Question

For each of the following equations you should locate all equilibrium points and investigate their stability properties for different values of the parameter u ER. You should then draw a summary/bifurcation diagram, with arrows, indicating the qualitative behaviour of the non-equilibrium solutions for all values of y, including at the bifurcation point u = u*. Note any bifurcations that occur and determine which of the following terms can be used to describe the nature of the bifurcations you find: transcritical, subcritical, super- critical, pitchfork, saddle-node/fold. Show that the relevant conditions in the three theorems discussed in the course notes hold for each bifurcation point you find.

Answer 1

solution 8 Given that Let f(x) =(x-1) (4+ (+) Let y = x-1. This geven equation becomes Ý =ý (Mtoty? Let f (y) = y put i+y3 9=0 => Y=0 + y = Jiu f(Y) = (4+1) + 3y2 when illi-t, then there are sfixed points

.y=0 (X=1) ys + Fi-ue :(4x = 10-i-ue) t'co) Eltico, so yoo is stable fi I T ) =M+1+36-1-4)=-2-2470 1 so, Isoleu as unstable When Myl, there is only are field print y=0 (in = 4), which is unstable,

so at M = utal, there is Pitchfork, bituscation. +* HA - stable HH unstabde. 9HHHHHHHHH Eelari bifoscation diagram O-