Geodesic
Chaos and order in the multidimensional Frenkel–Kontorova model
Teoretičeskaâ i matematičeskaâ fizika, Tome 85 (1990) no. 3, pp. 349-367 Cet article a éte moissonné depuis la source Math-Net.Ru

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A study is made of the properties of minimal solutions (minimals) of a multidimensional discrete periodic variational problem for which the space of parameters is $Z^d$ and the space of values $R^q$. A one-dimensional example of such a problem is the well-known Frenkel–Kontorova model. The concept introduced earlier for the case ($d\geqslant1$, $q=1$) of a self-consistent minimal is extended to the general case ($q>1$), and the concept of a weakly self-consistent minimal is introduced. It is shown that every self-consistent (respectively, weakly self-consistent) minimal is in a finite neighborhood of the graph of a linear (respectively, polylinear) function. For self-consistent minimals, the complete analog of the one-dimensional Aubry–Mather theory is constructed. For $q=1$ it is shown that all minimals are weakly self-consistent. For $q>1$ an example is constructed that demonstrates order–chaos bifurcation corresponding to the appearance of completely disordered families of minimals. The connection between this problem and Kolmogorov–Arnol'd–Moser (KAM) theory is discussed. 
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     author = {M. L. Blank},
     title = {Chaos and order in~the multidimensional {Frenkel{\textendash}Kontorova} model},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {349--367},
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     volume = {85},
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     url = {http://geodesic.mathdoc.fr/item/TMF_1990_85_3_a1/}
}
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M. L. Blank. Chaos and order in the multidimensional Frenkel–Kontorova model. Teoretičeskaâ i matematičeskaâ fizika, Tome 85 (1990) no. 3, pp. 349-367. http://geodesic.mathdoc.fr/item/TMF_1990_85_3_a1/

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