Stickiness in Chaos

We distinguish two types of stickiness in systems of two degrees of freedom (a) stickiness around an island of stability and (b) stickiness in chaos, along the unstable asymptotic curves of unstable periodic orbits. In fact there are asymptotic curves of unstable orbits near the outer boundary of an island that remain close to the island for some time, and then extend to large distances into the surrounding chaotic sea. But later the asymptotic curves return close to the island and contribute to the overall stickiness that produces dark regions around the islands and dark lines extending far from the islands. We studied these effects in the standard map with a rather large nonlinearity K=5, and we emphasized the role of the asymptotic curves U, S from the central orbit O (x=0.5, y=0), that surround two large islands O 1 and O' 1 , and the asymptotic curves U + US + S-from the simplest unstable orbit around the island O 1. This is the orbit 4/9 that has 9 points around the island O 1 and 9 more points around the symmetric island O' 1. The asymptotic curves produce stickiness in the positive time direction (U,U + ,U-) and in the negative time direction (S,S + ,S-). The asymptotic curves U + ,S + are closer to the island O 1 and make many oscillations before reaching the chaotic sea. The curves US S-are further away from the island O 1 and escape faster. Nevertheless all curves return many times close to O 1 and contribute to the stickiness near this island. The overall stickiness effects of U + ,U-are very similar and the stickiness effects along S + ,S-are also very similar. However the stickiness in the forward time direction, along U + ,U-, is very different from the stickiness in the opposite time direction along S + ,S-. We calculated the finite time LCN (Lyapunov characteristic number) χ(t), which is initially smaller for U + ,S + than for US S-. However after a long time all the values of χ(t) in the chaotic zone approach the same final value LCN=). (lim t t χ ∞ → The stretching number (LCN for one iteration only) varies along an asymptotic curve going through minima at the turning points of the asymptotic curve. We calculated the escape times (initial stickiness times) for many initial points outside but close to …