Open system control of dynamical transitions under the generalized Kruskal-Neishtadt-Henrard theorem

TL;DR

This paper demonstrates how adding back-reaction to control parameters can enhance barrier-crossing in dynamical systems, and uses a generalized Neishtadt theorem to systematically understand and design self-controlling dynamical processes.

Contribution

It introduces a method to improve barrier-crossing probability by incorporating back-reaction, and extends the Neishtadt theorem for systematic design of self-controlling systems.

Findings

Back-reaction significantly increases crossing trajectories.

The generalized theorem predicts enhancement without solving equations.

Systematic design of self-controlling dynamical systems is enabled.

Abstract

Useful dynamical processes often begin through barrier-crossing dynamical transitions; engineering system dynamics in order to make such transitions reliably is therefore an important task for biological or artificial microscopic machinery. Here we first show by example that adding even a small amount of back-reaction to a control parameter, so that it responds to the system's evolution, can significantly increase the fraction of trajectories that cross a separatrix. We then explain how a post-adiabatic theorem due to Neishtadt can quantitatively describe this kind of enhancement without having to solve the equations of motion, allowing systematic understanding and design of a class of self-controlling dynamical systems.