Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems

TL;DR

This paper introduces a time lagged ordinal partition network method that transforms continuous dynamical system time series into networks, capturing system dynamics and transitions effectively, including chaos and crises.

Contribution

The paper presents a novel network transformation algorithm incorporating time lag, enhancing the analysis of dynamical systems and their transitions.

Findings

Network structures reflect system dynamics (rings for periodic, bands for chaotic)

Network measures track dynamical changes similar to Lyapunov exponents

Method detects interior crises in experimental data

Abstract

We investigate a generalised version of the recently proposed ordinal partition time series to network transformation algorithm. Firstly we introduce a fixed time lag for the elements of each partition that is selected using techniques from traditional time delay embedding. The resulting partitions define regions in the embedding phase space that are mapped to nodes in the network space. Edges are allocated between nodes based on temporal succession thus creating a Markov chain representation of the time series. We then apply this new transformation algorithm to time series generated by the R\"ossler system and find that periodic dynamics translate to ring structures whereas chaotic time series translate to band or tube-like structures -- thereby indicating that our algorithm generates networks whose structure is sensitive to system dynamics. Furthermore we demonstrate that simple network measures including the mean out degree and variance of out degrees can track changes in the dynamical behaviour in a manner comparable to the largest Lyapunov exponent. We also apply the same analysis to experimental time series generated by a diode resonator circuit and show that the network size, mean shortest path length and network diameter are highly sensitive to the interior crisis captured in this particular data set.