Stability at the linearization boundary

Determine the nonlinear stability behavior of the separating solution for the equivariant relative-gradient ICA mean dynamics in the borderline cases where some diagonal stability parameter satisfies γ_j=1 or where ζ_jζ_l=1 with ζ_j,ζ_l>0.

Background

The local stability theorem gives sufficient conditions for the separating solution R=I to be locally asymptotically stable and proves instability when certain inequalities are reversed strictly.

The theorem's linearization does not decide the boundary cases γ_j=1 and ζ_jζ_l=1 with positive ζ_j and ζ_l. The paper explicitly identifies these cases as unresolved by the linear analysis, leaving their nonlinear stability behavior to be determined.