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 λ 0 4 = 1 or λ 0 3 = 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 λ 0 4 = 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 λ 0 3 = 1 .