TITLE: Uniqueness of system of ODE - invariant set
QUESTION [0 upvotes]: I have a problem which I am struggling to solve. It goes like:
$$
x^\prime = x- \frac{x^3}{3}-y\\
y^\prime = x - 1 -y 
$$
I need to show for that systen, that there is an unique solution in $(0,\infty)$ for any starting values $(x_0,y_0)\in\mathbb{R}^2$. There is a hint: Show that suitable rectangles are invariant sets.
And I have no clue which rectangle or how to show invariance of the set.
I would be very grateful if someone could help me out.

REPLY [2 votes]: Here's an image of the phase portrait, with an idea that might get you started. The red curve is the $x$-nullcline and the orange line is the $y$-nullcline. Can you see that for a rectangle such as the one in the picture, sufficiently large and with its top-left and bottom-right cornes on the $x$-nullcline, the flow on the upper edge always has a downward component, the flow on the right edge always has a leftward component, and so on?
{"set_name": "stack_exchange", "score": 0, "question_id": 4144679}