Feigenbaum Constant

My last article was a very short introduction to Chaos theory where I mainly wrote about the Butterfly effect, which is, the concept from where chaos theory began. I had previously discussed about the population graph in one of my articles. I described the graph as a fractal called “the fig tree”. I had also mentioned that fractals was a part of chaos theory. So, how does chaos finally form this graph?

There is a really famous constant that is mentioned along with other famous mathematical constants such as π, sqrt{2}, e, i, etc. I, personally, had never heard of it before, until recently. This constant is called the “Feigenbaum Constant”, it’s value being δ = 4.6692016……., which means it is irrational like π or e. There are two Feigenbaum constants. The other one called is symbolised as α, but, that is another whole story I will not talk about in this article.

Around the 1970s, a scientist named Robert May, wrote a paper in which he had written an equation which modelled the population growth. The equation is as follows:

In this, x_(n+1) is the population next year, x_n is the present population and λ is the fertility. This equation is a logistic map or just a function for population growth. So, basically, using this equation, we can predict what the population is going to be for a community next year. I said that λ is like the fertility of the population. So, if it’s value is high, there is high breeding, but, if it is low, then there is low breeding. The value of λ is between 0 and 1 where, 0 means no breeding and 1 means complete breeding.

Now, the scientists who were interested in population growth iterated this graph to observe the variation of population in the future. In the RHS or Right hand Side of the given equation, x_n is life, while (1 — x_n) is death.

Okay. Let’s now take any value for x_1. Let the it be 0.5, that is let the population be half. I’m taking the value of λ as 2.3.

So, if we calculate the population of the following years using the equation, that is, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_10, x_11, will be

0.575, 0.5621, 0.5661, 0.5649, 0.5653, 0.5652, 0.5652, 0.5652, 0.5652, 0.5652, respectively.

You can observe that the value has become constant. In other words the population growth has stabilised. This is called as the fixed point in the iteration.

What happens if we change λ. Let’s pick a λ that is very small, somewhere between 0 and 1. Let’s say 0.65. Intuitively, it’s obvious what will happen if the fertility is very low. But, let’s still calculate retaining x_1 as 0.5. As I calculated x_2, x_3, x_4….. the following are the values I calculated.

0.1625, 0.0885, 0.0524, 0.0323, 0.0203, 0.0129, 0.0083, 0.0053, 0.0035, 0.0022, 0.0015, 0.0009, 0.0006, 0.0004, 0.0003, 0.0002, 0.0001, 0.0000.

The population is dead.

What would happen if I take a higher fertility value, say, 3.2 ?

I calculated it again with x_1 as 0.5, after many iterations, I noticed that the values were going on as,

0.79946, 0.51304, 0.79946, 0.51304, 0.79946, 0.51304, 0.79946, 0.51304, 0.79946, 0.51304,….. The population is stable, but, stable at 2 values.

Now I’ll take a carefully chosen value of λ, which is, 3.5.

With x_1 as 0.5, again going through the calculations, I noticed that the values, after many iterations, were going on as,

0.87499, 0.38281, 0.82694, 0.50088, 0.87499, 0.38281, 0.82694, 0.50088, 0.87499, 0.38281, 0.82694, 0.50088, 0.87499, 0.38281, 0.82694, 0.50088,…..

This time, the value is stable at 4 values.

Now let’s make graphs out of all the cases we saw.

a) When population became stable

b) When population died

c) When population bounced between two values

d) When population bounced between four values

Now, with the results we have, we’ll plot a graph with λ on the x-axis and the population on the y-axis. The following is what you would get:

When λ = 3.2 we had got two values that were iterating. Thus, you would notice that the graph bifurcates there. ‘Bifurcate’ is just a sophisticated way of saying that the graph forks out. Similarly, at about 3.5, it bifurcates again into four. This goes on but, at a much faster rate. The graph would bifurcate even faster, now, at very small changes of λ itself. After a while, the graph shows something extraordinary as we go further right. But, before that, let me define what I had started this article with, the Feigenbaum constant.

As shown in the above diagram, if I take any two consecutive lengths of each bifurcation of the graph and find its ratio, you would receive a constant irrational value, 4.6692016…….

This is the Feigenbaum’s constant. It’s saying the length of a bifurcation is 4.6692016……. times smaller than the previous one. Feigenbaum found out that if you take any quadratic equation like the population equation, you can create a period doubling graph by just fiddling with the parameters. And, by taking the ratio of the lengths of two consecutive bifurcations, you would get the same number for any quadratic equation.

The following is the fate of the graph after around λ = 3.59.

The graph becomes crazy, or rather, chaotic. Although this graph was discovered before the chaos theory was even known. This constant and graph has thus been used a lot during it’s study. Chaos is sensitive to initial conditions that produces massive changes, as explained by the Butterfly effect. Similarly, here, a very tiny change in λ can cause crazy changes in the graph. Along with the Butterfly effect, this was the beginning of chaos theory.