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

We study the mode competition in a Hamiltonian system of two parametrically driven pendulums, linearly coupled by a torsion spring. First we make a classification of all the periodic motions in four main types: the trivial motion, two ‘normal modes’, and a mixed motion. Next we determine the stability regions of these motions, i.e., we calculate for which choices of the driving parameters (angular frequency ω and amplitude A) the respective types of motion are stable. To this end we take the (relatively simple) uncoupled case as our starting point and treat the coupling K as a control parameter. Thus we are able to predict the behaviour of the pendulums for small coupling, and find that increasing the coupling does not qualitatively change the situation anymore. One interesting result is that we find stable (and also Hopf bifurcated) mixed motions outside the stability regions of the other motions. Another remarkable feature is that there are regions in the (A, ω)-plane where all four motion types are stable, as well as regions where all four are unstable. As a third result we mention the fact that the coupling (i.e. the torsion spring) tends to destabilize the normal mode in which the pendulums swing in parallel fashion. The effects of the torsion spring on the stability region of this mode is, suprisingly enough, not unlike the effect of dissipation.