Logistic Equation

f[x_]:=r*x*(1-x);

Calculation of Fixed Points

Solve[f[x]==x,x]

Simplify[D[f[x],x]]

g[x_]:=r-2 r x;

g[0]

Simplify[g[((-1+r)/r)]]

Calculation of Period 2-orbits

Solve[f[f[x]]==x,x]

Simplify[g[((r+(r^2)-r (Sqrt[-3-2 r+(r^2)]))/(2 (r^2)))]*g[((r+(r^2)+r (Sqrt[-3-2 r+(r^2)]))/(2 (r^2)))]]

Solve[4+2 r-(r^2)==1,r]

Solve[4+2*r-r^2==-1,r]

N[1+Sqrt[6]]