I was given the following discrete dynamical system and asked to analyse its behaviour.
$ x_j - \frac{1}{3} (x_{j+1}+x_j+x_{j-1}) + \gamma (x_{j+1}^2 - x_{j-1}^2) = 0 $
where $ \gamma $ is a positive constant and $ x = \pm \alpha $ are its fixed points.
MY ATTEMPT:
I tried to write the system into a one-step two dimensional system; but the quadratic term is giving me a lot of difficulty. I have been given the hint of using the center manifold, but I do not know how to link to it.
Any hints?