Find the orbits of the system: $\dot{x} = y(1+ x + y)$, $\dot{y} = -x(1 + x + y)$.

Question

The answer from the book is: The points $x = x_0, y = y_0$ with $x_0 + y_0 = -1$, and the circles $x^2 + y^2 = c^2$, minus these points.

How can you find the orbits? Why do we need to minus the points?

Answer 1

Hint: Observe \begin{align} \dot x y = y^2(1+x+y) \ \ \text{ and } \ \ \dot y x = -x^2(1+x+y)\\ \dot x x = xy(1+x+y) \ \ \text{ and } \ \ \dot y y = -xy(1+x+y) \end{align} which means \begin{align} \dot \theta =& \frac{\dot yx-y\dot x}{r^2} = -\frac{r^2(1+x+y)}{r^2}= -(1+r\cos\theta+r\sin\theta)\\ \dot r =& \frac{x\dot x+y\dot y}{r} = 0 \end{align}