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

  • Martin L. Tanaka
  • Shane D. Ross
Original Paper


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.


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



Finite-time Lyapunov exponent


Lagrangian coherent structure(s)


State transition matrix


State transition matrix (method)


Nearest neighbor (method)


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  1. 1.
    Dingwell, J.B., Cusumano, J.P.: Nonlinear time series analysis of normal and pathological human walking. Chaos 10(4), 848–863 (2000) zbMATHCrossRefGoogle Scholar
  2. 2.
    Akay, M.: Wiley Encyclopedia of Biomedical Engineering. Wiley–Interscience, Hoboken (2006) CrossRefGoogle Scholar
  3. 3.
    England, S.A., Granata, K.P.: The influence of gait speed on local dynamic stability of walking. Gait Posture 25(2), 172–178 (2007) CrossRefGoogle Scholar
  4. 4.
    Pierrehumbert, R.T.: Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves. Geophys. Astrophys. Fluid Dyn. 58, 285–319 (1991) CrossRefGoogle Scholar
  5. 5.
    Pierrehumbert, R.T.: Large-scale horizontal mixing in planetary-atmospheres. Phys. A-Fluids Fluid Dyn. 3(5), 1250–1260 (1991) CrossRefGoogle Scholar
  6. 6.
    Shadden, S.C., Lekien, F., Marsden, J.E.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys. D Nonlinear Phenom. 212(3–4), 271–304 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Haynes, P.: Stratospheric dynamics. Annu. Rev. Fluid Mech. 37, 263–293 (2005) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Falconer, I., Gottwald, G.A., Melbourne, I., Wormnes, K.: Application of the 0-1 test for chaos to experimental data. SIAM J. Appl. Dyn. Syst. 6(2), 395–402 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica 15, 9–20 (1980) zbMATHCrossRefGoogle Scholar
  10. 10.
    Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.: Lyapunov exponents for smooth dynamical systems and Hamiltonian systems; a method for computing all of them. Part II: Numerical application. Meccanica 15, 21–30 (1980) CrossRefGoogle Scholar
  11. 11.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenom. 16(3), 285–317 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Eckmann, J.P., Kamphorst, S.O., Ruelle, D., Ciliberto, S.: Lyapunov exponents from time-series. Phys. Rev. A 34(6), 4971–4979 (1986) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Rosenstein, M.T., Collins, J.J., Deluca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 65(1–2), 117–134 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis. Cambridge University Press, Cambridge (2004) zbMATHGoogle Scholar
  15. 15.
    Haller, G., Yuan, G.: Lagrangian coherent structures and mixing in two-dimensional turbulence. Phys. D Nonlinear Phenom. 147(3–4), 352–370 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Haller, G.: Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos 10(1), 99–108 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Haller, G.: Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149(4), 248–277 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Haller, G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids 13(11), 3365–3385 (2001) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Haller, G.: Lagrangian coherent structures from approximate velocity data. Phys. Fluids 14(6), 1851–1861 (2002) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Casdagli, M., Eubank, S., Farmer, J.D., Gibson, J.: State-space reconstruction in the presence of noise. Physica D 51(1–3), 52–98 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Ellner, S., Turchin, P.: Chaos in a noisy world—new methods and evidence from time-series analysis. Am. Nat. 145(3), 343–375 (1995) CrossRefGoogle Scholar
  22. 22.
    Franca, L.F.P., Savi, M.A.: Distinguishing periodic and chaotic time series obtained from an experimental nonlinear pendulum. Nonlinear Dyn. 26(3), 253–271 (2001) CrossRefGoogle Scholar
  23. 23.
    Wang, Y., Haller, G., Banaszuk, A., Tadmor, G.: Closed-loop Lagrangian separation control in a bluff body shear flow model. Phys. Fluids 15(8), 2251–2266 (2003) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Aldridge, B., Haller, G., Sorger, P., Lauffenburger, D.: Direct Lyapunov exponent analysis enables parametric study of transient signalling governing cell behaviour. IEE Proc. Syst. Biol. 153, 425–432 (2006) CrossRefGoogle Scholar
  25. 25.
    El Rifai, K., Haller, G., Bajaj, A.K.: Global dynamics of an autoparametric spring-mass-pendulum system. Nonlinear Dyn. 49(1–2), 105–116 (2007) zbMATHCrossRefGoogle Scholar
  26. 26.
    Winter, D.A., Patla, A.E., Rietdyk, S., Ishac, M.G.: Ankle muscle stiffness in the control of balance during quiet standing. J. Neurophysiol. 85(6), 2630–2633 (2001) Google Scholar
  27. 27.
    Morasso, P.G., Sanguineti, V.: Ankle muscle stiffness alone cannot stabilize balance during quiet standing. J. Neurophysiol. 88(4), 2157–2162 (2002) Google Scholar
  28. 28.
    Kuo, A.D., Zajac, F.E.: A biomechanical analysis of muscle strength as a limiting factor in standing posture. J. Biomech. 26(Suppl. 1), 137–150 (1993) CrossRefGoogle Scholar
  29. 29.
    Pai, Y.C., Patton, J.: Center of mass velocity–position predictions for balance control. J. Biomech. 30(4), 347–354 (1997) CrossRefGoogle Scholar
  30. 30.
    Iqbal, K., Pai, Y.C.: Predicted region of stability for balance recovery: motion at the knee joint can improve termination of forward movement. J. Biomech. 33(12), 1619–1627 (2000) CrossRefGoogle Scholar
  31. 31.
    Edwards, W.T.: Comments on “Predicted region of stability for balance recovery: motion at the knee joint can improve termination of forward movement”. J. Biomech. 34(6), 831–833 (2001) CrossRefGoogle Scholar
  32. 32.
    Granata, K.P., Orishimo, K.F.: Response of trunk muscle co-activation to changes in spinal stability. J. Biomech. 34(9), 1117–1123 (2001) CrossRefGoogle Scholar
  33. 33.
    Cholewicki, J., McGill, S.M.: Mechanical stability of the in vivo lumbar spine: implications for injury and chronic low back pain. Clin. Biomech. (Bristol, Avon) 11(1), 1–15 (1996) CrossRefGoogle Scholar
  34. 34.
    Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Springer, Berlin (2004) Google Scholar
  35. 35.
    Fung, Y.: Biomechanics: Mechanical Properties of Living Tissues. Springer, New York (1993) Google Scholar
  36. 36.
    Strang, G.: Linear Algebra and its Applications. Harcourt Brace Jovanovich, San Diego (1998) Google Scholar
  37. 37.
    Lekien, F., Shadden, S.C., Marsden, J.E.: Lagrangian coherent structures in n-dimensional systems. J. Math. Phys. 48(6) (2007) Google Scholar
  38. 38.
    Tanaka, M.L., Granata, K.P.: Methods & nonlinear analysis for measuring torso stability. In ASCE 18th Engineering Mechanics Division Conference Blacksburg, VA, 3–6 June 2007 Google Scholar
  39. 39.
    Lee, H., Granata, K.P.: Process stationarity and reliability of trunk postural stability. Clin. Biomech. (Bristol, Avon) 23(6), 735–742 (2008) CrossRefGoogle Scholar
  40. 40.
    Lee, H., Granata, K.P., Madigan, M.L.: Effects of trunk exertion force and direction on postural control of the trunk during unstable sitting. Clin. Biomech. (Bristol, Avon) 23(5), 505–509 (2008) CrossRefGoogle Scholar
  41. 41.
    Slota, G.P., Granata, K.P., Madigan, M.L.: Effects of seated whole-body vibration on postural control of the trunk during unstable seated balance. Clin. Biomech. (Bristol, Avon) 23(4), 381–386 (2008) Google Scholar
  42. 42.
    Cholewicki, J., Polzhofer, G.K., Radebold, A.: Postural control of trunk during unstable sitting. J. Biomech. 33(12), 1733–1737 (2000) CrossRefGoogle Scholar
  43. 43.
    Tanaka, M.L.: Biodynamic analysis of human torso stability using finite time Lyapunov exponents. Ph.D. thesis, Virginia Polytechnic Institute and State University (2008) Google Scholar
  44. 44.
    Murray, R., Hauser, J.A.: Case: Study on Approximate Linearization: The Acrobot Example. Electronics Research Laboratory, College of Engineering, University of California, Berkeley, pp. 1–43 (1991) Google Scholar
  45. 45.
    Spong, M.W.: The swing up control problem for the acrobot. IEEE Control Syst. Mag. 15(1), 49–55 (1995) CrossRefGoogle Scholar
  46. 46.
    Collins, J.J., De Luca, C.J.: Open-loop and closed-loop control of posture: a random-walk analysis of center-of-pressure trajectories. Exp. Brain Res. 95(2), 308–318 (1993) CrossRefGoogle Scholar
  47. 47.
    Delignieres, D., Deschamps, T., Legros, A., Caillou, N.: A methodological note on nonlinear time series analysis: is the open- and closed-loop model of Collins and De Luca (1993) a statistical artifact? J. Mot. Behav. 35(1), 86–97 (2003) CrossRefGoogle Scholar
  48. 48.
    Bottaro, A., Casadio, M., Morasso, P.G., Sanguineti, V.: Body sway during quiet standing: is it the residual chattering of an intermittent stabilization process? Hum. Mov. Sci. 24(4), 588–615 (2005) CrossRefGoogle Scholar

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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