Dynamics of Cutting Near Double Hopf Bifurcation

Bifurcation analysis of the orthogonal cutting model with cutting force nonlinearity is presented with special attention to double Hopf bifurcations. The normal form of the system in the vicinity of the double Hopf point is derived analytically by means of center manifold reduction. The dynamics is restricted to a four-dimensional center manifold, and the long-term behavior is illustrated on simplified phase portraits in two dimensions. The topology of the phase portraits reveal the coexistence of periodic and quasi-periodic solutions, which are computed by approximate analytical formulas.