DOUBLE HOPF BIFURCATION OF PERIODIC MOTION OF THE MULTI-DEGREE-OF-FREEDOM VIBRATORY SYSTEM WITH A CLEARANCE

A multi-degree-of-freedom vibratory system having symmetrically placed rigid stops and subjected to periodic excitation is considered. Local codimension two bifurcation of the vibro-impact system, concerning two complex conjugate pairs of eigenvalues of linearized map escaping the unit circle simultaneously, is analyzed using the center manifold theorem and normal form method of maps. Local behavior of the system, near the point of double Hopf bifurcation, is investigated using qualitative analysis and numerical simulation. Near the value of double Hopf bifurcation there exist period-one double-impact symmetrical motion, Hopf bifurcation and torus bifurcation. The quasi-periodic impact motions are represented by the closed circle and “tire-like” attractor in projected Poincaré sections. With change of system parameters, the quasi-periodic impact motions usually lead to chaos via “tire-like” tori doubling.