mathnet{http://mi.mathnet.ru/mmj10}
\crossref{https://doi.org/10.17323/1609-4514-2001-1-1-27-47}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1852932}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

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    1. Moehlis J., “Canards in a surface oxidation reaction”, J. Nonlinear Sci., 12:4 (2002), 319–345  crossref  mathscinet  zmath  adsnasa  isi
    2. Suckley R., Biktashev V.N., “The asymptotic structure of the Hodgkin-Huxley equations”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13:12 (2003), 3805–3825  crossref  mathscinet  zmath  isi
    3. Ilyashenko Y., “Selected topics in differential equations with real and complex time”, Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, NATO Science Series, Series II: Mathematics, Physics and Chemistry, 137, 2004, 317–354  crossref  mathscinet  isi
    4. V. M. Buchstaber, O. V. Karpov, S. I. Tertychnyi, “Rotation number quantization effect”, Theoret. and Math. Phys., 162:2 (2010), 211–221  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Schurov I.V., “Ducks on the torus: existence and uniqueness”, J Dynam Control Systems, 16:2 (2010), 267–300  crossref  mathscinet  zmath  isi  elib
    6. Schurov I., “Duck Farming on the Two-Torus: Multiple Canard Cycles in Generic Slow-Fast Systems”, Discret. Contin. Dyn. Syst., 2011, no. S, SI, 1289–1298  mathscinet  zmath  isi  elib
    7. P. I. Kaleda, “Singular systems on the plane and in space”, J. Math. Sci. (N. Y.), 179:4 (2011), 475–490  mathnet  crossref  zmath  elib
    8. V. M. Buchstaber, S. I. Tertychnyi, “Explicit solution family for the equation of the resistively shunted Josephson junction model”, Theoret. and Math. Phys., 176:2 (2013), 965–986  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. A. Klimenko, O. Romaskevich, “Asymptotic properties of Arnold tongues and Josephson effect”, Mosc. Math. J., 14:2 (2014), 367–384  mathnet  mathscinet
    10. A. A. Glutsyuk, V. A. Kleptsyn, D. A. Filimonov, I. V. Shchurov, “On the Adjacency Quantization in an Equation Modeling the Josephson Effect”, Funct. Anal. Appl., 48:4 (2014), 272–285  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    11. V. M. Buchstaber, S. I. Tertychnyi, “Holomorphic solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction”, Theoret. and Math. Phys., 182:3 (2015), 329–355  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    12. Desroches M., Krupa M., Rodrigues S., “Spike-Adding in Parabolic Bursters: the Role of Folded-Saddle Canards”, Physica D, 331 (2016), 58–70  crossref  mathscinet  zmath  isi
    13. Parshin D.V., Ufimtseva I.V., Cherevko A.A., Khe A.K., Orlov K.Yu., Krivoshapkin A.L., Chupakhin A.P., “Differential Properties of Van der Pol - Duffing Mathematical Model of Cerebrovascular Haemodynamics Based on Clinical Measurements”, All-Russian Conference on Nonlinear Waves: Theory and New Applications (Wave16), Journal of Physics Conference Series, 722, IOP Publishing Ltd, 2016, UNSP 012030  crossref  mathscinet  isi
    14. Schurov I., Solodovnikov N., “Duck Factory on the Two-Torus: Multiple Canard Cycles Without Geometric Constraints”, J. Dyn. Control Syst., 23:3 (2017), 481–498  crossref  zmath  isi  scopus
    15. Glutsyuk A., Rybnikov L., “On Families of Differential Equations on Two-Torus With All Phase-Lock Areas”, Nonlinearity, 30:1 (2017), 61–72  crossref  zmath  isi  scopus
    16. Lucas M., Newman J., Stefanovska A., “Stabilization of Dynamics of Oscillatory Systems By Nonautonomous Perturbation”, Phys. Rev. E, 97:4 (2018), 042209  crossref  isi  scopus
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