STICKINESS IN CHAOS

Abstract

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 have 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 + U - S + 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 U -, 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 U -, S -. However, after a long time all the values of χ( t ) in the chaotic zone approach the same final value LCN = lim t → ∞ χ(t). 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 the island O 1. The lines that separate the regions of the fast from the slow escape time follow the shape of the asymptotic curves S +, S -. We explained this phenomenon by noting that lines close to S + on its inner side (closer to O 1) approach a point of the orbit 4/9, say P 1, and then follow the oscillations of the asymptotic curve U +, and escape after a rather long time, while the curves outside S + after their approach to P 1 follow the shape of the asymptotic curves U - and escape fast into the chaotic sea. All these curves return near the original arcs of U +, U - and contribute to the overall stickiness close to U +, U -. The isodensity curves follow the shape of the curves U +, U - and the maxima of density are along U +, U -. For a rather long time, the stickiness effects along U +, U - are very pronounced. However, after much longer times (about 1000 iterations) the overall stickiness effects are reduced and the distribution of points in the chaotic sea outside the islands tends to be uniform. The stickiness along the asymptotic curve U of the orbit O is very similar to the stickiness along the asymptotic curves U +, U - of the orbit 4/9. This is related to the fact that the asymptotic curves of O and 4/9 are connected by heteroclinic orbits. However, the main reason for this similarity is the fact that the asymptotic curves U , U +, U - cannot intersect but follow each other.