Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators

Harry Dankowicz, Xiaopeng Zhao

Research output: Contribution to journalArticlepeer-review


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 associated 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 impacting 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 presented 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.

Original languageEnglish (US)
Pages (from-to)238-257
Number of pages20
JournalPhysica D: Nonlinear Phenomena
Issue number3-4
StatePublished - Mar 15 2005
Externally publishedYes


  • Discontinuity mappings
  • Grazing bifurcations
  • Impact microactuators
  • Local analysis

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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