A Natural Limit Of The Function Cos(x)

A few days ago, while attempting to remain awake in my Calculus class, I discovered something that surprised me.

This is my rough approximation, I know that the structure isn’t quite accurate.
Take the function Cos(n(x)), where n is an integer representing additional functions Cos(x) and Cos is in radians. Therefore Cos(2(x)) equals Cos(Cos(Cos(x)))

The limit of this function as n goes to infinity is:


Now what surprised me is that this appears to be a natural limit (excuse the terms). If x is any real number, the above number is the limit as n->inf.

Even more interesting, it appears (by my somewhat limited Ti-83) that if x is any natural number, the above is the limit if n is less than or equal to sixty.

This seemed remarkably similar to e.

My question is has this already been discovered? Does it have any signifcance? Is it in any way like e, such as Log[sub]Cos*(x) calculations possible?

What you’ve discovered is that the nth interates of the cosine function converge for any real x? The number you say is the limit must then be a fixed point of the cosine function, i.e., an x for which Cos(x)=x. You can approximate this number (which must be irrational since it solves a transcendental equation) on your TI-83 by finding the intersection point of the two graphs y1(x)=cos(x) and y2(x)=x, and you’ll find the same number you gave in your OP.

That should be “iterates”, not “interates”. Same root word as “iteration”.

Interesting. I hadn’t thought of it that way. Incidentally, that x is equal to 42.34 degrees.

42.34 degrees, or 0.739085133215160641655312087 radians if you want to use natural units.

Many pre-calc books have a discussion of orbital diagrams, which give a good visual intuition for the process of iterating functions and finding fixed points. If you no longer have access to such a pre-calc book, this site was the first result of a hasty Google search.

This is a good jumping off point into chaos and fractals. A lot has been done with iterated systems and periodioc points, and a lot of interesting stuff can happen.

In 1975 Li and York published their famous paper Period Three Implies Chaos, describing some of the chaotic behavior exhibited in certain systems containing a point of period three. As far as I know, this was the first time the word “chaos” was applied to such dynamics. Here’s the Period Three Theorem.

Part of the Period Three Theorem was merely a special case of a result previously proven by the Russian mathematician Sarkovskii/Sharkovsky: Sarkovskii’s Theorem.

Many of the fractals you see are constructed by taking a certain function, observing what happens to points in the complex plane by iterating the function at that point indefinitely. “Color” each point in the complex plane different colors, according to how that particular point behaved, and you’ve colored yourself a fractal!

Any of these things may be of interest to you, but this is only scratching the surface.