Explorations of the phase-space linearization method for deterministic and stochastic nonlinear dynamical systems

Nonlinear Dynamics

Volumn 23, Issue 3, 2000, Pages 225-228

  Roy, D ^a  

^a INDIAN INSTITUTE OF TECHNOLOGY   (India)

Author keywords

[No Author keywords available]

Indexed keywords

DYNAMICS; INTEGRATION; LINEARIZATION; LYAPUNOV METHODS; MARKOV PROCESSES; MONTE CARLO METHODS; OSCILLATIONS; PHASE SPACE METHODS;

NONLINEAR OSCILLATORS; PHASE-SPACE LINEARIZATION (PSL); TAYLOR EXPANSION;

OSCILLATORS (MECHANICAL);

EID: 0034325967     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1008304408643     Document Type: Article

References (34)

1. Mitropolski, Y. A., Problems of the Asymptotic Theory of Non-Stationary Vibrations, Daniel Davey, New York, 1965.
2. Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1981.
3. Lichtenberg, A. J. and Lieberman, M. A., Regular and Stochastic Motion, Springer-Verlag, New York, 1982.
4. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
5. Nayfeh, A. H., 'Combination tones of single-degree-of-freedom systems with quadratic and cubic non-linearities', Journal of Sound and Vibration 88, 1984, 1-10.
6. Lau, S. L. and Cheung, Y. K., 'Amplitude incremental variational principle for nonlinear vibration of elastic systems', Journal of Applied Mechanics 48, 1981, 959-964.
7. Leung, A. Y. T. and Ge, T., 'Toeplitz Jacobian matrix non-linear periodic vibration', Journal of Applied Mechanics 62, 1998, 709-717.
8. Ortega, J. M. and Rheinboldt, W. C., Iterative Solutions of Non-Linear Equations in Several Variables, Academic Press, New York, 1970.
9. Liao, S. J., 'A second order approximate analytical solution of a simple pendulum by the process analysis method', Journal of Applied Mechanics 59, 1992, 970-975.
10. Roy, D., 'Non-chaotic response of non-linear oscillators under combined deterministic and weak stochastic excitations', Journal of Sound and Vibration 225, 1999, 741-766.
11. He, J.-H., 'A coupling method of a homotopy technique and a perturbation technique for non-linear problems', International Journal of Non-Linear Mechanics 35, 2000, 37-43.
12. Yamaguti, M. and Ushiki, S., 'Chaos in numerical analysis of ordinary differential equations', Physica D 3, 1981, 618-626.
13. Lorenz, E. N., 'Computational chaos: A prelude to computational instability', Physica D 35, 1989, 299-317.
14. Narayanan, S. and Shekar, P., 'A frequency domain based numeric-analytical method for non-linear dynamical systems', Journal of Sound and Vibration 211, 1998, 409-424.
15. Jimenez, J.C., Shoji, I., and Ozaki, T., 'Simulation of stochastic differential equations through the local linearization method. A comparative study', Journal of Statistical Physics 94, 1999, 587-602.
16. Iyengar, R. N. and Roy, D., 'New approaches for the study of nonlinear oscillators', Journal of Sound and Vibration 211, 1998, 843-875.
17. Iyengar, R. N. and Roy, D., 'Extensions of the phase space linearization method', Journal Sound and Vibration 211, 1998, 877-906.
18. Roy, D., 'Phase space linearization for non-linear oscillators: Deterministic and stochastic systems', Journal of Sound and Vibration, 1999 (to appear).
19. Iyengar, R.N. and Roy, D., 'Application of conditional linearization in the study of nonlinear systems', in Proceedings of the IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics, S. Narayanan and R. N. Iyengar (eds.), Kluwer, Dordrecht, 2000 (to appear).
20. Naess, A. and Johnson, J. M., 'Response statistics of nonlinear dynamic systems by path integration', in Proceedings of the IUTAM Symposium on Nonlinear Stochastic Mechanics, N. Bellomo and F. Casciati (eds.), Kluwer, Dordrecht, 1991, pp. 401-414.
21. Risken, H., The Fokker Planck Equation: Methods of Solution and Applications, Springer-Verlag, Berlin, 1989.
22. Chillingworth, D. R. J., Differential Topology with a View to Applications, Pitman, London, 1976.
23. Hale, J., Ordinary Differential Equations, Springer-Verlag, New York, 1980.
24. Banaszuk, A. and Hauser, J., 'Approximate feedback linearization: A homotopy approach', SIAM Journal of Optimization and Control 34, 1998, 1533-1554.
25. Goffman, C. and Pedrick, G., First Course in Functional Analysis, Prentice Hall, Englewood Cliffs, NJ, 1965,
26. Kapitaniak, T., Chaos in Systems with Noise, World Scientific, Singapore, 1990.
27. Kloeden, P. E. and Platen, E., Numerical Solutions of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
28. Milstein, G. N., Numerical Integration of Stochastic Differential Equations, Kluwer, Dordrecht, 1988.
29. Kloeden, P. E., Platen, E., and Schurz, H., Numerical Solution of Stochastic Differential Equations through Computer Experiments, Springer-Verlag, Berlin, 1994.
30. Nigam, N. C., Introduction to Random Vibrations, MIT Press, Cambridge, MA, 1983.
31. Askar, A., Koyluoglu, S. R., Nielsen, K., and Cakmak, A. S., 'Faster simulation methods for nonstationary random vibrations of nonlinear MDOF systems', Probabilistic Engineering Mechanics 11, 1996, 63-72.
32. Moon, F. C., Chaotic Vibrations: An Introduction to Applied Scientists and Engineers, Wiley, New York, 1987.
33. Lin, Y. K. and Cai, G. Q., Probabilistic Structural Dynamics: Advanced Theory and Applications, McGraw Hill, New York, 1995.
34. Dynkin, E. B., Markov Processes, Academic Press, New York, 1965 (translated from Russian by J. Fabius).