Creating one-dimensional maps is a very easy and straightforward process that can be used to explore chaotic behavior.

Given some function $f(x)$ we take an initial value $x_0$ and use the iterative process

$$x_{n+1} = f\left(x_n\right)$$

One popular map to explore is the Logistic Map, defined as

$$x_{n+1} = \mu x_n (1 – x_n)$$

# Cobweb Diagrams

We can construct Cobweb Diagrams using our logistic map and the diagonal $y=x$. These diagrams are made by “bouncing” around between our map and the diagonal to construct “cobwebs”.

The logistic map expresses chaotic behavior for certain values of $\mu$. We can examine “orbits” of this system by looking at what values the map bounces around to. A cobweb diagram is a good way to see these “orbits”.

We can find the parameter values at which the stable period $2^1$, $2^2$, and $2^3$ orbits are first created and label these $\mu_1$, $\mu_2$, $\mu_3$. We’ll use Cython for this process as we need to quickly evaluate a large amount of iterations.

Now we can use this function to find our cobwebs and plot.

And here they are.

# Bifurcation Diagrams

We’ve already noted that the accuracy of finding these bifurcation points was low, let’s instead examine a bifurcation diagram. A bifurcation diagram is essentially a probabilistic view of our map for different values of $\mu$. For the following plots, the $x$-axis is differing values of $\mu$, and the $y$-axis is a large number of plotted values after the transient.

The full bifurcation diagram for the logistic map follows.

Now that we can plot the bifurcation diagram, let’s examine the first several bifurcations.

So what are these values that we’ve found? Let’s record them.

We can plot our cobweb diagrams with these more accurate values.

We could keep recording these values if we wanted, as this will keep going infinitely, and appears to be approaching $\mu_\infty \approx 3.57$.

# Period 3 and 5 Orbits

We found all powers of $2^n$, but we haven’t found any odd-numbered orbits. Let’s track these down.

Now we can perform the same process to find the odd-numbered orbits. I’m not doing 9 different levels of this, because they’re hard to find, and it’s tedious work.

We can see that at each level of odd-numbered orbits, there’s an additional bifurcation series made up of $n \cdot 2 m$, where $n$ is the number of the initial bifurcation $(3, 5, 7, 9, \ldots)$, and $m$ is the next number in the series. This is especially clear for $n=3$. In other words, each odd numbered bifurcation follows the same pattern that the base-$2$ orbits do, they increase exponentially, while converging to a number, and the devolve into chaos as soon as you’re outside their fixed orbit values.

This means that for any $n$, there are an infinite number of corresponding bifurcations.

To show where these values actually occur in the entire bifurcation plot, we can plot our red lines.

It seems that these odd-numbered orbits approach the supercritical point at around around $3.6$.

Let’s look at the implications of these orbits on our cobweb diagram. We need to rewrite our cobweb function slightly, as it’s not precise enough.

Here’s an interactive version to play with.

Here are our $\mu$ values with $f(x)$.

We can clearly see that we have period $\{3, 5, 7, 9\}$ cycles here, but if we plot with $n=3$ we obtain something else entirely.

In the $f^{3}$ case, since we’re plotting $f(f(f(x)))$, our period 3 or bit becomes a fixed point, and all orbits that are share a root of $3$ become the previous orbit. So in this case our period 3 orbit becomes a period 0 (fixed point) orbit, and our period $9$ orbit becomes our period 3 orbit.