Chaos and order in the multidimensional Frenkel–Kontorova model

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.