Separatrices and basins of stability from time series data

TL;DR

This paper introduces a novel method for identifying phase space separatrices and basins of stability directly from noisy time series data by using finite-time Lyapunov exponents, applicable even with limited trajectories.

Contribution

The method uniquely identifies Lagrangian coherent structures from trajectories without requiring an analytical vector field, suitable for experimental data with noise and few trajectories.

Findings

Successfully applied to biological simulation data

Revealed basins of stability through separatrices

Effective with limited and noisy data

Abstract

An approach is presented for identifying separatrices in phase space generated from noisy time series data sets representative of measured experimental data. These separatrices are identified as ridges in the phase space distribution of finite-time Lyapunov exponents, i.e., Lagrangian coherent structures (LCS). As opposed to previous approaches, the LCS is identified using only trajectories since no analytical or data-defined vector field is available. The method is applied to a biological simulation in which the separatrix reveals a basin of stability. These results suggest that the method will be a fruitful approach to time series analysis, particularly in cases where a limited number of trajectories are available as might be encountered in experiments.