Topology trivialization transition in random non-gradient autonomous ODEs on a sphere

We calculate the mean total number of equilibrium points in a system of N random autonomous ODEs introduced by Cugliandolo et al [17] to describe non-relaxational glassy dynamics on the high-dimensional sphere. In doing it we suggest a new approach which allows such a calculation to be done most straightforwardly, and is based on efficiently incorporating the Langrange multiplier into the Kac-Rice framework. Analysing the asymptotic behaviour for large N we confirm that the phenomenon of 'topology trivialization' revealed earlier for other systems holds also in the present framework with nonrelaxational dynamics. Namely, by increasing the variance of the random 'magnetic field' term in dynamical equations we find a 'phase transition' from the exponentially abundant number of equilibria down to just two equilibria. Classifying the equilibria in the nontrivial phase by stability remains an open problem.