How to approximate a pseudo-periodic curve?

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My data describes a pseudo-periodical behaviour: the amplitude seems stable, but the frequence decreases with X, making the pseudo-period increasing exponentially. What I mean by that is that I measured that for any X I could expect a full pseudo-period to occur within the interval [X, X+exp(aX+b)], where a=7.27e-07 and b=10.12 a detail of my data. I am mostly interested in what happens past 4e5 I would like to approximate this with a pseudo-sinusoid (a function like F:X->C*sin(g(X)) where g is monotonous).

I know that I would need g(X+exp(aX+b)) = g(X)+2*Pi, but I am stuck there.

My goal is then to do something like a Fourier transform (not really, of course, since the function is not truly periodic, but you should get the idea), and see if it gets interesting.