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Zbl 0964.35165
Romanenko, E.Yu.; Sharkovsky, A.N.
From boundary value problems to difference equations: A method of investigation of chaotic vibrations. (English)
[J] Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, No.7, 1285-1306 (1999). ISSN 0218-1274

Among evolutionary boundary value problems (BVP) for partial differential equations (PDE), there is a wide class of problems which can be reduced to difference, differential-difference and other relevant equations. This class consists mainly of problems marked by the familiar representation of general solutions for the corresponding PDE. Examples have been well known for a long time, but their effective study by the ``reduction method'' has become possible only in the last 20-30 years owing to the advances made in the theory of difference equations. In this paper, the authors try to indicate how the reduction method may be used and how much this method might be profitable with the simplest nonlinear BVP involving the linear equation $$\frac{\partial w}{\partial t}=c\frac{\partial w}{\partial x}+aw,\quad x\in [0,1],\quad t\in \bbfR^+,\quad c>0$$ and the nonlinear boundary condition $$w|_{x=1}=f(w)|_{x=0},$$ or $$\frac{\partial w}{\partial t}|_{x=1}=g(w)\frac{\partial w}{\partial t} |_{x=0},$$ where $f\in C^1(\bbfR,\bbfR)$, $g\in C^0(\bbfR,\bbfR) $ are given functions. In consequence of the nonlinearity of the boundary conditions, these one dimensional problems offer solutions whose dynamics is extremely complex. It is the reduction to difference equations which has given people a tool for a more penetrating insight into the properties of such chaotic solutions. This paper presents the overall picture in the study of BVP reducible to difference equations and demonstrates the potentialities that such a reduction opens up.
[Peixuan Weng (Guangzhou)]

MSC 2000:: *35R10 Difference-partial differential equations
74H65 Chaotic behavior
39A10 Difference equations

Keywords: boundary value problem; partial differential equation; difference equation; chaotic vibrations

Cited in: Zbl 1101.37318 Zbl 0964.35164

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