# Can a local extremum of a function be an asymptotically stable equilibrium of corresponding gradient dynamics?

Let's first describe the setup: we consider a(say smooth enough) function $$f: \mathbb{R}^d \to \mathbb{R}$$ and write it as $$(x,y) \to f(x,y)$$, where $$x \in \mathbb{R}^{d_x}$$, $$y \in \mathbb{R}^{d_y}$$ and $$d=d_x+d_y$$. The Hessian of $$f$$ can then be written as $$H=\left( \begin{array}{rr} D_{xx}^2 f(x,y) & D_{yx}^2 f(x,y) \\ D_{xy}^2 f(x,y) & D_{yy}^2 f(x,y) \\ \end{array}\right).$$ On the other hand we consider the gradient dynamics of a game between $$x$$ and $$y$$, where $$x$$ wants to minimize, and $$y$$ to maximize $$f$$. If we write $$w(z)=\left( \begin{array}{rr} -D_x f(x,y) \\ D_y f(x,y) \end{array}\right) ,$$ the corresponding differential equation can be written as $$\dot{z}=w(z)$$. The flow in the neighborhoods of equilibria can then be analyzed by examining the Jacobian of $$w$$: $$J=\left( \begin{array}{rr} -D_{xx}^2 f(x,y) & -D_{yx}^2 f(x,y) \\ D_{xy}^2 f(x,y) & D_{yy}^2 f(x,y) \\ \end{array}\right).$$ My question is now the following: Is it possible, that a critical point of $$f$$ is on the one hand a local minimum of $$f$$, but simultaneously a locally asymptotically stable equilibrium of the gradient dynamics from above. Or in other words: Is it possible, that simultaneously $$H$$ is positive definite and all eigenvalues of $$J$$ have negative real parts at a critical point of $$f$$? I know that positive definiteness of $$H$$ is equivalent to all principal minors of $$H$$ being positive, and the stability of $$J$$ can be analyzed by the Routh-Hurwitz criterion, unfortunately, I couldn't find a (probably very easy) argument or explicit example yet to answer the question. Any help is highly appreciated!