On stability analysis via lyapunov exponents calculated from a time series using nonlinear mapping-a case study

Nonlinear Dynamics

Volumn 59, Issue 1-2, 2010, Pages 239-257

  Yang, Caixia ^a   Wu, Qiong ^a  

^a UNIVERSITY OF MANITOBA   (Canada)

Author keywords

Lyapunov exponents; Nonlinear mapping; Stability analysis; Time series

Indexed keywords

BIPEDAL ROBOT; EVOLUTION TIME; LINEAR MAPPING; LOCAL LINEAR MAPPING; LYAPUNOV EXPONENT; NONLINEAR APPROXIMATION; NONLINEAR MAPPINGS; POTENTIALLY STABLE; ROBOTIC CONTROL SYSTEM; STABILITY ANALYSIS; STABLE SYSTEMS; STATE SPACE; TIME LAG;

CHAOTIC SYSTEMS; COMPUTER SIMULATION; CONTROL SYSTEM ANALYSIS; DIFFERENTIAL EQUATIONS; LYAPUNOV METHODS; MAPPING; NONLINEAR ANALYSIS; STABILITY; SYSTEM STABILITY; TIME SERIES; TIME SERIES ANALYSIS;

CONTROL SYSTEM STABILITY;

EID: 77949422942     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-009-9535-7     Document Type: Article

References (41)

1. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physics D 16, 285-317 (1985)
2. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos, an Introduction to Dynamical Systems. Springer, New York (1997)
3. Williams, G.P.: Chaos Theory Tamed. Joseph Henry Press, Washington, D.C. (1997)
4. Sekhavat, P., Sepehri, N., Wu, Q.: Calculation of Lyapunov exponents using nonstandard finite difference discretization scheme: a case study. J. Differ. Equ. Appl. 10(4), 369-378 (2004)
5. Müller, P.C.: Calculation of Lyapunov exponents for dynamic systems with discontinuities. Chaos Solitons Fractals 5, 1671-1681 (1995)
6. Oseledec, V.I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc. 19, 197 (1968)
7. Asokanthan, S.F., Wang, X.H.: Characterization of torsional instability in a Hooke's joint-driven system via maximal Lyapunov exponents. J. Sound Vib. 194(1), 83-91 (1996)
8. Gilat, R., Aboudi, J.: Parametric stability of non-linearly elastic composite plates by Lyapunov exponents. J. Sound Vib. 235(4), 627-637 (2000)
9. Zevin, A.A., Pinsky, M.A.: Absolute stability criteria for a generalized Lur'e problem with delay in the feedback. SIAM J. Control Optim. 43(6), 2000-2008 (2005)
10. Awrejcewicz, J., Kudra, G.: Stability analysis and Lyapunov exponents of a multi-body mechanical system with rigid unilateral constraints. Nonlinear Anal. Theory Methods Appl. 63, 909-918 (2005)
11. Castillo, Rogelio, C., Gustavo, A., Javier, C.P.: Determination of limit cycles using both the slope of correlation integral and dominant Lyapunov methods. Nucl. Technol. 145(2), 139-149 (2004)
12. Wu, Q., Sekhavat, P., Sepehri, N., Peles, S.: On design of continuous Lyapunov's feedback control. J. Franklin Inst. Eng. Appl. Math. 342(6), 702-723 (2005)
13. Sekhavat, P., Sepehri, N., Wu, Q.: Impact control in hydraulic actuators with friction: theory and experiments. IFAC J. Control Eng. Pract. 14(12), 1423-1433 (2006)
14. Yang, C., Wu, Q.: On stabilization of bipedal robots during disturbed standing using the concept of Lyapunov exponents. In: Proceedings of the 2006 American Control Conference, Minneapolis,MN, pp. 2516-2521, 14-16 June 2006
15. Yang, C., Wu, Q.: On stabilization of bipedal robots during disturbed standing using the concept of Lyapunov exponents. Robotica 24(5), 621-624 (2006)
16. Sano, M., Sawada, Y.: Measurement of the Lyapunov spectrum from chaotic time series. Phys. Rev. Lett. 55, 1082 (1985)
17. Kunze, M.: Non-smooth Dynamical Systems. Springer, Berlin (2000)
18. Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, 2nd edn. Cambridge University Press, Cambridge (2004)
19. Zeng, X., Pielke, R.A., Eykholt, R.: Extracting Lyapunov exponents from short time series of low precision. Mod. Phys. Lett. B 6(2), 55-75 (1992)
20. Rosenstein, M.T., Collins, J.J., DeLuca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D, Nonlinear Phenom. 65, 117-134 (1993)
21. Hegger, R., Kantz, H., Schreiber, T.: Practical implementation of nonlinear time series analysis: the TISEAN package. Chaos, Interdiscip. J. Nonlinear Sci. 9(2), 413-435 (1999)
22. Kantz, H.: A robust method to estimate the maximal Lyapunov exponent of a time series. Phys. Lett. A 185(1), 77-87 (1994)
23. Yasuaki, O., Muhammad, A., Akihiro, S., Kazuki, F., Hikaru, I., Ryoichi, N., Ichiro, T.: Assessment of walking stability of elderly by means of nonlinear time-series analysis and simple accelerometry. JSME Int. J. Ser. C, Mech. Syst. Mach. Elem. Manuf. 48(4), 607-612 (2006)
24. Burdet, E., Tee, K.P.,Mareels, I.,Milner, T.E., Chew, C.M., Franklin, D.W., Osu, R., Kawato, M.: Stability and motor adaptation in human arm movements. Biol. Cybern. 94(1), 20-32 (2006)
25. Dingwell, B.J., Cusumano, J.P.: Nonlinear time series analysis of normal and pathological human walking. Chaos 10(4), 848-863 (2000)
26. Dingwell, J.B., Cusumano, J.P., Sternad, D., Cavanagh, P.R.: Slower speeds in patients with diabetic neuropathy lead to improved local dynamic stability of continuous overground walking. J. Biomech. 33, 1269-1277 (2000)
27. Dingwell, J.B., Cusumano, J.P., Cavanagh, P.R., Sternad, D.: Local dynamic stability versus kinematic variability of continuous overground and treadmill walking. ASME J. Biomech. Eng. 123, 27-32 (2001)
28. Dingwell, J.B.,Marin, L.C.: Kinematic variability and local dynamic stability of upper body motions when walking at different speeds. J. Biomech. 39, 444-452 (2006)
29. Dingwell, B.J.: Lyapunov exponents. In: Wiley Encyclopedia of Biomedical Engineering. Wiley, New York (2006)
30. Brown, R., Bryant, P., Abarbanel, H.D.: Computing the Lyapunov spectrum of a dynamical system from an observed time series. Phys. Rev. A 43, 2787-2806 (1991)
31. Yang, C., Wu, Q.: Effects of constraints on bipedal balance control. In: Proceedings of the 2006 American Control Conference, Minneapolis, MN, pp. 2510-2515, 14-16 June 2006
32. Haken, H.: At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point. Phys. Lett. A 94, 71-72 (1989)
33. Holzfuss, J., Lauterborn, W.: Lyapunov exponents from a time series of acoustic chaos. Phys. Rev. A 39, 2146-2152 (1989)
34. Vukobratovíc, M., Borovác, B.: Zero-moment point-thirty years of its life. Int. J. Humanoid Robot. 1(1), 157-173 (2004)
35. Abarbanel, H.D.I., Brown, R., Kennel, M.B.: Local Lyapunov exponents computed from observed data. J. Nonlinear Sci. 2, 343-365 (1992)
36. Takens, F.: Detect strange attractors in turbulence. In: Rand, D.A., Young, L.-S. (eds.) Dynamical Systems and Turbulence. Lecture Notes in Mathematics, vol.898, pp. 366- 381. Springer, Berlin (1981)
37. Fraser, A.M.: Reconstructing attractors from scalar time series: a comparison of singular system analysis and redundancy criteria. Physica D 34, 391-404 (1989)
38. Broomhead, D.S., King, G.P.: Extracting qualitative dynamics from experimental data. Physica D 20, 217-236 (1986)
39. Slotine, J.-J.E., Li,W.: Applied Nonlinear Control. Prentice Hall, New York (1991)
40. Nusse, H., Yorke, J.A.: Dynamics: Numerical Exploration. Springer, New York (1998)
41. Parlitz, U.: Identification of true and spurious Lyapunov exponents from time series. Int. J. Bifurc. Chaos 2, 155-165 (1992)