Numerical continuation, and computation of normal forms

In this paper, we analytically investigate the global dynamics associated with the nonlinear reversible systems that exhibit Hopf bifurcation in the presence of one-to-one nonsemisimple internal resonance. The effect of periodic parametric excitations is examined on such systems near the principal subharmonic resonance in presence of dissipation. The nonlinear and nonautonomous system is simplified considerably by reducing it to the corresponding four-dimensional normal form. The normal form associated with the reversible systems is obtained as a special case from the general normal form equations obtained in [N. Sri Namachchivaya, M.M. Doyle, W.F. Langford and N. Evans, Normal form for generalized hopf bifurcation with non-semisimple 1:1 resonance, Z. Angew. Math. Phys. (ZAMP) 45 (1994) 312–335]. Under small perturbations arising from parametric excitations and nonreversible dissipation, two mechanisms are identified in such systems that may lead to chaotic dynamics. Explicit restrictions on the system parameters are obtained for both of these mechanisms which lead to this complex behavior. Finally, the results are demonstrated through a two-degree-of-freedom model of a thin rectangular beam vibrating under the action of a pulsating follower force.