Nonanalytic twist maps and Frenkel-Kontorova models

Abstract

We give a review of our recent works on a class of twist maps and Frenkel-Kontorova (FK) models in which the external potential function is nonanalytic (C 1) and endowed with a variable degree of inflection z. When z > 3, recurrence of invariant curves has been observed. An “inverse residue criterion” for the reappearance of an invariant curve — complementary to the “residue criterion” for the disappearance of an invariant curve — is introduced to make a precise determination of the reappearance point. We have also studied the local and global scaling behaviors of these curves. The critical exponents, the singularity spectrum, and the generalized dimensions are found to vary with z when 2 ⩽ z < 3, but remain the same for z ⩾ 3. The degree of inflection thus plays a role reminiscent of that of dimensionality in phase transitions with z = 2 and 3 corresponding respectively to the lower and upper critical dimensions. When the same nonanalytic function is substituted for the sinusoidal external potential in the standard FK model, the same conclusions are reached: there exists a sequence of pinning-depinning transitions, and the critical exponents of the phonon gap, the coherence length, and the Peierls-Nabarro barrier all show the same dependence on z.