Nonlinear dynamics and bifurcations in external feedback control of microcantilevers in atomic force microscopy

We study nonlinear dynamics and bifurcations in external feedback control of vibrating microcantilevers in atomic force microscopy. The efficiency and validity of the control methodology for microcantilevers was demonstrated and abundant nonlinear phenomena were observed in a previous numerical study. Using the averaging method and center manifold theory, we analyze a degenerate bifurcation of codimension two to explain a key feature of the previous numerical results. Numerical examples are also given, in which theoretical results are compared with numerical computations for bifurcations and unstable manifolds.