  We study a system of $N\gg 1$ degrees of freedom coupled via a smooth
homogeneous Gaussian vector field with both gradient and divergence-free
components. In the absence of coupling, the system is exponentially relaxing to
an equilibrium with rate $\mu$. We show that, while increasing the ratio of the
coupling strength to the relaxation rate, the system experiences an abrupt
transition from a topologically trivial phase portrait with a single
equilibrium into a topologically non-trivial regime characterised by an
exponential number of equilibria, the vast majority of which are expected to be
unstable. It is suggested that this picture provides a global view on the
nature of the May-Wigner instability transition originally discovered by local
linear stability analysis.
