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

This article is cited in 19 scientific papers (total in 19 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.


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

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
\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

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    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
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