The one-dimensional mapping x(n+1) =ax^2(n)(1 − x(n))

has been used as a toy model for the sunspot cycle.

0 < x(n) < 1 and 0 < a< 6.75; here x(n) is the phase space variable, and a

is a

parameter.

(a) Show algebraically that there are one, two or three fixed points, giving

the values of

corresponding to each case. State the type of bifurcation

that occurs at

a = 4.

(b) Calculate the derivative of the RHS function at the fixed point(s). Hence

show that the fixed point at the origin is always stable. Show further

that the fixed point obtained by taking a negative square root is always

unstable, but that the one that features a positive square root is stable

for 4 < a < 16/3. By noting which stability inequality is violated at

a = 16/3, state what kind of bifurcation occurs at this value, and what

behavior you think will ensue just above it.

This is one of my past exam papers that i'm currently studying but I don't know how to start it off. I've done part a except "state the type of bifurcation at a=4".

If you could give me the steps necessary it would be great :)