Cost-Effective Robust Stabilization and Bifurcation Suppression

A novel technique of robust stabilization and bifurcation suppression is proposed. The proposed method, the center probe method, stabilizes an equilibrium point of a delay differential equation at a bifurcation point by introducing an impulsive controller that minimizes a given cost functional. The cost functional can weight certain structural properties of the controller, such as the number of nodes controlled (in the stabilization of a complex network). The method takes advantage of the dimension reduction properties of the center manifold, which makes the method notably efficient to implement. A numerical example is provided to demonstrate its effectiveness in suppressing a Hopf bifurcation and robustly stabilizing a nonlinear network model with 100 linearly coupled nodes, while simultaneously keeping the number of controlled nodes to a minimum and minimizing a cost function that assigns higher cost to nodes with higher degree. The strengths and weaknesses of the method compared to other impulsive stabilization techniques are discussed.