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Chaos and order in the multidimensional Frenkel-Kontorova model. (English. Russian original) Zbl 0723.49032
Theor. Math. Phys. 85, No. 3, 1255-1268 (1990); translation from Teor. Mat. Fiz. 85, No. 3, 349-367 (1990).
Summary: For the multidimensional discrete periodic variational problem with \(Z^ d\) being the parameter space and \(R^ q\) being the value space properties of minimal solutions (minimals) are studied. The well-known Frenkel-Kontorova model serves as a one-dimensional example of the problem. A self-conformed minimal notion, introduced earlier for the case (d\(\geq 1\), \(q=1)\), is extended to the general case \((q>1)\) and a notion of weakly self-conformed minimal is introduced. It is proved that every (weakly) self-conformed minimal is in a finite neighbourhood of a linear (polylinear) function graph. An analog of the one-dimensional Aubry- Mather theory is constructed for the self-conformed minimals. It is proved for \(q=1\) that all minimals are weakly self-conformed ones. For \(q>1\) an example, which demonstrates a bifurcation order-chaos, corresponding to perfectly disorder families of minimals, is constructed. A connection of the problem with the Kolmogorov-Arnold-Moser theory is discussed.

49Q10 Optimization of shapes other than minimal surfaces
Full Text: DOI
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