CODIMENSION TWO BIFURCATION AND CHAOS OF A VIBRO-IMPACT FORMING MACHINE ASSOCIATED WITH HOPF-FLIP CASE

A vibro-impact forming machine with double masses is considered. The stability and bifurcation of single-impact periodic motions of the impact-forming system are analyzed based on impact mapping. A center manifold theorem technique is applied to reduce the impact mapping of the system to a three-dimensional one. Local behavior of the normal form mapping, associated with Hopf-flip bifurcation, is analyzed. Dynamical behavior of the impact-forming machine, on the condition of codimension two bifurcation, is investigated using qualitative analysis and numerical simulation. Transition of different forms of fixed points close to the point of Hopf-flip bifurcation is demonstrated, and transition routes from periodical single-impact motion to chaos are also discussed.