Interaction of Hopf and period doubling bifurcations of a vibro-impact system

An inertial shaker as a vibratory system with impact is considered. By means of differential equations, periodicity and matching conditions, the theoretical solution of periodic n−1 impacting motion can be obtained and the Poincaré map is established. Dynamics of the system are studied with special attention to interaction of Hopf and period doubling bifurcations corresponding to a codimension-2 one when a pair of complex conjugate eigenvalues crosses the unit circle and the other eigenvalue crosses −1 simultaneously for the Jacobi matrix. The four-dimensional map can be reduced to a three-dimensional normal form by the center manifold theorem and the theory of normal forms. The two-parameter unfoldings of local dynamical behavior are put forward and the singularity is investigated. It is proved that there exist curve doubling bifurcation (a torus doubling bifurcation), Hopf bifurcation of 2–2 fixed points as well as period doubling bifurcation and Hopf bifurcation of 1–1 fixed points near the critical point. Numerical results indicate that the vibro-impact system presents complicated and interesting curve doubling bifurcation and Hopf bifurcation as the two controlling parameters vary.