Is there any research about a function with changing "period" like sin(1/x)?

Question

I'm encountering a function with a "changing period". It has some sense of period but not exactly. For example: f(x) = cos(1/x). Generally, it has the form of f(x + g(x)) = f(x). I cannot find any information about it. Is there any research about this kind of functions? Thanks!

Answer 1

I second the "chirp" answer. Chirp is the derivative of frequency, just as acceleration is the derivative of velocity and velocity of position. Namely, if we have a periodic function $f_P$ and chirp $c(t)$, then

$$y(t) := f_P\left(t\int_{a}^{t} c(t')\ dt'\right)$$

chirps with that chirp.

The name is pretty good: think about what $\sin\left(\frac{1}{x}\right)$ must sound like (freakie, huh?) - suppose say the "interesting bit" plays for 1 second. It would have to first sound like nothing, then once it rises to a frequency your ear can hear would start low, then VERY fast go way up in pitch to past where you could hear it again, so it would have a very short "wooeeeEEEEEP!" at the beginning, and then after you cross 0, it would come back down (after infinite frequency which, of course, is not possible physically, but also fortunately is not relevant to your ears) making a "PEEEeeeoow!" sound - so all-total you'd get a real short "woeEEEP-PEEEeow!" sound, almost like a birdie chirping - even more if you played a few in succession at slightly randomized close intervals.