Avoiding fold bifurcations with the help of new proximity indices

This work studies the problem of monitoring and avoiding fold (saddle-node) bifurcations of equilibria without solving any difficult optimization problems, such as the closest bifurcation problem. New scalar indices are proposed to monitor the proximity to fold bifurcations. These indices are based on the QR factorization of the Jacobian matrix, and they are directionally differentiable in the interior of the feasibility boundary. This allows us to compute steepest descent directions in the uncontrollable parameter space and steepest ascent directions in the controllable parameter space. The former are used to estimate the distance to the feasibility boundary, while the latter are used by a simple algorithm which designs the controllable parameters in a way that the proximity indices maintain values above a certain threshold, thus avoiding the feasibility boundary. One of the proposed indices is asymptotically linear with respect to the distance to the bifurcation value being approached, resulting in accurate estimates of the distance to the bifurcation.