论文标题
计算不稳定性指数的大型复杂系统的平衡
Counting equilibria of large complex systems by instability index
论文作者
论文摘要
我们考虑了$ n \ gg 1 $自由度的非线性自主系统，由弛豫（'梯度）和非释放（“ solenatal”）随机相互作用随机耦合。我们表明，随着相互作用强度的提高，这种系统通常会从一个稳定平衡的微不足道肖像发生突然过渡到“绝对不稳定性”的拓扑非平凡的状态，平均平衡是平均丰富的，但通常所有这些都是不稳定的，除非动态均无梯度，否则纯粹是不稳定的。当相互作用进一步增加时，稳定的平衡最终平均变为指数丰富，除非相互作用纯粹是螺旋螺旋。我们进一步计算具有不稳定方向的固定部分的平衡比例。
We consider a nonlinear autonomous system of $N\gg 1$ degrees of freedom randomly coupled by both relaxational ('gradient') and non-relaxational ('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 non-trivial 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 which have a fixed fraction of unstable directions.
