King's College London

TY - JOUR

T1 - Counting equilibria of large complex systems by instability index

AU - Arous, Gérard Ben

AU - Fyodorov, Yan V.

AU - Khoruzhenko, Boris A.

N1 - Publisher Copyright: © 2021 National Academy of Sciences. All rights reserved. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/8/24

Y1 - 2021/8/24

N2 - We consider a nonlinear autonomous system of N ≫ 1 degrees of freedom randomly coupled by both relaxational (“gradient”) and nonrelaxational (“solenoidal”) random interactions. We show that with increased interaction strength, such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically nontrivial regime of “absolute instability” where equilibria are on average exponentially abundant, but typically, all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further, the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. We further calculate the mean proportion of equilibria that have a fixed fraction of unstable directions.

AB - We consider a nonlinear autonomous system of N ≫ 1 degrees of freedom randomly coupled by both relaxational (“gradient”) and nonrelaxational (“solenoidal”) random interactions. We show that with increased interaction strength, such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically nontrivial regime of “absolute instability” where equilibria are on average exponentially abundant, but typically, all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further, the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. We further calculate the mean proportion of equilibria that have a fixed fraction of unstable directions.

KW - Complex systems

KW - Equilibrium

KW - Random matrices

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=85113275797&partnerID=8YFLogxK

U2 - 10.1073/pnas.2023719118

DO - 10.1073/pnas.2023719118

M3 - Article

AN - SCOPUS:85113275797

VL - 118

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 34

M1 - e2023719118

ER -