Lyapunov Exponents and Information Dimensions of Multi-Degree-of-Freedom Systems Under Deterministic and Stationary Random Excitations

  • C. W. S. To
  • M. L. Liu
Conference paper
Part of the Solid Mechanics and its Applications book series (SMIA, volume 47)


The important concept of Lyapunov exponent has emerged in many fields in the last decade. It plays a crucial role in the determination of bifurcations and chaotic motions in nonlinear systems. Strategies for its numerical computation of multi-degree-of-freedom (MDOF) nonlinear systems under deterministic excitations are available in the literature [1–2]. For nonlinear systems under stochastic excitations, techniques available for the determination of Lyapunov exponents are very limited. They are confined to single degree-of-freedom (DOF) systems under stationary random excitations. For two DOF systems with small nonlinearities and under stationary random excitations of small intensities it is restricted to non-resonant cases. Essentially, these techniques have their basis on the work due to Khasminskii [3].


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Shimada, I., and Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamical systems, Progress of Theoretical Physics 61(6) (1979), 1605-1616.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Wolf, A., Swift, J.B., Swiney, H.L., and Vasano, J.A.: Determining Lyapunov exponents from a time series, Physica 16D (1985), 285-317.Google Scholar
  3. 3.
    Khasminiskii, R.Z.: Sufficient and necessary conditions of almost sure asymptotic stability of a linear stochastic system, Theory of Probability and Application 12(1) (1967), 144-147.CrossRefGoogle Scholar
  4. 4.
    To, C.W.S., and Zhang, S.W.: On the techniques for digital simulation of random response of nonlinear oscillators, Journal of Sound and Vibration 131(1) (1989), 168-173.CrossRefGoogle Scholar
  5. 5.
    Nigam, N.C.: Introduction to Random Vibrations, The MIT Press, Cambridge, Massachusetts, 1983.Google Scholar
  6. 6.
    Kimura, K., and Sakata, M.: Nonstationary response analysis of a nonsymmetric nonlinear multi-degree-of-freedom system to nonwhite random excitation, Japan Society ofMechanical Engineers, International Journal 31(4) (1988), 690-697.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • C. W. S. To
    • 1
  • M. L. Liu
    • 2
  1. 1.Department of Mechanical EngineeringUniversity of Western OntarioLondonCanada
  2. 2.Department of Mechanical EngineeringLakehead UniversityThunder BayCanada

Personalised recommendations