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Caterpillar solutions in coupled pendula

Published online by Cambridge University Press:  10 December 2009

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Abstract

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The single pendulum is one of the fundamental model problems in the theory of dynamical systems; coupled pendula, or equivalently, two elastically coupled particles in a periodic potential on a line, are a natural extension of intrinsic interest. The system arises in various physical applications and it inherits some rudiments of the behaviour exhibited by its finite-dimensional parent, the sine-Gordon equation. Among these phenomena are the so-called caterpillar solutions, whose behaviour is reminiscent of solitons. These solutions turn out to have a transparent geometrical explanation. There is an interesting bifurcation picture associated with the system: the parameter region is broken up into the set of ‘pyramids’ parametrized by pairs of integers; these integers characterize the behaviour of the associated solutions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Volume 8: Charles Conley Memorial Issue , December 1988 , pp. 153 - 174
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

Andronov, A. A., Vitt, E. A. & Khaikin, S. E.. Theory of Oscillations. Princeton University Press, Princeton, NJ (1949).Google Scholar
Arnold, V. I.. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1983).10.1007/978-1-4684-0147-9Google Scholar
Campbell, D. K., Peyrard, M. & Sodano, P.. Kink-antikink interactions in the double sine-Gordon equation. Physica 19D (1986), 165–205.Google Scholar
Chua, L., Yao, Y. & Yang, Q.. The devil's staircase in circuits. Int. J. Circuit Th. Appl. (1986).10.1002/cta.4490140405Google Scholar
Cirillo, M., Parmentier, R. D. & Savo, B.. Mechanical analog studies of a perturbed sine-Gordon equation. Physica 3D (1981), 565–576.Google Scholar
Grüner, G. & Zettl, A.. CDW conduction: a novel collective transport phenomenon in solids. Phys. Rep. 119 (1985), 119–232.10.1016/0370-1573(85)90073-0Google Scholar
Imry, Y. & Schulman, L.. Qualitative theory of the nonlinear behavior of coupled Josephson junctions. J. Appl. Phys. 49 (1978), 749–758.10.1063/1.324654Google Scholar
Josephson, B. D.. Phys. Lett. 1 (1962), 251.10.1016/0031-9163(62)91369-0Google Scholar
Landberg, D. N., Scalapino, D. J. & Taylor, B. N.. The Josephson effects. Sci Amer. 21, 30–39.Google Scholar
Levi, M.. Beating modes in the Josephson junction. In Chaos in Nonlinear Dynamical Systems, ed. Chandra, J.. SIAM, Philadelphia (1984), 56–73.Google Scholar
Levi, M., Hoppensteadt, F. C. & Miranker, W. L.. Dynamics of the Josephson junction. Q. Appl. Math. (1978), 167–198.10.1090/qam/484023Google Scholar
Marcus, P. M. & Imry, Y.. Steady oscillatory states of a finite Josephson junction. Solid State Commun. 33 (1980), 345–349.10.1016/0038-1098(80)91167-9Google Scholar
Marcus, P. M., Imry, Y. & Ben-Jacob, E.. Characteristic modes and the transition to chaos of a resonant Josephson circuit. Solid State Commun. 41 (1982), 161–166.10.1016/0038-1098(82)91058-4Google Scholar
Matisoo, J.. Josephson-type superconductive tunnel junctions and applications. IEEE Trans. Mdgn. 5 (1969), 848–873.Google Scholar
Miracky, R. F., Devoret, M. H. & Clarke, J.. Deterministic hopping in a Josephson circuit described by a one-dimensional mapping. Phys. Rev. A 31 (1985), 2509–2519.10.1103/PhysRevA.31.2509Google Scholar
Nakajima, , Yamashita, K. T. & Onodera, Y.. Mechanical analogue of active Josephson transmission line. J. Appl. Phys. 45 (1974), 3141–3145.10.1063/1.1663738Google Scholar
Odeh, F.. On existence, uniqueness and stability of solutions of the Josephson phase equation. Preprint.Google Scholar
Scott, A. C.. A nonlinear Klein-Gordon equation. Amer. J. Phys. 37 (1969), 52–61.10.1119/1.1975404Google Scholar
Soerensen, M. P., Arley, N., Christiansen, P. L., Parmentier, R. D. & Skovgaard, O.. Intermittent switching between soliton dynamic states in a perturbed sine-Gordon model. Phys. Rev. Lett. (1983), 1919–1922.10.1103/PhysRevLett.51.1919Google Scholar
Stoker, J. J.. Nonlinear Vibrations. Interscience Publishers, New York (1950).Google Scholar
Sullivan, D. B. & Zimmerman, J. E.. Mechanical analogs of time dependent Josephson phenomena. Amer. J. Phys. 39 (1971), 1504–1517.10.1119/1.1976705Google Scholar
Tricomi, F.. Integrazione di un'equazione differenzale presentasi in elettrotecnica. Ann. Scuolo Norm. Sup. Pisa 2 (1933).Google Scholar
Vincent, M. St.. Preprint.Google Scholar
Zimmerman, J. E. & Sullivan, D. B.. High-frequency limitations of the double-junction SQUID amplifier. Appl. Phys. Lett. 31 (1977), 360–362.10.1063/1.89699Google Scholar