Topology trivialization transition in random non-gradient autonomous ODE's on a sphere

Abstract

We calculate the mean total number of equilibrium points in a system of $N$ random autonomous ODE's introduced by Cugliandolo et al. in 1997 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.

22 pages, no figures

References in corpus (5)

• Topological and Dynamical Complexity of Random Neural Networks
• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• Disorder chaos in the spherical mean-field model
• Matrix optimization under random external fields
• Energy Landscape of the Finite-Size Mean-field 2-Spin Spherical Model and Topology Trivialization

Cited by in corpus (1)

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