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Mat. Zametki, 1998, Volume 63, Issue 5, Pages 697–708 (Mi mz1336)  

Diffusion instability of a uniform cycle bifurcating from a separatrix loop

A. Yu. Kolesov

P. G. Demidov Yaroslavl State University

Abstract: We consider the boundary value problem
$$ \frac{\partial u}{\partial t} =D\frac{\partial^2u}{\partial x^2}+F(u,\mu), \qquad\frac{\partial u}{\partial x}|_{x=0} =\frac{\partial u}{\partial x}|_{x=\pi}=0. $$
Here $u\in\mathbb R^2$, $D=\operatorname{diag}\{d_1,d_2\}$, $d_1,d_2>0$, and the function $F$ is jointly smooth in $(u,\mu)$ and satisfies the following condition: for $0<\mu\ll1$ the boundary value problem has a homogeneous (independent of $x$) cycle bifurcating from a loop of the separatrix of a saddle. We establish conditions for stability and instability of this cycle and give a geometric interpretation of these conditions.

DOI: https://doi.org/10.4213/mzm1336

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English version:
Mathematical Notes, 1998, 63:5, 614–623

Bibliographic databases:

UDC: 517.926
Received: 04.12.1996

Citation: A. Yu. Kolesov, “Diffusion instability of a uniform cycle bifurcating from a separatrix loop”, Mat. Zametki, 63:5 (1998), 697–708; Math. Notes, 63:5 (1998), 614–623

Citation in format AMSBIB
\Bibitem{Kol98}
\by A.~Yu.~Kolesov
\paper Diffusion instability of a uniform cycle bifurcating from a separatrix loop
\jour Mat. Zametki
\yr 1998
\vol 63
\issue 5
\pages 697--708
\mathnet{http://mi.mathnet.ru/mz1336}
\crossref{https://doi.org/10.4213/mzm1336}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1683643}
\zmath{https://zbmath.org/?q=an:0919.35064}
\transl
\jour Math. Notes
\yr 1998
\vol 63
\issue 5
\pages 614--623
\crossref{https://doi.org/10.1007/BF02312842}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000076726600008}


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