Find equilibrium points of this system?

Question

$\dot x_1 = x_1(1 + x_2)$

$\dot x_2 = -x_2 + x_2^2 + (x_1x_2) - x_1^3$

I have set the left hand side to zero to solve and came out with $x_1 = 0$ everywhere and $x_2=0, +1, -1$.

I was wondering if someone could confirm this for me, or provide me a way to check my answers? Thanks :)

Answer 1

Setting the first equation to zero yields $x_1=0$ or $x_2=-1$:

• In the first case this simplifies the right hand side of equation two to $-x_2+x_2^2$. When equating that to zero yields $x_2=0$ or $x_2=1$. So equilibrium points are $(0,0)$ and $(0,1)$.
• In the second case this simplifies the right hand side of equation two to $2-x_1-x_1^3$. When equating that to zero yields $x_1=1$ and two complex values for $x_1$. So the last equilibrium point is $(1,-1)$.