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Locating oscillatory orbits of the parametrically-excited pendulum

Published online by Cambridge University Press:  17 February 2009

M. J. Clifford  and
S. R. Bishop
M. J. Clifford
Affiliation:
Centre for Nonlin. Dyn. and its Appl'ns, University College London, WC1E 6BT, U.K.
S. R. Bishop
Affiliation:
Centre for Nonlin. Dyn. and its Appl'ns, University College London, WC1E 6BT, U.K.
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Abstract

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A method is considered for locating oscillating, nonrotating solutions for the parametrically-excited pendulum by inferring that a particular horseshoe exists in the stable and unstable manifolds of the local saddles. In particular, odd-periodic solutions are determined which are difficult to locate by alternative numerical techniques. A pseudo-Anosov braid is also located which implies the existence of a countable infinity of periodic orbits without the horseshoe assumption being necessary.

Type
Research Article
Information
The ANZIAM Journal , Volume 37 , Issue 3 , January 1996 , pp. 309 - 319
Copyright
Copyright © Australian Mathematical Society 1996

References

[1] Birman, J. S., Braids, links, and linking class groups, Annals of Math. Study 82 (Princeton University Press, 1974).Google Scholar
[2] Boyland, P. L. and Franks, J., Notes on dynamics of surface homomorphisms (Mathematics Institute, Warwick University, 1989).Google Scholar
[3] Bryant, P. J. and Miles, J. W., “On a periodically forced, weakly damped pendulum. Part 1: Applied torque”, J. Austral. Math. Soc. Ser. B 32 (1990) 1–22.CrossRefGoogle Scholar
[4] Bryant, P. J. and Miles, J. W., “On a periodically forced, weakly damped pendulum. Part 2: Horizontal forcing”, J. Austral. Math. Soc. Ser. B 32 (1990) 23–41.CrossRefGoogle Scholar
[5] Bryant, P. J. and Miles, J. W., “On a periodically forced, weakly damped pendulum. Part 3: Vertical forcing”, J. Austral. Math. Soc. Ser. B 32 (1990) 42–60.CrossRefGoogle Scholar
[6] Capecchi, D. and Bishop, S. R., “Periodic oscillations and attracting basins for a parametrically excited pendulum”, Dynamics and Stability of Systems, 9 (1994) 123–143.CrossRefGoogle Scholar
[7] Clifford, M. J. and Bishop, S. R., “Generic features of escape from a potential well under parametric excitation”, Physics Letters A 184 (1993) 57–63.Google Scholar
[8] Feigenbaum, M. J., “Universal behaviour in nonlinear systems”, Los Alamos Science 1 (1980) 4–27.Google Scholar
[9] Foale, S. and Thompson, J. M. T., “Geometrical concepts and computational techniques of nonlinear dynamics”, Computer methods in Applied Mechanics and Engineering 89 (1991) 381–394.CrossRefGoogle Scholar
[10] Guckenheimer, J. and Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Springer-Verlag, New York, 1983).Google Scholar
[11] Gwinn, E. G. and Westervelt, R. M., “Horseshoes in the driven, damped pendulum”, Physica D 23 (1986) 396–401.CrossRefGoogle Scholar
[12] Hayashi, C., Nonlinear oscillations in physical systems (McGraw-Hill, New York, 1964).Google Scholar
[13] Hsu, C. S., Cell to cell mapping: a method of global analysis for nonlinear systems (Springer-Verlag, New York, 1987).CrossRefGoogle Scholar
[14] Jordan, D. W. and Smith, P., Nonlinear ordinary differential equations (Oxford University Press, New York, 1987).Google Scholar
[15] Koch, B. P. and Leven, R. W., “Subharmonic and homoclinic bifurcations in a parametrically forced pendulum”, Physica D 16 (1985) 1–13.CrossRefGoogle Scholar
[16] Li, T. and Yorke, J. A., “Period three implies chaos”, Amer. Math. Monthly 82 (1975) 481–485.CrossRefGoogle Scholar
[17] Mahwin, J., “The forced pendulum: a paradigm for nonlinear analysis and dynamical systems”, Expositiones Mathematicae 6 (1988) 271–287.Google Scholar
[18] McRobie, F. A., “Bifurcational precedences in the braids of periodic orbits of spiral 3-shoes in driven oscillators”, Proc. R. Soc. Lond. A 438 (1992) 545–568.Google Scholar
[19] McRobie, F. A. and Thompson, J. M. T., “An introduction to braids and knots in driven oscillators”, Int. J. Bifurcation and Chaos, 3 (1993) 1343–1361.Google Scholar
[20] Ott, E., Grebogi, C. and Yorke, J. A., “Controlling chaos”, Phys. Rev. Lett. 64 (11) (1990) 1196–1199.CrossRefGoogle ScholarPubMed
[21] Parker, T. S. and Chua, L. O., Practical numerical algorithms for chaotic systems (Springer-Verlag, New York, 1989).CrossRefGoogle Scholar
[22] Rainey, R. C. T., “The dynamics of tethered platforms”, Trans. Royal Institute of Naval Architects 120 (1978) 59–80.Google Scholar
[23] Salam, F. M. and Sastry, S. S., “Dynamics of the forced Josephson junction: the region of chaos”, IEEE Trans. Circuits and Systems CAS-30 (1985) 784–796.CrossRefGoogle Scholar
[24] Smale, S., “Differential dynamical systems”, Bull. Amer. Math. Soc. 73 (1967) 747–817.CrossRefGoogle Scholar
[25] Thompson, J. M. T. and Stewart, H. B., Nonlinear dynamics and chaos (Wiley, Chichester, 1986).Google Scholar