SLOWLY PULSATING SEPARATRICES SWEEP HOMOCLINIC TANGLES WHERE ISLANDS MUST BE SMALL - AN EXTENSION OF CLASSICAL ADIABATIC THEORY

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, we 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. These intersections define parallelograms whose O(epsilon) area provides an upper bound to that of any island possibly trapped in the tangle. The area of the homoclinic lobe enclosed by the constructed branches is almost equal to that swept by the separatrix since the preceding extremum. The paper is divided into two parts: our results are first presented on a simple model, emphasizing their physical and pictorial aspects; full mathematical statements and proofs for the general case follow.