  The theory of stochastic resetting asserts that restarting a stochastic
process can expedite its completion. In this paper, we study the escape process
of a Brownian particle in an open Hamiltonian system that suffers
noise-enhanced stability. This phenomenon implies that under specific noise
amplitudes the escape process is delayed. Here, we propose a new protocol for
stochastic resetting that can avoid the noise-enhanced stability effect. In our
approach, instead of resetting the trajectories at certain time intervals, a
trajectory is reset when a predefined energy threshold is reached. The
trajectories that delay the escape process are the ones that lower their energy
due to the stochastic fluctuations. Our resetting approach leverages this fact
and avoids long transients by resetting trajectories before they reach low
energy levels. Finally, we show that the chaotic dynamics (i.e., the sensitive
dependence on initial conditions) catalyzes the effectiveness of the resetting
strategy.
