So, I have to find all the equilibrium solutions of each of the following systems of equations and determine, if possible, whether they are stable or unstable.
\begin{align*} x' &= \,y + \cos y - 1 \\ y' &= - \sin x + x^3 \end{align*}
And I know already that $y=0$ and $x=0$ is a solution, but I don't know what to do when $(-\sin(x)-x^3)=0$. I thought maybe the Taylor series of $\sin(x)$ might help, but I got nowhere.
Any insight might be great.
From $|\sin x|\le 1$ you get directly $|x|^3\le1\implies |x|\le 1$. Then the sine series due to its alternating nature provides the bounds for $\sqrt6>x>0$ (and symmetrically also for $x<0$) $$ x-\frac16x^3\le\sin x\le x \\~\\ x^3-x\le x^3-\sin x\le \frac76x^3-x $$ This implies a sign change from negative to positive on the interval $[\sqrt{\frac67},1]$, thus a root there. There will be no symbolic formula for the solution $0.9286263..$ of the transcendental equation $0=x^3-\sin x$ in this interval.