Escape Rates of the H\'enon-Heiles System

A particle in the H\'enon-Heiles potential can escape when its energy is above the threshold value $E_{th}={1/6}$. We report a theoretical study on the the escape rates near threshold. We derived an analytic formula for the escape rate as a function of energy by exploring the property of chaos. We also simulated the escaping process by following the motions of a large number of particles. Two algorithms are employed to solve the equations of motion. One is the Runge-Kutta-Fehlberg method, and another is a recently proposed fourth order symplectic method. Our simulations show the escape of H$\mathrm{\acute{e}}$non-Heiles system follows exponential laws. We extracted the escape rates from the time dependence of particle numbers in the H$\mathrm{\acute{e}}$non-Heiles potential. The extracted escape rates agree with the analytic result.