Persistence of equilibria for locally asymptotically stable systems

In this paper, we derive a necessary condition for local asymptotic stabili ty of equilibria of nonlinear systems with parameters. As a corollary of ou r general result, we deduce Brockett's necessary condition for local asympt otic stability of equilibria of nonlinear autonomous systems. The proof we give, however, is quite different from the existing proofs in the literatur e. We show by way of an example that our new necessary condition is stronge r than Brockett's necessary condition, but retains much of the user-friendl y appeal of Brockett's criterion. We further illustrate our result with mor e examples. Copyright (C) 2001 John Wiley & Sons, Ltd.