Stationary points of a system of differential equations

Question

I have the following system of differential equations:

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The questions I have to answer are the following:

1) Find the stationary points of the system

2) Draw the vector space of the system using matlab in a rectangle containing the stationary points.For $t\in[0,10]$ make animation of the movement of the point $(x(t),y(t))$ which at time $t=0$ starts from point (x1,y1) which is enetered with the mouse by clicking in on the rectangle.

Now to the point. I have found two stationary points (0,0) and (-2,-2). My first question is: Am I right?

Answer 1

Yes, these are both stationary points. To deduce this analytically, note that if $y$ is stationary then the second equation implies $\dot{y}=e^{y-x}-e^{x-y}=0\implies y=x$. But $x$ itself must be stationary, and then $y=x$ implies $\dot{x}=-2x-2y-2x^2=-4x-2x^2=0\implies x=0,-2$. Therefore $x=y=0$ and $x=y=-2$ are indeed stationary solutions.