×

zbMATH — the first resource for mathematics

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.

MSC:
49Q10 Optimization of shapes other than minimal surfaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Aubry and P. Y. Le Daeron, Physica (Utrecht) D,8, 381 (1983).
[2] V. Bangert, ?Mather sets for twist maps and geodesics on tori,? Preprint, Bern University (1986). · Zbl 0664.53021
[3] L. A. Bunimovich, Ya. G. Pesin, Ya. G. Sinai, and M. V. Yakobson, ?Ergodic theory for smooth dynamical systems,? in: Modern Problems of Mathematics, Fundamental Directions, Vol. 2 [in Russian], VINITI Moscow (1985). · Zbl 0781.58018
[4] M. L. Byalyi and L. V. Polterovich, Funktsional Analiz i Ego Prilozhen.,20, 9 (1986).
[5] J. N. Mather, Comment. Math. Helv.,60, 508 (1985). · Zbl 0597.58015
[6] M. L. Blank, in: Nonlinearity, Vol. 2 (1989), pp. 1-22.
[7] I. R. Sagdeev and V. M. Vinokur, J. Phys.,48, 1395 (1987).
[8] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM Studies in Applied Maths., Vol. 4), Philadelphia (1981). · Zbl 0472.35002
[9] J. Moser, Ann. Inst. H. Poincaré,3, 229 (1986).
[10] J. Moser, Erg. Th. Dyn. Sys.,6, 401 (1986).
[11] V. Bangert, ?Minimal geodesics,? Preprint, Bern University (1987). · Zbl 0645.58017
[12] G. Hedlund, Ann. Math., Ser. 2,33, 719 (1932). · Zbl 0006.32601
[13] V. Bangert, Ann. H. Inst. H. Poincaré,6, 95 (1986).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.