Monotone gradient dynamics and location of stationary (p,q)-configurations

Exploiting the monotone property of the gradient dynamics of the Frenkel-Kontorova model, we locate in the space of (p,q)-configurations the ordered and unordered stationary states, as well as forbidden regions for such states. Moreover we show that some generalized Frenkel--Kontorova models (associated to multiharmonic standard maps) can have ordered (p,q)--configurations that are neither action minimizing nor minimaximizing, and give their location with respect to the set of (p,q)--minimizers and minimaximizers.