Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates

We consider a nonlinear autonomous random dynamical system of N degrees of freedom coupled by Gaussian random interactions and characterized by a continuous spectrum n (mu) (lambda) of real positive relaxation rates. Using Kac-Rice formalism, the computation of annealed complexities (both of stable equilibria and of all types of equilibria) is reduced to evaluating the averages involving the modulus of the determinant of the random Jacobian matrix. In the limit of large system N >> 1 we derive exact analytical results for the complexities for short-range correlated coupling fields, extending results previously obtained for the 'homogeneous' relaxation spectrum characterised by a single relaxation rate. We show the emergence of a 'topology trivialisation' transition from a complex phase with exponentially many equilibria to a simple phase with a single equilibrium as the magnitude of the random field is decreased. Within the complex phase the complexity of stable equilibria undergoes an additional transition from a phase with exponentially small probability to find a single stable equilibrium to a phase with exponentially many stable equilibria as the fraction of gradient component of the field is increased. The behaviour of the complexity at the transition is found only to depend on the small lambda behaviour of the spectrum of relaxation rates n (mu) (lambda) and thus conjectured to be universal. We also provide some insights into a counting problem motivated by a paper of Spivak and Zyuzin of 2004 about wave scattering in a disordered nonlinear medium.

Automated summary

We study a nonlinear autonomous random dynamical system with Gaussian random interactions. By computing the average modulus of the determinant of the random Jacobian matrix, we obtain the annealed complexities of stable equilibria and all types of equilibria. For short-range correlated coupling fields, we derive exact analytical results for the complexities in the large system limit, extending previous results for homogeneous relaxation spectrum. We find a "topology trivialisation" transition from a complex phase with exponentially many equilibria to a simple phase with a single equilibrium as the magnitude of the random field decreases. Within the complex phase, the complexity of stable equilibria undergoes an additional transition from a phase with exponentially small probability of finding a single stable equilibrium to a phase with exponentially many stable equilibria as the fraction of gradient component of the field increases. The behavior of the complexity at the transition is conjectured to be universal, depending only on the small lambda behavior of the relaxation rate spectrum. We also provide insights into a counting problem related to wave scattering in a disordered nonlinear medium.