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

Abstract 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 Poincare 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 Poincare 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 T 1 and T 2 bifurcation under some parameter combinations. Numerical simulation results reveal that the vibro-impact system may present different types of complicated invariant tori T 1 and T 2 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.