Boris Khoruzhenko, Queen Mary University of London

*Title: *How many stable equilibria will a large complex system have?

Abstract: In 1972 Robert May argued that (generic) complex systems become unstable to small displacements from equilibria as the system complexity increases. In search of a global signature of this instability transition, we consider a class of nonlinear dynamical systems whereby N degrees of freedom are coupled via a smooth homogeneous Gaussian vector field. Our analysis shows that with the increase in complexity, as measured by the number of degrees of freedom and the strength of interactions relative to the relaxation strength, such systems undergo an abrupt change from a simple set of equilibria (a single stable equilibrium for N large) to a complex set of equilibria. Typically, none of these equilibria are stable and their number is growing exponentially with N. This suggests that the loss of stability manifests itself on the global scale in an exponential explosion in the number of equilibria. [My talk based on a joint paper with Yan Fyodorov and on an unpublished work with Gerard Ben Arous and Yan Fyodorov]