We can also take a look at the stability of this equation by adding:

In this case for u(t), we have . It is the well known pitch fork bifurcation.

The pitch fork has the following characteristics.

We have taken a few points to check if the numerical results are coherent with the expected values. With an initialization with u=1 on all the domain, the solution converged towards the expected points.

With an sinusoidal initialization we have the following result that indicate the same stability pattern.