Nonlinearity-generated resilience in large complex systems
Sirio Belga Fedeli, Yan V. Fyodorov, J. R. Ipsen
Abstract
We consider a generic nonlinear extension of May's 1972 model by including all higher-order terms in the expansion around the chosen fixed point (placed at the origin) with random Gaussian coefficients. The ensuing analysis reveals that as long as the origin remains stable, it is surrounded by a "resilience gap": there are no other fixed points within a radius r_{*}>0 and the system is therefore expected to be resilient to a typical initial displacement small in comparison to r_{*}. The radius r_{*} is shown to vanish at the same threshold where the origin loses local stability, revealing a mechanism by which systems close to the tipping point become less resilient. We also find that beyond the resilience radius the number of fixed points in a ball surrounding the original point of equilibrium grows exponentially with N, making systems dynamics highly sensitive to far enough displacements from the origin.