The fast Lyapunov indicator: a simple tool to detect weak chaos. Application to the structure of the main asteroidal belt

A new method is presented, based on the variation with time of the length of vectors evolving in tangential space, which distinguishes very quickly between regular and chaotic motion. This method is closely related to the computation of the Lyapunov characteristic exponents, but because of the speed of computation it can be easily applied to the study of a large set of orbits. This method is tested for the 2-dimensional standard mapping, and the structure of the phase space is explored for the 4-dimensional standard map simulating the conditions of the distribution of asteroids. Then the distribution of the 716 asteroids orbiting between the 31 and 52 Kirkwood gaps is studied.