= Solution
{c}
Applying <time reversal of an autonomous differential equation> to the system from part (i), and renaming its coordinates $(x,y)$, gives
$$
\boxed{\widetilde X(x,y)=-(y+1),\qquad \widetilde Y(x,y)=x^2-1.}
$$
The <equilibrium points> remain $(1,-1)$ and $(-1,-1)$. <Time reversal of an autonomous differential equation> preserves a <saddle equilibrium>, interchanging its stable and unstable directions; it also reverses the direction around a <centre equilibrium>. Therefore this system has precisely the required \b[counterclockwise centre at $(1,-1)$ and saddle at $(-1,-1)$].
Back to article page
