Nonlinear analogue of the May-Wigner instability transition.
We study a system of [Formula: see text] 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.
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom y.fyodorov@qmul.ac.uk b.khoruzhenko@qmul.ac.uk.
Yan V
YV
Fyodorov
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom y.fyodorov@qmul.ac.uk b.khoruzhenko@qmul.ac.uk.
Boris A
BA
Khoruzhenko
complex systems; equilibrium; model ecosystems; random matrices