The stability of the equilibrium position in some critical cases

The stability of the equilibrium position of general systems of differential equations is usually treated as an asymptotic stability criterion in the form of a set of algebraic inequalities on the Taylor coefficients in the right-hand side. A change in sign in any inequality leads to instability. With a change of any inequality to an equality the stability criterion fails and the degree of degeneracy (codimension) of the problem increases by unity. The present analysis deals with the problem of establishing asymptotic stability criteria for the remaining three still unresolved cases where the codimension is equal to 3.