Take the logistic map
$$x(i+1) = f(xi), f(x) = rx(1 − x)$$
Find numerically the $r$ values for the first two bifurcations.
I know I have to set $f(x)=0$, solve for $x$, and then use the $x$ values to find $r$. So, in Matlab I did
syms x r;
solve(r*x*(1-x)==0,r);
From this I've found that $r = 0$ and $r = 1$. Is this correct? If not, what did I do wrong here? Or, just help me find the bifurcation values here.
The bifurcations occur for the Logistic map at the points when the period doubles.
For instance, suppose that $b$ is the first bifurcation point. Then, for $b<r$ the Logistic map will have a fixed point (i.e. period-1 point) and for $b>r$ it will have period-2 points. You need to find the particular value for $b$ that makes this true.
Similarly, the second bifurcation point occurs in the change from period-2 to period-4 points.
Hope that helps!