Nonlinear analogue of the May−Wigner instability transition

Significance 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. We study a system of N≫1 degrees of freedom coupled via a smooth homogeneous Gaussian vector field with both gradient and divergence-free components. In the absence of coupling, the system is exponentially relaxing to an equilibrium with rate μ. We show that, while increasing the ratio of the coupling strength to the relaxation rate, the system experiences an abrupt transition from a topologically trivial phase portrait with a single equilibrium into a topologically nontrivial regime characterized by an exponential number of equilibria, the vast majority of which are expected to be unstable. It is suggested that this picture provides a global view on the nature of the May−Wigner instability transition originally discovered by local linear stability analysis.