"Stickiness" in mappings and dynamical systems

We present results of a study of the so-called "stickiness" regions where o rbits in mappings and dynamical systems stay for very long times near an is land and then escape to the surrounding chaotic region. First we investigat ed the standard map in the form Xi + 1 = x(i) + y(i+1) and y(i+1) = y(i) K/2 pi . sin (2 pi x(i)) with a stochasticity parameter K = 5, where only t wo islands of regular motion survive. We checked now many consecutive point s-for special initial conditions of the mapping-stay within a certain regio n around the island. For an orbit on an invariant curve all the points rema in forever inside this region, but outside the "last invariant curve" this number changes significantly even for very small changes in the initial con ditions. In our study we found out that there exist two regions of "sticky" orbits around the invariant curves : A small region I confined by Cantori with small holes and an extended region II is outside these cantori which h as an interesting fractal character. Investigating also the Sitnikov-Proble m where two equally massive primary bodies move on elliptical Keplerian orb its, and a third massless body oscillates through the barycentre of the two primaries perpendicularly to the plane of the primaries-a similar behaviou r of the stickiness region was found. Although no clearly defined border be tween the two stickiness regions was found in the latter problem the fracta l character of the outer region was confirmed. (C) 1998 Elsevier Science Lt d. All rights reserved.