Analysis of system structure around degenerate critical points

In this paper, we state the new problem of system structures around degenerate critical points in the viewpoint of control engineering. Then, two fundamental schemes are presented to construct a weak Lyapunov function systematically. The procedures are based on the following deformations. One of them is a collapsing residual singularity defined by the formal local coordinate of Thorn´s splitting lemma. The other is a blowing-up for the resolution of singularity