Abstract
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.
Highlights
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
We introduced a model describing generic large complex systems and examined the dependence of the total number of equilibria in such systems on the system complexity as measured by the number of degrees of freedom and the
Our outlook is complementary to that of May’s in that it adopts a global point of view, which is not limited to the neighborhood of the presumed equilibrium
Summary
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 context of generic systems, the Hartman−Grobner theorem asserts that the neighborhood stability of a typical equilibrium can be studied by replacing the nonlinear interaction functions near the equilibrium with their linear approximations. It is along these lines that May suggested looking at the linear model dyj dt. May considered an ensemble of community matrices J assembled at random, whereby the matrix elements Jjk are sampled from a probability distribution with zero mean and a prescribed variance α2. A detailed review of May’s model in the light of recent advances in random matrix theory can be found in ref. 4
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