Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators Harry Dankowicz and Xiaopeng Zhao Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA Abstract Impact microactuators rely on repeated collisions to generate gross displacements of a microelectromechanical machine element without the need for large applied forces. Their design and control rely on an understanding of the critical transition between non-impacting and impacting long-term system dynamics and the associ- ated changes in system behavior. In this paper, we present three co-dimension-one, characteristically distinct transition scenarios associated with grazing conditions for a periodic response of an impact microactuator: a discontinuous jump to an impact- ing periodic response (associated with parameter hysteresis), a continuous transition to an impacting chaotic attractor, and a discontinuous jump to an impacting chaotic attractor. Using the concept of discontinuity mappings, a theoretical analysis is pre- sented that predicts the character of each transition from a set of quantities that are computable in terms of system properties at grazing. Specifically, we show how this analysis can be applied to predict the bifurcation behavior on neighborhoods of two co-dimension-two bifurcation points that separate the co-dimension-one bifurcation scenarios. The predictions are validated against results from numerical simulations of a model impact microactuator. Key words: Impact microactuators, grazing bifurcations, local analysis, discontinuity mappings PACS: 1 Introduction Precise displacement control and manipulation is required in microscopes, optical devices, and nanoscale data storage as well as during microsurgery. Adhesive forces, such as electrostatic forces, van der Waals forces, and surface Preprint submitted to Physica D 28 February 2005