Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion |
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Authors: | I B Bokolishvily S A Kaschenko G G Malinetskii A B Potapov |
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Institution: | (1) Georgian University of Technology, Tbilisi, Georgia;(2) Yaroslavl State University, Yaroslav, Russia;(3) Keldysh Institute for Applied Mathematics, Miusskaya Square 4, 125047 Moscow, Russia |
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Abstract: | Summary We consider four models of partial differential equations obtained by applying a generalization of the method of normal forms
to two-component reaction-diffusion systems with small diffusionu
t=εDu
xx+(A+εA
1)u+F(u),u ∈ ℝ2. These equations (quasinormal forms) describe the behaviour of solutions of the original equation forε → 0.
One of the quasinormal forms is the well-known complex Ginzburg-Landau equation. The properties of attractors of the other
three equations are considered. Two of these equations have an interesting feature that may be called asensitivity to small parameters: they contain a new parameterϑ(ε)=−(aε
−1/2 mod 1) that influences the behaviour of solutions, but changes infinitely many times whenε → 0. This does not create problems in numerical analysis of quasinormal forms, but makes numerical study of the original
problem involvingε almost impossible. |
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Keywords: | normal forms dynamical chaos Ginzburg-Landau equation reaction-diffusion models |
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