Consider the system $$x' = (\epsilon x+2y)(z+1)$$ $$y' = (-x+\epsilon y)(z+1)$$
$$ z' = -z^3$$
Show that the origin is not asymptotically stable when $\epsilon =
0$.
I am told if we start with z = 0, z remains at 0 and the system reduces to $$x' = 2y$$
$$y' = −x$$
(1). My question comes how do we know we need to set z = 0?
This vector field points normal to $\begin{bmatrix} x \\ 2y \end{bmatrix}$ and hence tangent to any ellipse $x^2 + 2y^2 = r^2$ .
(2). Another question is how we derive the ellipse equation here?
Thus given any neighborhood of (0, 0, 0), for r sufficiently small, the solution $$x(t) = \sqrt2 cos t$$ $$y(t) = sin t$$ $$z(t) = 0$$ starts in the given neighborhood and does not approach the origin.
(3). How do we derive the solution to x,y,and z here and how do we see the solution does not approach the origin?