Nonlinear Dynamics

, Volume 58, Issue 1–2, pp 1–21 | Cite as

Separatrices and basins of stability from time series data: an application to biodynamics

Original Paper

Abstract

An approach is presented for identifying separatrices in state space generated from noisy time series data sets which are representative of those generated from experiments. We demonstrate how these separatrices can be found using Lagrangian coherent structures (LCSs), ridges in the state space distribution of the maximum finite-time Lyapunov exponent. As opposed to the current approach which requires a vector field in the state space at each instant of time, this method can be performed using only trajectories reconstructed from time series. As such, this paper forms a bridge connecting methods for evaluating time series data with methods used to evaluate LCSs in vector fields. The methods are applied to a problem in musculoskeletal biomechanics, considered as an exemplar of a class of experimental systems that contain separatrices. In this case, the separatrix reveals a basin of stability for a balancing task, outside of which is a zone of failure. We demonstrate that LCSs calculated from only trajectory data, which samples only portions of the state space, align well with LCSs found using a known vector field. In general, we believe this method provides a fruitful approach for extracting information from noisy experimental data regarding boundaries between qualitatively different kinds of behavior.

Keywords

Separatrices Basin of stability Time series analysis Lagrangian coherent structures LCS Lyapunov exponents Recovery envelope 

Abbreviations

FTLE

Finite-time Lyapunov exponent

LCS(s)

Lagrangian coherent structure(s)

Φ

State transition matrix

STM

State transition matrix (method)

NN

Nearest neighbor (method)

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Orthopaedic SurgeryWake Forest University School of MedicineWinston-SalemUSA
  2. 2.Virginia Tech—Wake Forest University School of Biomedical Engineering and SciencesBlacksburgUSA
  3. 3.Department of Engineering Science and MechanicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA

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