A complicated bifurcation sequence of the inertial shaker near the 1∶4 strong resonance point

The primary objectives of the investigation are to analyze the dynamical behavior of an inertial shaker near the 1:4 strong resonance point by theoretical analyses and numerical simulation. Based on deriving the Poincaré mapping of the inertial shaker, a center manifold theorem technique is applied to reduce the Poincaré mapping to a two-dimensional one, and the normal form mapping associated with the 1:4 strong resonance is obtained. So many bifurcation sequences which can reflect the dynamical behavior of the inertial shaker can be obtained by theoretical analyses. In this paper, a type of the complicated bifurcation sequence of the inertial shaker near the 1:4 strong resonance point is mainly investigated by theoretical analyses and numerical simulation. The results from numerical simulation accord with the ones from theoretical analyses. The results illustrate that there are Neimark-Sacker bifurcation of periodic motion and some complicated bifurcations, such as tangent bifurcations (T_on and T_out types) of period-4 orbits, in the inertial shaker near the 1:4 strong resonance point.