I have a time-varying system
\begin{cases} \dot{x_{1}} = -2t x_{1} + 2x_{1}^{3} x_{2} \sin^{2}(t) \\ \dot{x_{2}} = -t x_{2} + 3x_{1}x_{2}^{7} e^{-t} \end{cases}
My attempt is to first rewrite this one into a linear time-varying system with $x = (x_{1}(t), x_{2}(t))$ of the form:
\begin{equation} \dot{x(t)} = \begin{bmatrix} -2t & 2x_{1}^{3} \sin^{2}(t)\\ 3x_{2}^{7} e^{-t} & - t\\ \end{bmatrix} x \end{equation}
I choose the Lyapunov function of the form $V(x) = \frac{1}{2} || x||^{2}$. However, the derivative of this gives me:
\begin{equation} \dot{V}(t,x) = -t (2x_{1}^{2} + x_{2}^{2}) + 2 x_{1}^{4}x_{2} \sin^{2}(t) + 3x_{1} x_{2}^{8} e^{-t} \end{equation}
I am stuck on dealing with the first term $"(2x_{1}^{2} + x_{2}^{2})"$ because it depends on the sign of $t$ to get the derivative of $V$ is negative. How can I choose a Lyapunov function to prove the stability of the origin for this system?
Note that in the square $-1<x_1<1$ and $-1<x_2<1$ we have the bounds \begin{cases} \ -2t x_{1} - 2\sin^{2}(t)< \dot{x_{1}} < -2t x_{1} + 2\sin^{2}(t) \\ \ -t x_{2} - 3e^{-t}< \dot{x_{2}} < -t x_{2} + 3e^{-t} . \end{cases} So you can bound your solutions with solutions of linear equations in this region.