Global bifurcations and chaos in externally excited cyclic systems

The global bifurcations in mode interaction of a nonlinear cyclic system subjected to a harmonic excitation are investigated with the case of the primary resonance, the averaged equations representing the evolution of the amplitudes and phases of the interacting normal modes exhibit complex dynamics. The energy-phase method proposed by Haller and Wiggins is employed to analyze the global bifurcations for the cyclic system. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case in both Hamiltonian and dissipative perturbations, which imply that chaotic motions occur for this class of systems. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found and the visualizations of these complicated structures are presented.