A Numerically Accessible Criterion for the Breakdown of Quasi-periodic Solutions and Its Rigorous Justification

We formulate and justify rigorously a numerically efficient criterion for the computation of the analyticity breakdown of quasi-periodic solutions in Symplectic maps (any dimension) and 1-D Statistical Mechanics models. Depending on the physical interpretation of the model, the analyticity breakdown may correspond to the onset of mobility of dislocations, or of spin waves (in the 1-D models) and to the onset of global transport in symplectic twist maps in 2-D. The criterion proposed here is based on the blow-up of Sobolev norms of the hull functions. We prove theorems that justify the criterion. These theorems are based on an abstract implicit function theorem, which unifies several results in the literature. The proofs also lead to fast algorithms, which have been implemented and used elsewhere. The method can be adapted to other contexts.