Nonlinearity-generated Resilience in Large Complex Systems

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.

arXiv comment17 pages, 3 figures

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• Generalized Lotka-Volterra equations with random, non-reciprocal interactions: the typical number of equilibria
• Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates
• Quenched complexity of equilibria for asymmetric Generalized Lotka-Volterra equations
• Dynamically selected steady states and criticality in non-reciprocal networks
• Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems
• Glass--like transition described by toppling of stability hierarchy
• Counting equilibria of large complex systems by instability index