# Counting of equilibria of large complex systems by instability index

## Importance

This paper sheds light on the phase portrait structure of a “minimal model” for a large nonlinear system of many interacting random degrees of freedom equipped with a stability feedback mechanism. In this way, it significantly extends the analysis of local stability of large complex systems, undertaken by Robert May in 1972. We show that the transition from stability to instability is characterized by the exponential explosion of the number of unstable equilibria, with considerably reduced probability. to find truly attractive balances locally. At the same time, we demonstrate an abundance of equilibria with a large proportion of stable directions, which can arguably slow down the dynamics of the system for long enough and induce aging effects.

## Abstract

We consider a nonlinear autonomous system of

$\mathrm{NOT}??1$degrees of freedom coupled randomly by relaxational (“gradient”) and non-relaxing (“solenoids”) random interactions. We show that with increased interaction strength, such systems generally undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium to a topologically non-trivial regime of “absolute instability” where the equilibria are on average. exponentially abundant, but generally all are unstable, unless the dynamics are purely gradient. As interactions increase even more, stable equilibria end up becoming on average exponentially abundant unless the interaction is purely solenoid. We further calculate the average proportion of equilibria which have a fixed fraction of unstable directions.

## Footnotes

- ??
^{1}To whom correspondence can be addressed. E-mail: benarous {at} cims.nyu.edu, yan.fyodorov {at} kcl.ac.uk, Where b.khoruzhenko {at} qmul.ac.uk.

## Data availability

There is no data underlying this work.