Dynamical analysis of a two-parameter family for a vibro-impact system in resonance cases

Dynamics of a two-degree-of-freedom vibro-impact system in resonance is considered. The dynamical model and Poincaré maps are established. When a pair of complex conjugate eigenvalues of the Jacobian matrix cross the unit circle and satisfy the resonant condition λ04=1 or λ03=1, the four-dimensional map is reduced to a two-dimensional one by the center manifold theorem, and the reduced map is put into its normal form by the method of normal form. The two-parameter unfoldings of local dynamical behavior studied in this paper develop the results of one-parameter family analysis. The Hopf and subharmonic bifurcation conditions of period n-1 motion are given. The numerical simulation method confirms the theoretical analysis. It is shown that there exists an invariant torus via Hopf bifurcation and period 4-4 motions via subharmonic bifurcation as two controlling parameters varying near the critical point for the resonance λ04=1, and that there exists an invariant circle and unstable fixed points of order 3 bifurcating from the fixed point, and the system leads eventually to chaos in the resonance λ03=1.