22 - Spectral Criterion for Asymptotic Stability

Spectral Criterion for Asymptotic Stability

Fixed point ${\mathbf{y}}^{\ast }$ of an autonomous ODE is asymptotically stable if

$$\sigma (\mathbf{Df}({\mathbf{y}}^{\ast }))\subset {\mathbb{C}}^{-}:=\{z\in C:\mathrm{Re}(z)<0\}$$

where $\sigma (\mathbf{Df}({\mathbf{y}}^{\ast }))$ denotes set of eigenvalues of matrix $\mathbf{Df}({\mathbf{y}}^{\ast })$