Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory

Abstract

The universal description of orbits in the domain swept by a slowly varying separatrix is provided through a symplectic map derived by means of an extension of classical adiabatic theory. This map connects action-angle-like variables of an orbit when far from the instantaneous separatrix to time-energy variables at a reference point of the orbit very close to the corresponding separatrix. When the separatrix pulsates periodically with a small frequency epsilon , the authors combine this map with WKB theory to obtain a description of the structure underlying chaos: the homoclinic tangle related to the hyperbolic fixed point whose separatrix is pulsating. For each extremum of the area within the pulsating separatrix, an initial branch of length O(1/ epsilon ) of the stable manifold is explicitly constructed, and makes O(1/ epsilon ) transverse homoclinic intersections with a similar branch of the unstable manifold.