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Consider the following system of equations

$$\begin{cases} x'=-y-x^3 \\ y'=x-y^3 \\ \end{cases} $$ Then, its only equilibrium is $(x^*,y^*)=(0,0)$. Using the Lyapunov function

$$V(x,y)=x^2+y^2$$

one can prove that the equilibrium $(x^*,y^*)=(0,0)$ is asymptotically stable.

I also tried to use linearization, which gives the system $$ \begin{pmatrix} x' \\ y' \\ \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} $$ The matrix has two eigenvalues $\lambda_{1,2}=\pm i$ and their real part is zero. So, as I understand, we can't infer anything about stability of $(0,0)$ from the linearization of the system?

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    $\begingroup$ Yes that is true. When the real parts is zero one cannot deduce stability from linear dynamics alone. $\endgroup$ – Winther Jul 2 '18 at 17:22

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