MODE COMPETITION IN A SYSTEM OF 2 PARAMETRICALLY DRIVEN PENDULUMS - THE ROLE OF SYMMETRY

This paper is the final part in a series of four on the dynamics of tw o coupled, parametrically driven pendulums. In the previous three part s (Banning and van der Weele, Mode competition in a system of two para metrically driven pendulums; the Hamiltonian case, Physica A 220 (1995 ) 485-533; Banning et al., Mode competition in a system of two paramet rically driven pendulums; the dissipative case, Physica A 245 (1997) 1 1-48; Banning et al., Mode competition in a system of two parametrical ly driven pendulums with nonlinear coupling, Physica A 245 (1997) 49-9 8) we have given a detailed survey of the different oscillations in th e system, with particular emphasis on mode interaction. In the present paper we use group theory to highlight the role of symmetry. It is sh own how certain symmetries can obstruct period doubling and Hopf bifur cations; the associated routes to chaos cannot proceed until these sym metries have been broken. The symmetry approach also reveals the gener al mechanism of mode interaction and enables a useful comparison with other systems.