Nonlinear analogue of the May-Wigner instability transition

Complex systems equipped with stability feedback mechanisms become unstable to small displacements from equilibria as the complexity (as measured by the interaction strength and number of degrees of freedom) increases. This paper takes a global view on this transition from stability to instability. Our examination of a generic dynamical system whereby N degrees of freedom are coupled randomly shows that the phase portrait of complex multicomponent systems undergoes a sharp transition as the complexity increases. The transition manifests itself in the exponential explosion of the number of equilibria, and the transition threshold is quantitatively similar to the local instability threshold. Our model provides a mathematical framework for studying generic features of the global dynamics of large complex systems.