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A necessary condition for local asymptotic stability of equilibria of nonlinear systems with parameters

We derive a necessary condition for local asymptotic stability of equilibria of nonlinear systems with parameters. As a corollary of our general result, we deduce Brockett's necessary condition (1983) for local asymptotic stability of equilibria of nonlinear autonomous systems. We show by way of an example that our necessary condition is stronger than Brockett's necessary condition, but retains much of the user-friendly appeal of Brockett's criterion. We further illustrate our result with more examples.