  We consider a nonlinear autonomous system of $N\gg 1$ degrees of freedom
randomly coupled by both relaxational ('gradient') and non-relaxational
('solenoidal') random interactions. We show that with increased interaction
strength such systems generically undergo an abrupt transition from a trivial
phase portrait with a single stable equilibrium into a topologically
non-trivial regime of 'absolute instability' where equilibria are on average
exponentially abundant, but typically all of them are unstable, unless the
dynamics is purely gradient. When interactions increase even further the stable
equilibria eventually become on average exponentially abundant unless the
interaction is purely solenoidal. We further calculate the mean proportion of
equilibria which have a fixed fraction of unstable directions.
