# Proving a set is positive invariant for a dynamical system

I have the following dynamical system:

\begin{align} \dot{x}&=-x-2y^2, \\ \dot{y}&=-x^2y-y^3. \end{align} My task is to show that, for the dynamical system, the set $$S=\left\{ (x,y) \in \mathbb{R}^2:x \leq0 \right\}$$ is positive invariant.

My first thought is to use a Liapunov function defined by $$L:S \to \mathbb{R}, \: \: L(x,y)=-x,$$ which is positive definite. However, calculating $\dot{L}$ gives

$$\dot{L}=L_x\dot{x}+L_y\dot{y}=-\left(-x-2y^2\right)=x+2y^2,$$ from which I cannot seem to deduce anything.

Any help would be great!

• Might not simplified to something nice, but maybe try converting the system into polar coordinates? – Chee Han May 10 '17 at 21:01

Hint: Check how vector field points along the boundary of your domain of interest. If it points inside, then Bony-Brezis theorem can be applied. Or you just can say that vector field along the boundary doesn't let trajectories go out of domain $x \leqslant 0$.
• Oh right okay, so because on $x=0$, $\dot{x}=-2y^2<0$ for all $y$, the vector field will be pointing to the left, so trajectories can't 'escape' once they're in the domain $x \leqslant 0$? – Will May 12 '17 at 17:03