Local and Global Bifurcations in Parametrically Excited Nonlinear

Naresh Malhotra, Control & Dynamical Systems, California Institute of Technology

The effect of periodic parametric excitations is discussed on systems that exhibit double-Hopf bifurcation with one-to-one internal resonance along with principal parametric resonance with respect to the excitation frequency. For this purpose a generalized four dimensional, non-linear and non-autonomous system is considered. The linear operator is assumed to have a generic nonsemisimple structure, and the system is simplified considerably by reducing it to the corresponding four-dimensional normal form. The local behavior of the equilibrium solutions is studied along with their stability properties. Several codimension 1 and higher bifurcation varieties are observed using a combination of center manifold and normal form techniques. Some of the global bifurcations that are present, can be associated with the Bogdanov-Takens and Hopf-simple bifurcation varieties. The numerical results indicate the existence of homoclinic orbits along with the period doubling behavior which leads to chaos.

In the second part of the talk, the global dynamics associated with such systems will be analysed in the presence of reversible symmetry. The normal form associated with the reversible systems is obtained as a special case from the general normal form equations obtained previously. Under small perturbations arising from parametric excitations and non-reversible dissipation, two mechanisms have been identified in such systems that may lead to chaotic dynamics. Explicit restrictions on the system parameters have been obtained for both of these mechanisms which lead to complex behavior. Finally, the results are applied on a two-degree-of-freedom model of a thin rectangular beam vibrating under the action of a time-varying follower force. The results are also interpreted in terms of the flexural-torsional motion of the beam structure.