# Bifurcation diagrams

A bifurcation diagram is yet another geometric tool to help us understand the emergence of chaos in the logistic family - as well as in other parametrized families of functions. It is one of the iconic images of chaos.

## Explanation

Recall the logistic function: $f_r(x) = rx(1-x).$ The symbol $$r$$ is called a parameter. As we've learned, the long term, iterative behavior of the logistic function is very dependent on the parameter. When $$r$$ is small (smaller than $$3$$, in fact), the iterates of $$f_r$$ converge to a single fixed point. Once $$r$$ is larger than 3, the behavior is more complicated.

In order to understand the complications, we generate the bifurcation diagram as follows: Given $$r$$ in a certain range, perhaps $$2\leq r \leq 4$$, we iterate the corresponding logistic function $$f_r(x)=rx(1-x)$$ starting from the point $$x=1/2$$ where the maximum occurs. We iterate a lot - like $$1,000$$ times. We discard the first hundred or so iterates because we don't care about transient behavior. We then plot the remaining points $$(r,y)$$. These points lie in a vertical column with contant horizontal value $$r$$.

Hopefully, this annotated bifurcation diagram helps illustrate this all a bit more clearly:

We can see that as $$r$$ increases, the two cycle bifurcates into a four cycle at around $$r\approx3.45$$. That happens again and again and again as we go through a so called period doubling cascade. That infinite cascade actually completes around $$r\approx3.57$$ and chaos ensues afterward. There are, however, windows of periodic calm interspersed; the most prominent being the period 3 window around $$r\approx 3.832$$