Bifurcations and Stability

    The equilibrium points are defined by :  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.