A GEOMETRIC ANALYSIS OF STABILITY REGIONS FOR A LINEAR DIFFERENTIAL EQUATION WITH TWO DELAYS

We describe an algorithmic approach for determining the geometry of the region of stability for a linear differential equation with two delays. Numerous applications utilize two-delay differential equations and require a framework to assay stability. The imaginary and zero solutions of the characteristic equation, where bifurcations in stability occur, produce an infinite set of surfaces in the coefficient parameter space. A methodology is outlined for identifying which of these surfaces form the boundary of the stability region. For a range of delays, the stability region changes in only three ways, starting at an identified initial point and becoming more complex as one coefficient increases. Detailed graphical analyses, including three-dimensional plots, show the evolution of the stability surface for given ratios of delays, highlighting variations across delays. The results demonstrate that small changes in the delay ratio cause significant changes in the size and shape of the stability region.