I have the following system:
$$ \begin{cases} \dot{x_{1}}(t) = x_{2} - x_{1}^{3}\\ \dot{x_{2}}(t) = -0.5 x_{1} - x_{2} + d\sin(t) \end{cases} $$
where $d$ is some constant.
I want to show $(0,0)$ is asymptotically stable. I am trying to find the coefficients with the candidate $V(x) = A x_{1}^{2} + B x_{2}^{2}$. I find that $A = 1$ and $B = 2$.
This gives me $V(x) = x_{1}^{2} + 2x_{2}^{2}$.
Taking the derivative I get:
$$ \dot{V} = -2(x_{1}^{4} + 2 x_{2}^{2}) + 4d \sin(t) $$
I stuck with the term that is dependent on $t$.
Any hints or ideas I could get rid of the term dependent on $t$?