Stickiness and cantori

Abstract

We study the phenomenon of stickiness in the standard map. The sticky regions are limited by cantori. Most important among them are the cantori with noble rotation numbers, that are approached by periodic orbits corresponding to the successive truncations of the noble numbers. The size of an island of stability depends on the last KAM torus. As the perturbation increases, the size of the KAM curves increases. But the outer KAM curves are gradually destroyed and in general the island decreases. Higher-order noble tori inside the outermost KAM torus are also destroyed and when the outermost KAM torus becomes a cantorus, the size of an island decreases abruptly. Then we study the crossing of the cantori by asymptotic curves of periodic orbits just inside the cantorus. We give an exact numerical example of this crossing (non-schematic) and we find how the asymptotic curves, after staying for a long time near the cantorus, finally extend to large distances outwards. Finally, we find the relation between the forms of the sticky region and asymptotic curves.