Estimating Critical Hopf Bifurcation Parameters for a Second-Order Delay Differential Equation with Application to Machine Tool Chatter |
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Authors: | Gilsinn David E. |
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Affiliation: | (1) Mathematical and Computational Sciences Division, National Institute of Standards and Technology, 100 Bureau Drive, Stop 8910, Gaithersburg, MD, 20899-8910, U.S.A |
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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. |
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Keywords: | 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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