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In this paper, we use finite dimensional ordinary differential equation (ODE) approximations of delay differential equations (DDEs) with variable delays for stability and bifurcation studies. Two different approaches to handle variable delays in these systems have been presented. The first one involves embedding the variable time-delay in a fixed delay interval and is suitable for variable delays whose variation with time is explicitly defined and an upper-bound is known apriori. The efficacy of this procedure is demonstrated for the practical example of turning operation where periodic or random variation of spindle speed is employed for chatter control. For the practical application of a state-dependent delayed model for turning, this approach leads to a system of differential algebraic equations which are not easily amenable to stability and bifurcation studies. Hence, an alternate approach based on mapping the variable delay to a fixed delay using a dynamic scaling of the delayed variable is presented which results in system of ODEs facilitating the stability and bifurcation analysis. In all the cases, an excellent agreement has been achieved between the stability results from the approximation and the analytical/full numerical simulation studies. Bifurcation diagrams have been presented only for the state-dependent delay case where they match the existing results reported in the literature using DDE-biftool.

Stability and bifurcation studies of delayed systems with variable delays using Galerkin projections