I'm trying to make linearization for the following system:
$y_1'=(1+y_1)\sin y_2, y_2'=1-y_1-\cos y_2$ in the critical point $y_1=y_2=0$.
I'm not sure that I know what I need to do.
I believe that I need to solve the homogeneous system: $(1+y_1)\sin y_2=0, 1-y_1-\cos y_2$=0 so I got that $y=-1$ and $\sin y_2=0$, but the first option is not possible, so I got that $y_2= k\pi $ for all $k \in \mathbb{z}$ and $y_1=2$ or $y_1=0$.
For $y_1=0$, I get that the linear approximation equations are:
$y_1= \epsilon z_1 +o( \epsilon)$, $y_2=\pi k +\epsilon z_2 +o( \epsilon)$, but I don't manage to reach an equation for $z_1$ and $z_2$, Is this the way to do that?
Thanks!