On stability of boundary equilibria in systems with cosymmetry

Using the direct Lyapunov method, we study the stability of an equilibrium of a cosymmetric vector field in the case when the stability spectrum lies in the closure of the left half-plane and the neutral spectrum (lying on the imaginary axis) consists of simple eigenvalues zero and a pair of purely imaginary numbers. Owing to cosymmetry, this equilibrium state is a member of a continuous one-parameter family of equilibria with a variable stability spectrum. We use theorems on asymptotic stability with respect to part of variables. We find stability criteria in the case of general position, as well for all degenerations of codimension one and one case of codimension two. As a result, we give description for dangerous and safe stability boundaries.