I asked this on Math SE but received no replies. I hope this is a relevant forum to ask in.
I would like to analyze the spectrum of the following function: $$f(t)=\cos(t\cdot a(1+b\cos(ct)))$$ with $a,b,c$ some real parameters. As the function is non periodic (perhaps unless $a,b,c$ are fine-tuned), the most natural approach is to Fourier transform it. But doing it by hand is quite horrible, Wolfram Alpha aso struggles, and using Mathematica I get $$-ia\cdot\cos(\sqrt{2\pi})\cdot \delta'[\omega] - abc\cdot\cos^2(\sqrt{2\pi})\cdot \delta''[\omega]$$ I'm only interested in the real part of the result. I'm now trying to make sense of what I got.
1) Does the Mathematica result seem reasonable to you?
2) Looking at it from a generator function perspective (as in here), it seems there is a sharp peak centered at zero, accompanied by more minor peaks at $\pm\Omega$ for some $\Omega>0$. I find this result surprising, as I expected that the spectrum would be centered at $a$ as the "main" frequency followed by some smaller harmonics or some sort of $sinc$-like function.
3) Following a much simpler argument than the full Fourier transform, I try to think of it as a function that has a frequency that oscillates in "time" ($t$). Then maybe as an analogy I can ask myself which frequencies are most probable. In that case, I can think of the $\cos(ct)$ term as the cosine of a uniform random variable $ct:= Y\sim Uni(-\pi,\pi)$, and so the probability density function is actually U-shaped centered at $a$ and given by a function in the general form of $$f_Y(x)=\frac{1}{\pi\sqrt{1-x^2}}$$(normalized etc.)
But these two analyses are far off from one another, and I would like to understand what is going on.
A counter-argument to my probabilistic interpretation is that the term that determines the intensity of a frequency component is the coefficients of the Fourier decomposition (if it were periodic), and here this is not a linear combination of frequencies with different probabilities, but something else. Still, I need some help explaining the results. Thanks!