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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 3(357), Pages 81–114 (Mi umn737)  

This article is cited in 11 scientific papers (total in 11 papers)

Some physical models of the reaction-diffusion equation, and coupled map lattices

Ya. B. Pesina, A. A. Yurchenkob

a Pennsylvania State University
b Georgia Institute of Technology

Abstract: A number of models are surveyed which appear in physics, biology, chemistry, and other areas and which are described by a reaction-diffusion equation. The corresponding coupled map lattice (CML) system is obtained by discretizing this equation. These CMLs are classified by the type of the dynamics of the local map. Several different types of behavior are observed: Morse–Smale type systems, systems with attractors, and systems with Smale horseshoes.


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English version:
Russian Mathematical Surveys, 2004, 59:3, 481–513

Bibliographic databases:

UDC: 517.91/.93
MSC: Primary 35K57; Secondary 37D15, 37D45, 37N25, 37N10
Received: 24.06.2003

Citation: Ya. B. Pesin, A. A. Yurchenko, “Some physical models of the reaction-diffusion equation, and coupled map lattices”, Uspekhi Mat. Nauk, 59:3(357) (2004), 81–114; Russian Math. Surveys, 59:3 (2004), 481–513

Citation in format AMSBIB
\by Ya.~B.~Pesin, A.~A.~Yurchenko
\paper Some physical models of the reaction-diffusion equation, and coupled map lattices
\jour Uspekhi Mat. Nauk
\yr 2004
\vol 59
\issue 3(357)
\pages 81--114
\jour Russian Math. Surveys
\yr 2004
\vol 59
\issue 3
\pages 481--513

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    This publication is cited in the following articles:
    1. Nizhnik I., “Stable stationary solutions for a reaction-diffusion equation with a multi-stable nonlinearity”, Phys. Lett. A, 357:4-5 (2006), 319–322  crossref  zmath  adsnasa  isi  elib
    2. You Yuncheng, “Global dynamics of the Brusselator equations”, Dyn. Partial Differ. Equ., 4:2 (2007), 167–196  crossref  mathscinet  zmath  isi
    3. Richter H., “Coupled map lattices as spatio-temporal fitness functions: Landscape measures and evolutionary optimization”, Phys. D, 237:2 (2008), 167–186  crossref  mathscinet  zmath  adsnasa  isi
    4. Sotelo Herrera M.D., San Martin J., “Travelling waves associated with saddle-node bifurcation in weakly coupled CML”, Physics Letters A, 374:33 (2010), 3292–3296  crossref  zmath  adsnasa  isi
    5. Lijun Bo, Yongjin Wang, “On a stochastic interacting model with stepping-stone noises”, Statistics & Probability Letters, 2011  crossref  mathscinet  isi
    6. Fontich E., de la Llave R., Martin P., “Dynamical systems on lattices with decaying interaction I: A functional analysis framework”, J Differential Equations, 250:6 (2011), 2838–2886  crossref  mathscinet  zmath  adsnasa  isi
    7. Meurer T., “On the Extended Luenberger-Type Observer for Semilinear Distributed-Parameter Systems”, IEEE Trans. Autom. Control, 58:7 (2013), 1732–1743  crossref  mathscinet  zmath  isi
    8. Blazevski D., de la Llave R., “Localized Stable Manifolds for Whiskered Tori in Coupled Map Lattices with Decaying Interaction”, Ann. Henri Poincare, 15:1 (2014), 29–60  crossref  mathscinet  zmath  isi
    9. O. V. Pochinka, A. S. Loginova, E. V. Nozdrinova, “One-Dimensional Reaction-Diffusion Equations and Simple Source-Sink Arcs on a Circle”, Nelineinaya dinam., 14:3 (2018), 325–330  mathnet  crossref
    10. Manakova N.A., Gavrilova O.V., “About Nonuniqueness of Solutions of the Showalter-Sidorov Problem For One Mathematical Model of Nerve Impulse Spread in Membrane”, Bull. South Ural State U. Ser.-Math Model Program Comput., 11:4 (2018), 161–168  crossref  isi
    11. V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, O. V. Pochinka, “Klassifikatsiya sistem Morsa–Smeila i topologicheskaya struktura nesuschikh mnogoobrazii”, UMN, 74:1(445) (2019), 41–116  mathnet  crossref  elib
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