  RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE  General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

 Personal entry: Login: Password: Save password Enter Forgotten password? Register

 Mat. Zametki, 2002, Volume 71, Issue 6, Pages 818–831 (Mi mz387)  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  Full text: PDF file (259 kB) References: PDF file   HTML file

English version:
Mathematical Notes, 2002, 71:6, 749–760 Bibliographic databases:    UDC: 517.926
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
\Bibitem{BobKolRoz02} \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 \mathnet{http://mi.mathnet.ru/mz387} \crossref{https://doi.org/10.4213/mzm387} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1933103} \zmath{https://zbmath.org/?q=an:1087.34034} \transl \jour Math. Notes \yr 2002 \vol 71 \issue 6 \pages 749--760 \crossref{https://doi.org/10.1023/A:1015812727037} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000176477200020} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0141848562} 

• http://mi.mathnet.ru/eng/mz387
• https://doi.org/10.4213/mzm387
• http://mi.mathnet.ru/eng/mz/v71/i6/p818

 SHARE:      Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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       2. Xie Feng, Han Maoan, Zhang Weijiang, “Canard phenomena in oscillations of a surface oxidation reaction”, J. Nonlinear Sci., 15:6 (2005), 363–386        3. Bobkova A. S., “The behavior of solutions of multidimensional singularly perturbed systems with one fast variable”, Differ. Equ., 41:1 (2005), 22–32        4. J. Appl. Industr. Math., 4:2 (2010), 194–199   5. Xie Feng, Han Maoan, “Existence of Canards under Non-generic Conditions”, Chin. Ann. Math. Ser. B, 30:3 (2009), 239–250       6. L. I. Kononenko, “Relaksatsii v singulyarno vozmuschennykh sistemakh na ploskosti”, Vestn. NGU. Ser. matem., mekh., inform., 9:4 (2009), 45–50 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  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 •  Number of views: This page: 241 Full text: 61 References: 43 First page: 3 Contact us: math-net2019_03 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2019