A review of symbolic dynamics and symbolic reconstruction of dynamical systems

Abstract

Discretizing a nonlinear time series enables us to calculate its statistics fast and rigorously. Before the turn of the century, the approach using partitions was dominant. In the last two decades, discretization via permutations has been developed to a powerful methodology, while recurrence plots have recently begun to be recognized as a method of discretization. In the meantime, horizontal visibility graphs have also been proposed to discretize time series. In this review, we summarize these methods and compare them from the viewpoint of symbolic dynamics, which is the right framework to study the symbolic representation of nonlinear time series and the inverse process: the symbolic reconstruction of dynamical systems. As we will show, symbolic dynamics is currently a very active research field with interesting applications.

Identifying low-dimensional trajectories of mechanically-ventilated patient systems: Empirical phenotypes of joint patient+care processes to enhance temporal analysis in ARDS research

“…Ongoing work toward formulating hypotheses about system trajectories related to intervention and outcome motivated cohort-scale phenotypes, providing a shared low-dimensional basis for LVS comparison. This translates LVS evolution, and questions related to protocols governing its control, into forms representable by symbolic dynamics [36,37,38] that can be used to examine patterns arising within patient cohorts. occupying two cluster-identified similar breath shapes.…”

Symbolic dynamics for a kinds of piecewise smooth maps

“…Symbolic dynamics and topological horseshoe theory play an important role to prove the existence of chaotic invariant sets, see e.g. [15,20,25,8,9,32]. The subshift of finite type is an important branch of symbolic dynamical systems, which has been widely studied.…”