  The driven double-well Duffing oscillator is a well-studied system that
manifests a wide variety of dynamics, from periodic behavior to chaos, and
describing a diverse array of physical systems. It has been shown to be
relevant in understanding chaos in the classical to quantum transition. Here we
explore the complexity of its dynamics in the classical and semi-classical
regimes, using the technique of ordinal pattern analysis. This is of particular
relevance to potential experiments in the semi-classical regime. We unveil
different dynamical regimes within the chaotic range, which cannot be detected
with more traditional statistical tools. These regimes are characterized by
different hierarchies and probabilities of the ordinal patterns. Correlation
between the Lyapunov exponent and the permutation entropy is revealed that
leads to interpret dips in the Lyapunov exponent as transitions in the dynamics
of the system.
