Locate the bifurcation values for the oneparameter family an

Locate the bifurcation values for the one-parameter family and draw the phase lines for values of the parameter slightly smaller than, slightly larger than, and at the bifurcation values. dy/dt = y^3 - alpha y^2 + y

Solution

dy/dt = y^3 - ay^2 + y

for the bifurcation points we\'ll solve dy/dt=0

=> y^3 - ay^2 + y = 0
y(y^2-ay+1)=0
y=0 and y = [a+-sqrt(a^2-4)]/2

=> y1=0 , y2 = [a+sqrt(a^2-4)/2 and y3 = [a+sqrt(a^2-4)/2

for three real bifurcation points the domain for a is a E (-infinity , -2) U (2 ,infinity)

and when a E(-2,2) there would be 1 real bifurcation point y =0

and when a=+-2 , we\'ll have two bifurcation points y=0 and y=a/2