Strong Internal Resonance, $Z_2 \oplus Z_2 $ Symmetry, and Multiple Periodic Solutions

Abstract

Equal (strong internal resonance) or nearly equal natural frequencies in a Hamiltonian system are shown to lead to a higher multiplicity and secondary branching of periodic solutions. The role of $Z_2 \oplus Z_2 $ symmetry is emphasized as a basis for the analysis. The Lyapunov–Schmidt method is used to generate a set of bifurcation equations. The higher order terms in the bifurcation equations are formally neglected, leading to the basic normal form for coupled equations with $Z_2 \oplus Z_2 $ symmetry. Known results for this normal form are used to determine the nature of the periodic solutions in a neighborhood of a 1:1 resonance. A stability analysis is based on Floquet theory and the Lyapunov–Schmidt method. Regions of stability are established and particular singular points in parameter space are found where periodic solutions may not exist. The analysis is carried out on an example: the orthogonal planar pendulum.