The slow invariant manifold of a conservative pendulum-oscillator system

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 6, Issue 4, 1996, Pages 673-692

  Georgiou, Ioannis T ^a   Schwartz, Ira B ^a  

^a NAVAL RESEARCH LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0039928465     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127496000345     Document Type: Article

References (24)

1. Bryant, P. J. & Miles, J. W. [1990] "On a periodically forced, weakly damped pendulum. Part 1: Applied torque," J. Austral. Math. Soc. Ser. B32, 1-22.
2. Bryant, P. J. & Miles, J. W. [1990] "On a periodically forced, weakly damped pendulum. Part 2: Horizontal forcing," J. Austral. Math. Soc. Ser. B32, 22-41.
3. Bryant, P. J. & Miles, J. W. [1990] "On a periodically forced, weakly damped pendulum. Part 3: Vertical forcing," J. Austral. Math. Soc. Ser. B32, 42-60.
4. Clifford, M. J. & Bishop, S. R. [1994] "Bifurcation precedences for parametric escape from an asymmetric potential well," Int. J. Bifurcation and Chaos 4(3), 623-630.
5. D'Humieres, D., Beasley, M. R., Huberman, B. A. & Libchaber, A. [1982] "Chaotic states and routes to chaos in the forced pendulum," Phys. Rev. 26(6), 3483-3496.
6. Fenichel, N. [1979] "Geometric singular perturbation theory for ordinary differential equations," J. of Differential Equations 31, 53-98.
7. Georgiou, I. T. [1993] Nonlinear Dynamics and Chaotic Motions of a Singularly Perturbed Nonlinear Viscoelastic Beam, Ph.D. Thesis, Purdue University, West Lafayette, Indiana.
8. Georgiou, I. T. & Schwartz, I. B. [1995] "The dynamics of a pendulum-viscoelastic rod system," (In preparation).
9. Georgiou, I. T., Bajaj, A. K. & Corless, M. [1994a] "Slow normal modes in a singularly perturbed nonlinear oscillator with applications to structural dynamics," in Nonlinear and Stochastic Dynamics, ASME, AMD-192, DE-78, 17-30.
10. Georgiou, I. T., Bajaj, A. K. & Corless, M. [1994b] "Slow and fast invariant manifolds, and normal modes in a two-degree-of-freedom structural dynamical system with multiple equilibrium states," Int. J. of Non-Linear Mechanics (submitted).
11. Kelly, A. [1967] "On the Lyapunov Subcenter Manifold," J. Math. Anal. Appl. 18, 472-478.
12. Knobloch, H. W. & Aullback, B. [1984] "Singular perturbations and integral manifolds," J. Math. Phys. Sci. 18(5), 415-423.
13. Kokotovic̀, P. V., Khalil, H. K. & O'Reilly, J. [1986] Singular Perturbation Methods in Control: Analysis and Design, Chap. 2 (Academic Press, London), pp. 25-76.
14. Leven, R. W. & Koch, B. P. [1981] "Chaotic behavior of a parametrically excited damped pendulum," Phys. Lett. A86(2), 71-74.
15. Lichtenberg, A. J. & Lieberman, M. A. [1992] Regular and Chaotic Dynamics (Springer-Verlag, New York).
16. Miles, J. [1984] "Resonant motion of a spherical pendulum," Physica D11, 309-323.
17. Miles, J. [1988] "Resonance and symmetric breaking for the pendulum," Physica D31, 252-268.
18. Mustafa, G. & Ertas, A. [1993] "Nonlinear interaction of a parametrically-excited coupled column-pendulum oscillator," in Dynamics and Vibration of Time-Varying Systems and Structures, ASME, DE-56, 447-453.
19. Rosenberg, R. M. [1964] "On the existence of normal mode vibrations of nonlinear systems with two-degrees-of-freedom," Quarterly of Appl. Math. 22(3), 217-234.
20. Samaranayake, S. & Bajaj, A. K. [1993] "Bifurcations in the dynamics of an orthogonal double pendulum," Nonlinear Dynamics 4, 605-633.
21. Schmidt, D. S. [1978] "Hopf's bifurcation theorem and the center theorem of Lyapunov with resonance cases," J. Math. Anal. Appl., 63, 354-370.
22. Shaw, S. W. & Pierre, C. [1993] "Normal modes for nonlinear vibratory systems," J. Sound and Vibration 164, 85-124.
23. Shaw, S. W. & Wiggins, S. [1988] "Chaotic dynamics of a whirling pendulum," Physica D31, 190-211.
24. Wiggins, S. [1990] Introduction to Applied Nonlinear Dynamical Systems and Chaos, Chap. 4 (Springer-Verlag, New York), pp. 483-505.