Torus T2T2 and its routes to chaos of a vibro-impact system

A three-degree-of-freedom vibro-impact system is considered. The Poincaré map is established and the dynamical behavior of the system, in the case of a codimension-two bifurcation which is characterized by so-called Hopf–Hopf degeneracy, is investigated by theoretical analysis and numerical simulations. The six-dimensional map is reduced to a four-dimensional normal form by the center manifold theorem and theory of normal forms. The two-parameter unfoldings and bifurcation diagrams near the critical point are analyzed. It is proved that there exist the torus T2T2 bifurcation under some parameter combinations. Numerical simulations verify the theoretical solution. The system exhibits complicate and interesting dynamical behaviors such as some heteroclinic orbits bifurcating from fixed point. As the control parameters vary further, the torus T2T2 becomes deformed, folded, and one-frequency phase lock takes place and the system leads eventually to chaos.