Estimating Critical Hopf Bifurcation Parameters for a Second Order DelayDifferential Equation with Application to Machine Tool Chatter
David E. Gilsinn
Nonlinear time delay differential equations are well known to have arisen in models in physiology, biology and population dynamics. They have also arisen in models of metal cutting processes. Machine tool chatter, from a process called regenerative chatter, has been identified as self sustained oscillations for nonlinear delay differential equations. The actual chatter occurs when the machine tool shifts from a stable fixed point to a limit cycle and has been identified as a realized Hopf bifurcation. This paper demonstrates first that a class of nonlinear delay differential equations used to model regenerative chatter satisfies the Hopf conditions. It then gives a precise characterization of the critical eigenvalues on the stability boundary and continues with a complete development of the Hopf parameter, the period of the bifurcating solution and associated Floquet exponents. Several cases are simulated in order to show the Hopf bifurcation occurring at the stability boundary. A discussion of a method of integrating 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