I have this state model: $$\dot{x_1}=-x_1+{x_1}^3+2x_2$$ $$\dot{x_2}=-x_2-x_1^3$$
My Lyapunov candidate is $$x^TPx $$ where P is $$\frac{1}{2}\begin{bmatrix}1 & 1\\1 & 3\end{bmatrix} $$
I know that $$\dot{V}(x)=\dot{x}^TPx+x^TP\dot{x}=2x^TP\dot{x}$$
I don't know how to procede from here, so if anyone could give me a clue or hint, that would be very much appreciated.
Edit:
So I managed to solve this by solving this step first:
$$\dot{V}(x)=\dot{x}^TPx+x^TP\dot{x}=2x^TP\dot{x}$$
$$\begin{bmatrix}x_1 & x_2\end{bmatrix}\begin{bmatrix}1 & 1\\1 & 3\end{bmatrix}\begin{bmatrix}\dot{x}_1\\ \dot{x}_2\end{bmatrix}$$
Then I was able to show, by solving the Lyapunov candidate, that the origin is asymptotically stable.