Given the following system of nonlinear ODEs,
$$x_1'=-x_1-x_2$$ $$x_2'=2x_1-x_2^3$$
I need to use the quadratic Lyapunov function
$$V(x) = x^TQx$$
where $Q$ is a positive definite matrix such that
$$A^TQ+QA=-I$$
and where $A=Df(0,0)$, to find a neighbourhood $U$ around the origin, as large as possible, such that $U$ is positively invariant and all solutions starting in $U$ tend to the origin as $t\to\infty$.
I've carefully examined the course notes and the book by Perko (Dynamical Systems and Differential Equations), but couldn't find a clue how to do this problem.
I'd appreciate it very much if someone could please explain:
How is this neighbourhood found?
How is $V$ related to all this?
Write $$\dot{x}=Ax+g(x)=\left[\matrix{-1 & -1\\ 2 & 0}\right]x+\left[\matrix{0\\-x_2^3}\right]$$ The symmetric positive definite solution to $$QA+A^TQ=-I$$ is (by direct calculation) $$Q=\frac{1}{2}\left[\matrix{3 & 1\\ 1 & 2}\right]$$ Now for the Lyapunov function candidate $V=x^TQx$ we have $$\dot{V}=\dot{x}^TQx+x^TQ\dot{x}=(Ax+g(x))^TQx+x^TQ(Ax+g(x))\\ =x^T(QA+A^TQ)x+2x^TQg(x)\\ =-\|x\|^2+2x^TQg(x)\\ \leq -\|x\|^2+2\lambda_{\max}(Q)\|x\|\|g(x)\|$$ Note that $\|g(x)\|=|x_2|^3\leq \|x\|^3$ and therefore $$\dot{V}\leq -\|x\|^2+2\lambda_{\max}(Q)\|x\|^4\\ \leq -\|x\|^2+2\frac{\lambda_{\max}(Q)}{\lambda_{\min}(Q)}V\|x\|^2\\ \leq -\|x\|^2\left(1-2\frac{\lambda_{\max}(Q)}{\lambda_{\min}(Q)}V\right)$$ Thus $\dot{V}< 0$ for all $$V< \frac{\lambda_{\min}(Q)}{2\lambda_{\max}(Q)}$$ with $V\neq 0$ and therefore all trajectories staring within the neighborhood $$U:=\left\{x: V(x)< \frac{\lambda_{\min}(Q)}{2\lambda_{\max}(Q)}\right\}$$ remain in $U$ for all time thereafter and tend to the origin as $t\rightarrow\infty$.