Nonlinear Dynamics

, 55:239 | Cite as

Fractional derivative reconstruction of forced oscillators

  • G. Lin
  • B. F. Feeny
  • T. Das
Original Paper


Fractional derivatives are applied in the reconstruction from a single observable of the dynamics of a Duffing oscillator and a two-well experiment. The fractional derivatives of time series data are obtained in the frequency domain. The derivative fraction is evaluated using the average mutual information between the observable and its fractional derivative. The ability of this reconstruction method to unfold the data is assessed by the method of global false nearest neighbors. The reconstructed data is used to compute recurrences and fractal dimensions. The reconstruction is compared to the true phase space and the delay reconstruction in order to assess the reconstruction parameters and the quality of results.


Phase-space reconstructions Fractional derivatives Fractional calculus Chaos Embeddings 


  1. 1.
    Takens, F.: Detecting strange attractors in turbulence. In: Lecture Notes in Mathematics, vol. 898, pp. 366–381. Springer, Berlin (1981) Google Scholar
  2. 2.
    Packard, N.H., Crutchfield, J.P., Farmer, J.D., Shaw, R.S.: Geometry from a time series. Phys. Rev. Lett. 45(9), 712–716 (1980) CrossRefGoogle Scholar
  3. 3.
    Noakes, L.: The Takens embedding theorem. Int. J. Bifurc. Chaos 1(4), 867–872 (1991) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fraser, A.M., Swinney, H.L.: Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33(2), 1134–1140 (1986) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Kennel, M., Brown, R., Abarbanel, H.D.I.: Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A 45, 3403–3411 (1992) CrossRefGoogle Scholar
  6. 6.
    Cao, L.: Practical method for determining the minimum embedding dimension of a scalar time series. Physica D 110, 43–50 (1997) MATHCrossRefGoogle Scholar
  7. 7.
    Potopov, A.: Distortions of reconstruction for chaotic attractors. Physica D 101, 207–226 (1997) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Mindlin, G.B., Solari, H.G.: Topologically inequivalent embeddings. Phys. Rev. E 52(2), 1497–1502 (1995) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Gilmore, R.: Topological analysis of chaotic dynamical systems. Rev. Mod. Phys. 70(4), 1455–1526 (1998) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Gilmore, R., Lefranc, M.: The Topology of Chaos. Wiley-Interscience, New York (2002) MATHGoogle Scholar
  11. 11.
    Lin, G.: Phase-space reconstruction by alternative methods. M.S. Thesis, Michigan State University, East Lansing (2001) Google Scholar
  12. 12.
    Feeny, B.F., Lin, G.: Fractional derivatives applied to phase space reconstructions. Nonlinear Dyn. 38(1–4), 85–99 (2004). Special Issue: Fractional Derivatives and Their Applications, O.P. Agrawal, J.A. Tenreiro Machado, and J. Sabatier (eds.) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of viscoelastically damped structures. AIAA J. Guid. 14(2), 304–311 (1991) CrossRefGoogle Scholar
  14. 14.
    Padovan, J., Sawicki, J.T.: Nonlinear vibrations of fractionally damped systems. Nonlinear Dyn. 16(4), 321–336 (1998) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Zhang, W., Shimizu, N.: Numerical algorithm for dynamic problems involving fractional operators. JSME Int. J. Ser. C 41(3), 364–370 (1998) Google Scholar
  16. 16.
    He, J.H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998) MATHCrossRefGoogle Scholar
  17. 17.
    Stiassnie, M.: A look at fractal functions through their fractional derivatives. Fractals 5(4), 561–564 (1997) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Machado, J.A.T. (ed.): Nonlinear Dyn. 29(1–4) (2002). Special Issue of Fractional Order Calculs and its Applications Google Scholar
  19. 19.
    Agrawal, O.P., Machado, J.A.T., Sabatier, J.: Fractional derivatives and their applications. Introduction. Nonlinear Dyn. 38(1–4), 1–2 (2004) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974) Google Scholar
  21. 21.
    Tseng, C.-C., Pei, S.-C., Hsia, S.-C.: Computation of fractional derivatives using fourier transform and digital FIR differentiator. Signal Process. 80, 151–159 (2000) MATHCrossRefGoogle Scholar
  22. 22.
    Ewins, D.J.: Modal Testing: Theory and Practice. Research Studies Press, Letchworth (1984) Google Scholar
  23. 23.
    Ueda, Y.: The Road to Chaos. Aerial Press, Santa Cruz (1994) Google Scholar
  24. 24.
    Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983) MATHGoogle Scholar
  25. 25.
    Moon, F.C.: Chaotic and Fractal Dynamics—An Introduction for Applied Scientists and Engineers. Wiley, New York (1992) Google Scholar
  26. 26.
    Tufillaro, N.B., Abbott, T., Reilly, J.: An Experimental Approach to Nonlinear Dynamics and Chaos. Addison-Wesley, Reading (1992) MATHGoogle Scholar
  27. 27.
    Yuan, C.-M., Feeny, B.F.: Parametric identification of chaotic systems. J. Vib. Control 4(4), 405–426 (1998) CrossRefGoogle Scholar
  28. 28.
    Auerbach, D., Cvitanovic, P., Eckmann, J.-P., Gunaratne, G., Procaccia, I.: Exploring chaotic motion through periodic orbits. Phys. Rev. Lett. 58, 2387–2389 (1987) CrossRefMathSciNetGoogle Scholar
  29. 29.
    Lathrop, D.P., Kostelich, E.J.: Characterization of an experimental strange attractor by periodic orbits. Phys. Rev. A 40, 4028–4031 (1989) CrossRefMathSciNetGoogle Scholar
  30. 30.
    Van de Water, W., Hoppenbrouwers, M., Christiansen, F.: Unstable periodic orbits in the parametrically excited pendulum. Phys. Rev. A 44(10), 6388–6398 (1991) CrossRefGoogle Scholar
  31. 31.
    Van de Wouw, N., Verbeek, G., Van Campen, D.H.: Nonlinear parametric identification using chaotic data. J. Vib. Control 1, 291–305 (1995) CrossRefGoogle Scholar
  32. 32.
    Feeny, B.F.: Fast multi-fractal analysis by recursive box covering. Int. J. Bifurc. Chaos 10(9), 2277–2287 (2000) MATHCrossRefGoogle Scholar
  33. 33.
    Molteno, T.C.A.: Fast O(N) box-counting algorithm for estimating dimensions. Phys. Rev. E 48(5), R3263–R3266 (1993) CrossRefGoogle Scholar
  34. 34.
    Grassberger, P., Proccacia, I.: Characterization of strange attractors. Phys. Rev. Lett. 50, 346–349 (1983) CrossRefMathSciNetGoogle Scholar
  35. 35.
    Malraison, G., Atten, P., Berge, P., Dubois, M.: Dimension of strange attractors: an experimental determination of the chaotic regime of two convective systems. J. Phys. Lett. 44, 897–902 (1983) CrossRefGoogle Scholar
  36. 36.
    Beck, J.V., Arnold, K.J.: Parameter Estimation in Engineering and Science. Wiley, New York (1977) MATHGoogle Scholar
  37. 37.
    Feder, J.: Fractals. Plenum Press, New York (1988) MATHGoogle Scholar
  38. 38.
    Gerschenfeld, N.: An experimentalist’s introduction to the observation of dynamical systems. In: Hao, B.-L. (ed.) Directions in Chaos, vol. II. World Scientific, Singapore (1988) Google Scholar
  39. 39.
    Cusumano, J.P.: The data used for this experiment was generated in the Engineering Science and Mechanics Department, Penn State University, College Station, PA (1994) Google Scholar
  40. 40.
    Cusumano, J.P., Kimble, B.: Experimental observation of basins of attraction and homoclinic bifurcation in a magneto-mechanical oscillator. In: Thompson, J.M.T., Bishop, S.R. (eds.) Nonlinearity and Chaos in Engineering Dynamics, pp. 71–89. Wiley, Chichester (1994) Google Scholar
  41. 41.
    Cusumano, J.P., Kimble, B.: A stochastic intorrogation method for experimental measurements of global dynamics and basin evolution—application to a 2-well oscillator. Nonlinear Dyn. 8(2), 213–235 (1994) CrossRefGoogle Scholar
  42. 42.
    Feeny, B.F., Yuan, C.M., Cusumano, J.P.: Parametric identification of an experimental magneto-elastic oscillator. J. Sound Vib. 247(5), 785–806 (2001) CrossRefGoogle Scholar
  43. 43.
    Feeny, B.F., Liang, J.-W.: Phase-space reconstructions and stick-slip. Nonlinear Dyn. 13, 39–57 (1997) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of MarylandUniversity ParkUSA
  2. 2.Department of Mechanical EngineeringMichigan State UniversityEast LansingUSA

Personalised recommendations