Engineering separatrix volume as a control technique for dynamical transitions

Abstract

Dynamical transitions, such as a change from bound to unbound motion, often occur as post-adiabatic crossings of a time-dependent separatrix. Whether or not any given orbit will include such a crossing transition typically depends sensitively on initial conditions, but a simple estimate for the fraction of orbits which will cross the separatrix, based on Liouville's theorem, has appeared several times in the literature. Post-adiabatic dynamical transitions have more recently been reconsidered as a control problem rather than an initial value problem: what forms of time-dependent Hamiltonian can most efficiently induce desired transitions, or prevent unwanted ones? We therefore apply the Liouvillian estimate for the transition fraction to show how engineering separatrix volumes in phase space can be a control technique for dynamical transitions.