Abstract:
The theory of reaction-diffusion systems in a neighborhood of a bifurcation point is considered. The basic types of space-time ordering, diffusion chaos in such systems, and sequences of bifurcations leading to complication of solutions are studied. A detailed discussion is given of a hierarchy of simplified models (one- and two-dimensional mappings, systems of ordinary differential equations, and others) which make it possible to carry out a qualitative analysis of the problem studied in the case of small regions. A number of generalizations of the equations studied and the simplest types of ordering in the two-dimensional case are described.
Citation:
T. S. Akhromeeva, S. P. Kurdyumov, G. G. Malinetskii, A. A. Samarskii, “On the classification of the solutions of a system of nonlinear diffusion equations in a neighborhood of a bifurcation point”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh., 28, VINITI, Moscow, 1986, 207–313; J. Soviet Math., 41:5 (1988), 1292–1356
\Bibitem{AkhKurMal86}
\by T.~S.~Akhromeeva, S.~P.~Kurdyumov, G.~G.~Malinetskii, A.~A.~Samarskii
\paper On the classification of the solutions of a~system of nonlinear diffusion equations in a~neighborhood of a~bifurcation point
\serial Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh.
\yr 1986
\vol 28
\pages 207--313
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/intd93}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=894260}
\zmath{https://zbmath.org/?q=an:0635.35007|0699.35017}
\transl
\jour J. Soviet Math.
\yr 1988
\vol 41
\issue 5
\pages 1292--1356
\crossref{https://doi.org/10.1007/BF01098786}
Linking options:
https://www.mathnet.ru/eng/intd93
https://www.mathnet.ru/eng/intd/v28/p207
This publication is cited in the following 11 articles:
S. D. Glyzin, “Raznostnye approksimatsii uravneniya «reaktsiya–diffuziya» na otrezke”, Model. i analiz inform. sistem, 16:3 (2009), 96–115
A. Yu. Kolesov, N. Kh. Rozov, V. A. Sadovnichii, “Mathematical aspects of the theory of development of turbulence in the sense of Landau”, Russian Math. Surveys, 63:2 (2008), 221–282
D. S. Kaschenko, I. S. Kaschenko, “Dinamika parabolicheskogo uravneniya s maloi diffuziei i otkloneniem prostranstvennoi peremennoi”, Model. i analiz inform. sistem, 15:2 (2008), 89–93
A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Buffer Phenomenon in Nonlinear Physics”, Proc. Steklov Inst. Math., 250 (2005), 102–168
A. Yu. Kolesov, N. Kh. Rozov, “Characteristic features of the dynamics of the Ginzburg–Landau equation in a plane domain”, Theoret. and Math. Phys., 125:2 (2000), 1476–1488
G. G. Malinetskiǐ, A. V. Potapov, G. Z. Tsvertsvadze, “Some qualitative features of diffusion-induced chaos”, Comput. Math. Math. Phys., 34:4 (1994), 471–478
I. D. Chueshov, “Global attractors for non-linear problems of mathematical physics”, Russian Math. Surveys, 48:3 (1993), 133–161
A. V. Razgulin, “Self-excited oscillations in the nonlinear parabolic problem with transformed argument”, Comput. Math. Math. Phys., 33:1 (1993), 61–70
G. G. Malinetskii, G. Z. Tsertsvadze, “The investigation of the Lyapunov spectrum of the Kuramoto–Tsuzuki equation”, Comput. Math. Math. Phys., 33:7 (1993), 919–927
G. Z. Tsertsvadze, “On the convergence of difference schemes for the Kuramoto–Tsuzuki equation and reaction-diffusion type systems”, U.S.S.R. Comput. Math. Math. Phys., 31:5 (1991), 40–47
S. A. Kashchenko, “Asymptotic behaviour of rapidly oscillating contrasting spatial structures”, U.S.S.R. Comput. Math. Math. Phys., 30:1 (1990), 186–197
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