I have this equation $y_1' = y_1 * (1 + y_1^2 + y_2^2),\, y_2' = y_2 * (1 - y_1^2 -y_2^2)$ and I need to prove that if I take a initial point from the set $ \Gamma = \left \{ (y_1,y_2) \in \mathbb{R} : y_1^2 + y_2^2 > 1 \right \}$ then its orbit is contained in the same set.
I know I have to use the proposition that says that two orbits can never intersect but I have tried calculating the orbits of initial points taken in the frontier of $\Gamma$ and they are vertical lines, so I don't know what will happen if I select $y_0$ just over the circle.