Recurrence of KAM tori and their novel critical behavior in nonanalytic twist maps

Abstract

A novel type of critical phenomena is discovered in a class of nonanalytic twist maps with a variable degree of inflection z. It is found that when z≳3 a KAM torus can reappear after it has disappeared. An ‘‘inverse residue criterion’’ is introduced to locate the precise reappearance point. We have also studied the local and global scaling behaviors of these tori. The critical exponents, the singularity spectrum and the generalized dimension all vary with z when 2≤z<3 but are independent of z when z≳3. In this sense the degree of inflection plays a role quite similar to that of dimensionality in phase transitions with z=2 and 3 corresponding respectively to the lower and upper critical dimensions. The resemblance to phase transitions in remarkable.