Showing that a system has a unique steady state at $(0, 0).$

Question

Consider a system:

$$ dx/dt = y + x(2 − x^2 − y^2 ), $$ $$dy/dt = −x + y(1 − x^2 − y^2)$$

(i) Show that the system has a unique steady state at $(0, 0).$

My immediate thought is to simply plug $(x,y) = (0,0)$ into the above system and simply show we'd have $dx/dt = 0$ & $dy/dt = 0$, however this question is worth 6 marks leading me to believe there must be more work to be done that this seemingly obvious solution.

Do we perhaps have to linearise the system first?
Any thoughts or advice would be great, thanks!

Answer 1

You do need to verify that $dx/dt$ and $dy/dt$ are both zero at $(x,y)=(0,0)$ (which you can do by plugging in), but you also need to verify that this is the only point at which $dx/dt$ and $dy/dt$ are both zero. To do this, set both $dx/dt$ and $dy/dt$ equal to zero and manipulate those equations to show that $x=y=0$.

One method of proving that is as follows.

You have $y+x(2-x^2-y^2)=0$ and $-x+y(1-x^2-y^2)=0$. Multiplying the first equation by $y$ and the second by $x$ yields $y^2+xy(2-x^2-y^2)=0$ and $-x^2+xy(1-x^2-y^2)=0$. Subtracting the second equation from the first gives $x^2+y^2+xy=0$. Now complete the square to argue that $x=y=0$.