On The Escape Of 2D Chaotic Hamiltonian Systems

In recent years, the escape problem of chaotic Hamiltoniansystems are becoming an interesting subject. The studies of chaoticand non-chaotic studies have shown an exponential decay law forchaotic motion, whereas the non-chaotic system decays accordingto a power law. Subsequently, the experimental studies of stadiumbilliards and elbow cavity using microwaves have also shown theexponential decay laws. It is seems that the decay laws are relatedto the chaotic characteristics only, not influenced by the specificsystems.The study of escape rates of the Hénon-Heiles system shows,near escaping threshold, the escape rates are all linear in energy,both analytic and numerical approaches support this view. It is notedthe Hénon-Heiles system is a 2D chaotic Hamiltonian system withsmooth openings. There has a conjecture that the escape rates of alltwo dimensional chaotic Hamiltonian systems with smooth openingsare linear in energy, which are not influenced by the specificsystems.Based on the previous conjecture, we study the escape rates oftwo dimensional chaotic Hamiltonian systems. A barrier is added tothe Hénon-Heiles system to obtain a series of chaotic Hamiltonian systems with varying parameters for the location, the width and the heights of the barrier. The numerical extracted rates for thesesystems are consistent with the analysis and can be parameterizedusing simple formulas. Near escaping threshold, the escape ratesare all linear in energy. The results provide strong evidencesconfirming an earlier conjecture that the escape rates of all twodimensional chaotic Hamiltonian systems with smooth openings arelinear in energy.