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Splash effect and ergodic properties of solution of the classic difference-differential equation. (English) Zbl 0965.37005

The classical model of movement of \(N\) identical material points along a circle with large radius is considered. This model is going back to Bernoulli and Lagrange and can be described by the following difference-differential equation: \[ m\ddot{W}_j=c(W_{j+1}-2W_j+W_{j-1})+G_j,\qquad j=1,\ldots,N, \] with periodic boundary condition \(W_{j+N}=W_j. \) The relationship between ergodic properties of this model and the splash effect [see A. M. Filimonov, C. R. Acad. Sci., Paris, Sér. I 315, 957-961 (1992; Zbl 0761.73053)] is described.

MSC:

37A25 Ergodicity, mixing, rates of mixing
34K10 Boundary value problems for functional-differential equations
58K55 Asymptotic behavior of solutions to equations on manifolds
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)

Citations:

Zbl 0761.73053
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References:

[1] Anderson P.W., Nobel Lecture 8 (1977)
[2] Bernoulli D., Memoires de la’ academie Royale de Berlin 9 pp 147– (1753)
[3] Filimonov A.M., Computes Rendus Acad. Sci. Paris t 313 pp 961– (1991)
[4] Filimonov A.M., The case of multiple frequencies. Computes Rendus Acad. Sci. Paris t 315 pp 957– (1992) · Zbl 0761.73053
[5] DOI: 10.1080/10236199608808075 · Zbl 0882.34068 · doi:10.1080/10236199608808075
[6] DOI: 10.1080/10236199808808125 · Zbl 0907.34050 · doi:10.1080/10236199808808125
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