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Mat. Zametki, 2002, Volume 71, Issue 6, Pages 818–831 (Mi mz387)  

This article is cited in 8 scientific papers (total in 8 papers)

The “Duck Survival” Problem in Three-Dimensional Singularly Perturbed Systems with Two Slow Variables

A. S. Bobkovaa, A. Yu. Kolesovb, N. Kh. Rozovc

a M. V. Lomonosov Moscow State University
b P. G. Demidov Yaroslavl State University
c M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We consider the system of ordinary differential equations $\dot x = f(x,y)$, $\varepsilon\dot y=g(x,y)$, where $x\in\mathbb R^2$, $y\in\mathbb R$, $0<\varepsilon \ll 1$ and $f,g\in C^\infty$. It is assumed that the equation $g = 0$ determines two different smooth surfaces $y=\varphi(x)$ and $y=\psi(x)$ intersecting generically along a curve $l$. It is further assumed that the trajectories of the corresponding degenerate system lying on the surface $y=\varphi(x)$ are ducks, i.e., as time increases, they intersect the curve $l$ generically and pass from the stable part $\{y=\varphi(x), g'_y<0\}$ of this surface to the unstable part $\{y=\varphi(x), g'_y>0\}$. We seek a solution of the so-called duck survival problem, i.e., give an answer to the following question: what trajectories from the one-parameter family of duck trajectories for $\varepsilon=0$ are the limits as $\varepsilon\to 0$ of some trajectories of the original system.

DOI: https://doi.org/10.4213/mzm387

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English version:
Mathematical Notes, 2002, 71:6, 749–760

Bibliographic databases:

UDC: 517.926
Received: 22.02.2001
Revised: 05.11.2001

Citation: A. S. Bobkova, A. Yu. Kolesov, N. Kh. Rozov, “The “Duck Survival” Problem in Three-Dimensional Singularly Perturbed Systems with Two Slow Variables”, Mat. Zametki, 71:6 (2002), 818–831; Math. Notes, 71:6 (2002), 749–760

Citation in format AMSBIB
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\by A.~S.~Bobkova, A.~Yu.~Kolesov, N.~Kh.~Rozov
\paper The ``Duck Survival'' Problem in Three-Dimensional Singularly Perturbed Systems with Two Slow Variables
\jour Mat. Zametki
\yr 2002
\vol 71
\issue 6
\pages 818--831
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\jour Math. Notes
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\vol 71
\issue 6
\pages 749--760
\crossref{https://doi.org/10.1023/A:1015812727037}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Bobkova A. S., “Duck trajectories in multidimensional singularly perturbed systems with a single fast variable”, Differ. Equ., 40:10 (2004), 1373–1382  mathnet  crossref  mathscinet  zmath  isi  scopus  scopus
    2. Xie Feng, Han Maoan, Zhang Weijiang, “Canard phenomena in oscillations of a surface oxidation reaction”, J. Nonlinear Sci., 15:6 (2005), 363–386  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    3. Bobkova A. S., “The behavior of solutions of multidimensional singularly perturbed systems with one fast variable”, Differ. Equ., 41:1 (2005), 22–32  mathnet  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    4. J. Appl. Industr. Math., 4:2 (2010), 194–199  mathnet  crossref  mathscinet
    5. Xie Feng, Han Maoan, “Existence of Canards under Non-generic Conditions”, Chin. Ann. Math. Ser. B, 30:3 (2009), 239–250  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    6. L. I. Kononenko, “Relaksatsii v singulyarno vozmuschennykh sistemakh na ploskosti”, Vestn. NGU. Ser. matem., mekh., inform., 9:4 (2009), 45–50  mathnet
    7. L. I. Kononenko, E. P. Volokitin, “Parametrizatsiya i kachestvennyi analiz singulyarnoi sistemy v matematicheskoi modeli reaktsii kataliticheskogo okisleniya”, Sib. zhurn. industr. matem., 15:1 (2012), 44–52  mathnet  mathscinet
    8. L. I. Kononenko, “Pryamaya i obratnaya zadachi dlya singulyarnoi sistemy s medlennymi i bystrymi peremennymi v khimicheskoi kinetike”, Vladikavk. matem. zhurn., 17:1 (2015), 39–46  mathnet
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