On approximations of first integrals for a system of weakly nonlinear, coupled harmonic oscillators

Nonlinear Dynamics

Volumn 30, Issue 3, 2002, Pages 243-266

  Waluya, S B ^a   Van Horssen, W T ^a  

^a DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Asymptotic approximation of first integral; Bifurcation point; Coupled harmonic oscillators; First integral; Integrating factor; Integrating vector; Periodic solution; Perturbation method

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; INTEGRAL EQUATIONS; MATHEMATICAL MODELS; NONLINEAR SYSTEMS; PERTURBATION TECHNIQUES; RESONANCE; VIBRATIONS (MECHANICAL);

BIFURCATION POINT; COUPLED HARMONIC OSCILLATORS; FIRST INTEGRALS; INTEGRATING FACTOR; INTEGRATING VECTOR;

OSCILLATORS (MECHANICAL);

EID: 0036848065     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1020539621532     Document Type: Article

References (22)

1. van Horssen, W. T., 'On integrating factors for ordinary differential equations', Nieuw Archief voor Wiskunde 15, 1997, 15-26.
2. van Horssen, W. T., 'A perturbation method based on integrating factors', SIAM Journal on Applied Mathematics 59(4), 1999, 1427-1443.
3. van Horssen, W. T., 'A perturbation method based on integrating vectors and multiple scales', SIAM Journal on Applied Mathematics 59(4), 1999, 1444-1467.
4. Waluya, S. B. and van Horssen, W. T., 'Asymptotic approximations of first integrals for a nonlinear oscillator', Nonlinear Analysis TMA 51(8), 2002, 1327-1346.
5. Waluya, S. B. and van Horssen, W. T., 'On approximations of first integrals for a weakly nonlinear oscillator', in Proceedings of the National Mathematical Conference, ITB Indonesia, S. Adji et al. (eds.), MIHMI Vol. 6, No. 5, P4M-ITB 2000, pp. 529-537.
6. Waluya, S. B. and van Horssen, W. T., 'On asymptotic approximations of first integrals for a weakly nonlinear oscillator', in Proceedings of the 18th Biennal ASME Conference, Symposium on Dynamics and Control of Time-Varying Systems and Structures, Pittsburgh, PA, September 9-12, Vol. DETC2001/VIB-21426, S. C. Sinha et al. (eds.), ASME, New York, 2001, pp. 1-9.
7. Verros, G. and Natsiavas, S., 'Self-excited oscillators with asymmetric nonlinearities and one-to-two internal resonance', Nonlinear Dynamics 17, 1998, 325-346.
8. Natsiavas, S., 'Free vibration of two coupled nonlinear oscillators', Nonlinear Dynamics 6, 1994, 69-86.
9. Haaker, T. I. and van der Burgh, A. H. P., 'Rotational galloping of two coupled oscillators', Meccanica 33, 1998, 219-227.
10. Mitsi, S., Natsiavas, S., and Tsiafis, I., 'Dynamics of nonlinear oscillators under simultaneous internal and external resonances', Nonlinear Dynamics 16, 1998, 23-39.
11. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
12. Bajaj A. K., Chang, S. I., and Johnson, J. M., 'Amplitude modulated dynamics of a resonantly excited autoparametric two degree of freedom system', Nonlinear Dynamics 5, 1994, 433-457.
13. van der Beek, C. G. A., 'Normal form and periodic solutions in the theory of non-linear oscillations. Existence and asymptotic theory', International Journal of Non-Linear Mechanics 24(4), 1989, 263-279.
14. van der Beek, C. G. A., 'Analysis of a system of two weakly nonlinear coupled harmonic oscillators arising from the field of wind-induced vibrations', International Journal of Non-Linear Mechanics 27(4), 1992, 679-704.
15. Forsyth, A. R., Lehrbuch der Differential-Gleichungen, Vieweg, Braunschweig, 1912.
16. Kaplan, W., Ordinary Differential Equations, Addison-Wesley, MA, 1958.
17. Ince, E. L. and Sneddon, I. N., The Solution of Ordinary Differential Equations, Longman Scientific & Technical, Harlow Essex, 1987.
18. Forsyth, A. R., A Treatise on Differential Equations, sixth edition, Dover, New York, 1956.
19. Arnold, V. I., Ordinary Differential Equations, The MIT Press, Cambridge, MA, 1978.
20. Verhulst, F., Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Berlin, 1996.
21. Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Applied Mathematical Sciences, Vol. 42, Springer-Verlag, New York, 1983.
22. Kevorkian, J. and Cole, J. D., Multiple Scale and Singular Perturbation Methods, Applied Mathematical Sciences, Vol. 114, Springer-Verlag, New York, 1996.