Hopf-Hopf bifurcation and invariant torus T2 of a vibro-impact system

Hopf-Hopf bifurcation of a three-degree-of-freedom vibro-impact system is considered in this paper. The period n-1 motion is determined and its Poincaré map is established. When two pairs of complex conjugate eigenvalues of the Jacobian matrix of the map at fixed point cross the unit circle simultaneously, the six-dimensional Poincaré map is reduced to its four-dimensional normal form by the center manifold and the normal form methods. Two-parameter unfoldings and bifurcation diagrams near the critical point are analyzed. It is proved that there exist the torus T1 and T2 bifurcation under some parameter combinations. Numerical simulation results reveal that the vibro-impact system may present different types of complicated invariant tori T1 and T2 as two controlling parameters varying near Hopf-Hopf bifurcation points. Investigating torus bifurcation in vibro-impact system has important significance for studying global dynamical behavior and routes to chaos via quasi-period bifurcation.