Lubna Abu Rmaileh,
"Characterizing Island Structures in Area-Preserving Maps"
A newly suggested diagnostic family is used to study the structure of the phase space of area-preserving chaotic maps. The phase space typically consists of islands bounded by segments of the stable and unstable manifolds of two hyperbolic periodic points, and a chaotic sea. Islands can have a rich variety of structures, such as fixed points, periodic orbits, chaotic regions, and smaller islands. The diagnostic family encompasses a number of asymptotic extremal values denoted by M+(?), M?(?), Mshift(?), and Mmean(?), where ? is an observable. Using those extrema fields we deduce information about the dynamics of area-preserving chaotic maps, and we are able to detect smaller islands up to a certain size which depends on the resolution used. Furthermore, we use the cumulative distribution function (CDF) and the probability distribution function (PDF) of the extremal values to distinguish different islands along with some of their properties such as their number, sizes, and positions. We also examine the convergence in time of the extrema fields and their CDFs. We apply these methods in doubly-periodic domains (T^2) as well as in open domains (R^2). We test our results on several maps, such as the Chirikov standard map, the sawtooth map, Henon map, and other cubic maps.