Multi-pulse jumping orbits and homoclinic trees in motion of a simply supported rectangular metallic plate. (English) Zbl 1271.74230
Summary: The global bifurcations in mode interaction of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation are investigated with the case of the 1:1 internal resonance, the modulation equations representing the evolution of the amplitudes and phases of the interacting normal modes exhibit complex dynamics. The energy-phase method proposed by G. Haller and S. Wiggins [Arch. Ration. Mech. Anal. 130, No. 1, 25–101 (1995; Zbl 0829.58016)] is employed to analyze the global bifurcations for the rectangular metallic plate. 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 may occur for this class of systems. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. To illustrate the theoretical predictions, we present visualizations of these complicated structures and numerical evidence of chaotic motions.
Citations:
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