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Mode competition in a system of two parametrically driven pendulums; the dissipative case

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  • Banning, E.J.
  • van der Weele, J.P.
  • Ross, J.C.
  • Kettenis, M.M.
  • de Kleine, E.

Abstract

In this paper we study the dynamics of a system of two linearly coupled, parametrically driven pendulums, subject to viscous dissipation. It is a continuation of the previous paper (E.J. Banning and J.P. van der Weele (1995)), in which we treated the Hamiltonian case. The damping has several important consequences. For instance, the driving amplitude now has to exceed a threshold value in order to excite non-trivial motion in the system. Furthermore, dissipative systems (can) exhibit attraction in phase space, making limit cycles, Arnol'd tongues and chaotic attractors a distinct possibility. We discuss these features in detail. Another consequence of the dissipation is that it breaks the time-reversal symmetry of the system. This means that several, formerly distinct motions now fall within the same symmetry class and may for instance annihilate each other in a saddle-node bifurcation. Implications of this are encountered throughout the paper, and we shall pay special attention to its effect on the interaction between two of the normal modes of the system.

Suggested Citation

  • Banning, E.J. & van der Weele, J.P. & Ross, J.C. & Kettenis, M.M. & de Kleine, E., 1997. "Mode competition in a system of two parametrically driven pendulums; the dissipative case," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 245(1), pages 11-48.
    • Handle: RePEc:eee:phsmap:v:245:y:1997:i:1:p:11-48
    DOI: 10.1016/S0378-4371(97)00276-8
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    References listed on IDEAS

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    1. Banning, E.J & van der Weele, J.P, 1995. "Mode competition in a system of two parametrically driven pendulums; the Hamiltonian case," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 220(3), pages 485-533.
    2. Banning, E.J. & van der Weele, J.P. & Ross, J.C. & Kettenis, M.M., 1997. "Mode competition in a system of two parametrically driven pendulums with nonlinear coupling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 245(1), pages 49-98.
    3. Banning, E.J. & van der Weele, J.P. & Kettenis, M.M., 1997. "Mode competition in a system of two parametrically driven pendulums; the role of symmetry," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 247(1), pages 281-311.
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    Cited by:

    1. Banning, E.J. & van der Weele, J.P. & Kettenis, M.M., 1997. "Mode competition in a system of two parametrically driven pendulums; the role of symmetry," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 247(1), pages 281-311.
    2. Banning, E.J. & van der Weele, J.P. & Ross, J.C. & Kettenis, M.M., 1997. "Mode competition in a system of two parametrically driven pendulums with nonlinear coupling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 245(1), pages 49-98.

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    1. Banning, E.J. & van der Weele, J.P. & Ross, J.C. & Kettenis, M.M., 1997. "Mode competition in a system of two parametrically driven pendulums with nonlinear coupling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 245(1), pages 49-98.
    2. Banning, E.J. & van der Weele, J.P. & Kettenis, M.M., 1997. "Mode competition in a system of two parametrically driven pendulums; the role of symmetry," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 247(1), pages 281-311.

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