Visualization of a hyperbolic structure in area preserving maps
A. Giorgilli, V. F. Lazutkin, Carles Simó · 1997
. We present a simple method which displays a hyperbolic structure in the phase space of an area preserving map. The method is illustrated for the case of the Carleson standard map. As it follows from our experiments, the structure of the chaotic zone for the standard map is different from the one found for the systems of Anosov type. x1. Introduction. One of the most misterious and, probably, difficult problems in modern dynamics is that of the coexistence of regular and chaotic motions in Hamiltonian dynamical systems [S]. If the number of degrees of freedom exceeds 1, both types of motions appear in the phase space of most of known examples, such as the standard map, the double pendulum, the three body problem, etc. The famous KAM theory gives us plenty of information concerning the regular part of the motion whilst almost nothing is known about the structure of the chaotic part. In the case of area preserving maps the regular part is represented by invariant circles, or periodic c...