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

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.