Robustness of stability of linear time-invariant systems using the relationship between the structured compex stability radius and a prametrized algebraic Riccato equation is analysed. Our approach is based on the observation that the algebraic Riccati equation can bie viewed as a bifurcation problem. It ist proves that the stability radius ist, under certain assumptions, associated with a turning point of the bifurcation problem given by the parametrized algebraic Riccati equation. As a byproduct, the stability radius can be computed via path following. Some numerical examples are presented.