How to define an asymmetric oscillatory function with increasing amplitude and period?

Question

I am struggling to model the correlation shown in the figure so that I can predict the positive values of y beyond the observed range:

The correlation between y and x appears oscillatory and characterised by:

1. increasing amplitude;
2. increasing period;
3. asymmetric oscillations.

I was able to define functions that satisfy some of these features, but I could not find a function that satisfies all these features at once. Can you help me do it?

Here is what I was able to accomplish so far, using R:

A function characterised by increasing amplitude

x <- seq(0, 10, by=0.01)
a = 9 # controls amplitude change rate
b = 1.2 # controls frequency change rate
y <- x*b*sin(x*a) # increasing amplitude

The grey line shows what I am attempting to model, the red line is the function in the graph title.

A function characterised by increasing amplitude and period

b = 1.2 # b controls rate of amplitude increase
g = 10
h = 200
# g and h control rate of period increase
y4 <- x*b*sin(exp(g-x)/h) 

This function goes flat by x~7.

The grey line shows what I am attempting to model, the red line is the function in the graph title.

A function characterised by asymmetric oscillations

I was able to model this with a modified Clausen function of order 2:

d=4.5 # changes frequency
f=2 # changes amplitude
clausen2 <- f*sin(x*d) + f*sin(2*x*d)/4 + f*sin(3*x*d)/(3^2) + f*sin(4*x*d)/(4^2) +
f*sin(5*x*d)/(5^2) + f*sin(6*x*d)/(6^2) + f*sin(7*x*d)/(7^2) + f*sin(8*x*d)/(8^2) +
f*sin(9*x*d)/(9^2)

The grey line shows what I am attempting to model, the red line is the function in the graph title.

I am not sure if I am overcomplicating things and/or missing something obvious.

Answer 1

Following your idea of the Clausius function $$S_2(x)=\sum_{m=1}^N \frac{\sin(m x)}{m^2}$$ (cut off at $N$ terms) you can define a function whose amplitude $A(t)$ varies with $t$ and whose frequency, specified in terms of $x=\omega(t) t$, quantifies the rapidness of oscillation, scales with $t$. A first estimate (for constant $\omega(t)$) from your plot guided me to choose $A(t)=1.2 t$, $\omega(t) t=3.2 t$. Moreover, I took $N=20$ terms to obtain

$$f(t)=A(t)S_2(\omega(t) t)$$

given a plot similar to yours.

A version with decreasing frequency $\omega(t)= 20.2 t^{-0.7}$ and increasing amplitude (as in your plot), e.g. $A(t)=1.2 t$, yields

You can numerically try to fit other parametrized functions $A(t)$ and $\omega(t)$ to match your data exactly.

Answer 2

Try to use the following trick. Denote your Clausen function as, say, $g(x)$. Then try to plot something like $g(3.7 x^{0.4}) x^{1.8}/6.5$. (Equivalently, you can change every $x$ in your Clausen function to $3.7 x^{0.4}$ and then multiply the whole expression by $x^{1.8}/6.5$.) This is what I got using Desmos site: https://www.desmos.com/calculator/k86lqwvxvd

Certainly, you are free to play with all the included numbers (i.e., with $3.7, 0.4, 1.8, 6.5$) in order to provide better matching with the reference graph.

Here, the term $x^{1.8}/6.5$ provides the increase of the amplitude, while the modified argument $3.7 x^{0.4}$ of the Clausen function is responsible for the increase of period.