Destruction of Invariant Tori as an Eigenvalue Problem

Abstract

It is shown that the criterion for smoothness of invariant tori in certain dissipative dynamical systems approaching chaos can be formulated in terms of an eigenvalue problem---a discrete Schr\"odinger equation in a potential generated by the map. By this method it is shown that there exists a continuous line through parameter space---the "critical line"---on which the invariant circles lose smoothness. It is explained why this line should approach smoothness for the very-high-order mode-locking regions and this corroborates earlier numerical results relating global scaling properties of dynamical systems to those of circle maps. Further, this method yields effective approximation schemes for obtaining the critical line.