Universal properties of escape in dynamical systems

This paper summarizes a numerical study of the escape properties of three two-dimensional, time-independent potentials possessing different symmetries. It was found, for all three cases, that (i) there is a rather abrupt transition in the behaviour of the late-time probability of escape, when the value of a coupling parameter, e, exceeds a critical value, e2. For e > e2, it was found that (ii) the escape probability manifests an initial convergence towards a nearly time-independent value, po(e), which exhibits a simple scaling that may be universal. However, (iii) at later times the escape probability slowly decays to zero as a power-law function of time. Finally, it was found that (iv) in a statistical sense, orbits that escape from the system at late times tend to have short time Lyapounov exponents which are lower than for orbits that escape at early times.