Toy exemple : a disk inside a square

Here is an example where we know the viability kernel.

Indeed, let's consider two-dimensional dynamics where the orbit of the points on the circle centered at the origin of the coordinate system and with radius $a$ is the same circle; where the orbit of the points with a norm strictly less than $a$ is a spiral tending towards the origin, and where the orbit of the points with a norm strictly greater than $a$ is a spiral which diverges.

This is the case for the solutions of the equation $x'=f(x)$ with

$$
f(x_1,x_2):=(\frac{||x||-a}{||x||}x_1+x_2 ; -x_1+\frac{||x||-a}{||x||}x_2)
$$   

Let us consider a square as a constraint set  : 

$$
K := \{(x_1,x_2)\in R^2\,|\, |x_1|\leq c \text{ et } |x_2|\leq c \}.
$$

Clearly, the set of points from which trajectories originate that remain within $K$ indefinitely is the disk centered at the origin with radius $a$.

This example makes it possible to test the precision of approximations obtained using generic software.

Below is a comparison, for $a = 0.7$ and $c=1.5$, between the theoretical result (the disk bounded by the red circle) and the approximate result obtained using the ViabLab software on a regular 2001×2001 grid (the green set bounded by the blue line):