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Nonlinear Dynamics

, Volume 30, Issue 2, pp 103–154 | Cite as

Estimating Critical Hopf Bifurcation Parameters for a Second-Order Delay Differential Equation with Application to Machine Tool Chatter

  • David E. Gilsinn
Article

Abstract

Nonlinear time delay differential equations are well known to havearisen in models in physiology, biology and population dynamics. Theyhave also arisen in models of metal cutting processes. Machine toolchatter, from a process called regenerative chatter, has been identifiedas self-sustained oscillations for nonlinear delay differentialequations. The actual chatter occurs when the machine tool shifts from astable fixed point to a limit cycle and has been identified as arealized Hopf bifurcation. This paper demonstrates first that a class ofnonlinear delay differential equations used to model regenerativechatter satisfies the Hopf conditions. It then gives a precisecharacterization of the critical eigenvalues on the stability boundaryand continues with a complete development of the Hopf parameter, theperiod of the bifurcating solution and associated Floquet exponents.Several cases are simulated in order to show the Hopf bifurcationoccurring at the stability boundary. A discussion of a method ofintegrating delay differential equations is also given.

center manifolds delay differential equations exponential polynomials Hopf bifurcation limit cycle machine tool chatter normal form semigroup of operators subcritical bifurcation 

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Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • David E. Gilsinn
    • 1
  1. 1.Mathematical and Computational Sciences DivisionNational Institute of Standards and TechnologyGaithersburgU.S.A

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