Continuity of depinning force

Abstract

Monotone recurrence relations give rise to a class of dynamical systems on the high-dimensional cylinder which generalizes the class of monotone twist maps. A solution of a monotone recurrence relation corresponds to an equilibrium of the generalized Frenkel–Kontorova (FK) model. The Aubry–Mather theory for monotone recurrence relations says that for each ω∈R there is a Birkhoff minimizer with rotation number ω. Whether all Birkhoff minimizers of rotation number ω form a foliation is a question like whether there is an invariant circle with rotation number ω for a monotone twist map. In this paper we give a criterion for the existence of minimal foliations and study its continuity.The depinning force Fd(ω), depending on rotation numbers, is a critical value of the driving force for the FK model, under which there are Birkhoff equilibria and hence the system is pinned, and above which there are no Birkhoff equilibria of rotation number ω and the system is sliding. We show that all Birkhoff minimizers with irrational rotation number ω form a foliation if and only if Fd(ω)=0. If ω=p/q is rational, then Fd(p/q)=0 if and only if all (p,q)-periodic Birkhoff minimizers constitute a foliation. Moreover, we show that Fd(ω) is continuous at irrational ω and Hölder continuous at Diophantine points, and it also depends continuously on system parameters.