Unified Analysis of Complex Nonlinear Motions via Densities
Abstract
In this study analyses of deterministic and randomly perturbed complex nonlinear responses of nonlinear systems using densities are illustrated from an engineering perspective. Motivations to examine deterministic nonlinear dynamical systems via densities are first discussed. Pertinent mathematical backgrounds and techniques common to the analyses of both deterministic chaos and random chaotic processes are reviewed. Probability densities of nonlinear responses are computed by solving the Fokker-Planck equation (FPE) numerically to examine stochastic properties of random chaotic responses. It is shown that, by introducing random perturbations in the otherwise deterministic (periodic) excitation, the boundaries separating co-existing attractors associated with the corresponding deterministic system become blurred, and the existence of the attractors can be depicted by the evolution of a unique density in (physical) phase space. Transient and asymptotic behaviors of the densities reveal the relative stability of the various attractors. Mathematical theories useful for stability analysis of the attractors are discussed. Finally, asymptotically stable co-existing “periodic” and “chaotic” motions of a nonlinear dynamical system subjected to periodic excitation with random perturbations are demonstrated via an engineering example.
Keywords
Chaotic Motion Random Perturbation Deterministic System Unique Density Stochastic PropertyPreview
Unable to display preview. Download preview PDF.
References
- 1.Moon, F. C., Chaotic Vibrations, John Wiley & Sons (1987)zbMATHGoogle Scholar
- 2.Thompson, J.M.T. and Stewart, H.B., Nonlinear Dynamics and Chaos, John Wiley & Sons (1986)zbMATHGoogle Scholar
- 3.Guckenheimer, J. and Holmes P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag (1983)zbMATHGoogle Scholar
- 4.Wiggins, S., Global Bifurcations and Chaos: Analytical Methods, Springer-Verlag (1988)zbMATHCrossRefGoogle Scholar
- 5.Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag (1990)zbMATHGoogle Scholar
- 6.Thompson, J.M.T., Rainey, R.C.T. and Soliman, M.S., Phil Trans Roy Soc London, 332, 149 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
- 7.Falzarano, J., Shaw S.W. and Troesch, A.W., Int J of Bifurcation and Chaos, 2, 101, (1992)MathSciNetzbMATHCrossRefGoogle Scholar
- 8.Meunier, C., and Verga, A.D., J Stats Phys, 50, 345 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
- 9.Frey, M. and Simiu, E., Proc 9 th Engrg Mech Conf, ASCE, 660 (1992)Google Scholar
- 10.Frey, M. and Simiu, E., Proc 2 nd Comp Stoch Mech Conf 113 (1995)Google Scholar
- 11.Hsieh, S-R., Troesch, A.W. and Shaw S.W., Proc R Soc Lond. A., 446, 195, (1994)CrossRefGoogle Scholar
- 12.Lin, H. and Yim, S.C.S., Applied Ocean Research, (to appear)Google Scholar
- 13.Yim, S.C.S. and Lin, H., Applied Chaos, John Wiley & Sons (1992)Google Scholar
- 14.Risken, H., The Fokker-Planck Equation, Springer-Verlag (1984)zbMATHGoogle Scholar
- 15.Gardiner, C.W., Handbook of Stochastic Methods, Springer-Verlag (1985)Google Scholar
- 16.Lin, Y.K., Probabilistic Theory of Structural Dynamics, McGraw-Hill (1967)Google Scholar
- 17.Stratonovich, R.L., Topics in the Theory of Random Noise (1967)zbMATHGoogle Scholar
- 18.Wissel, C., Z Physik B, 35, 185 (1979)MathSciNetCrossRefGoogle Scholar
- 19.Lasota, A., and Mackey, M.C., Chaos, Fractals, and Noise, 2nd Ed., Springer-Verlag (1994)zbMATHGoogle Scholar