MODE COMPETITION IN A SYSTEM OF 2 PARAMETRICALLY DRIVEN PENDULUMS - THE HAMILTONIAN CASE

We study the mode competition in a Hamiltonian system of two parametri cally 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 d etermine the stability regions of these motions, i.e., we calculate fo r which choices of the driving parameters (angular frequency Omega and amplitude A) the respective types of motion are stable. To this end w e take the (relatively simple) uncoupled case as our starting point an d treat the coupling K as a control parameter. Thus we are able to pre dict the behaviour of the pendulums for small coupling, and find that increasing the coupling does not qualitatively change the situation an ymore. One interesting result is that we find stable (and also Hopf bi furcated) mixed motions outside the stability regions of the other mot ions. Another remarkable feature is that there are regions in the (A, Omega)-plane where all four motion types are stable, as well as region s where all four are unstable. As a third result we mention the fact t hat the coupling (i.e. the torsion spring) tends to destabilize the no rmal mode in which the pendulums swing in parallel fashion. The effect s of the torsion spring on the stability region of this mode is, surpr isingly enough, not unlike the effect of dissipation.