# Bifurcations in maps

1. May 16, 2012

### Rubik

1. The problem statement, all variables and given/known data
Prove carefully that the following map has a horseshoe for one positive value of $\mu$
xn+1 = xn2 - $\mu$
For what positive values of $\mu$ will the map have a horseshoe?
2. Relevant equations

3. The attempt at a solution
xn+1 = xn2 - $\mu$ = g(xn,$\mu$)
∂g/∂x = 2x
→ turning pt at 2x = 0 → turning pt at x=0

Intersects the x axis at 0 = x2 - $\mu$
x = ±√$\mu$

I am not even sure if I am on the right track at all. It feels very wrong so any help would be greatly appreciated