The equilibrium points are defined
hence the point
is clearly a solution.
Another solution of this system is a point with x=y=a, that is the new system to solve is :
Hence, we find and
For r<1 there is only one equilibrium point 0 because there no solution for
For r>1 there is three equilibrium points 0, X+ and X-
The Matlab calculation of the bifurcation gives, for s=10, b=2.66 and 0<r<30 :
The equivalent diagram for 0<r<5 :
The Matlab program used to calculate this is situated here. For coming back to this part of th report click on Bifurcations and Stability in the upper window.