QLunch: Kac-Rice fixed point analysis for large complex systems

Speaker: Jesper Ipsen, University of Melbourne

Title: Kac-Rice fixed point analysis for large complex systems

Abstract:
How many fixed points does a large complex system have? We consider a class random nonlinear dynamical systems which allows us to answer this question explicitly. The method used is based on the multi-variate Kac-Rice formula and is valid for sufficiently regular Gaussian functions.

Two main properties of this class of random models will be emphasized: (i) the average number of fixed points is an universal quantity in the sense that it does not depend on the finer details of how the models is constructed, and (ii) these models contain a phase transition between a phase with single fixed point and phase where the number of fixed points grows exponentially with the dimension.