A good function to fit this data

Question

I'm computing the angle of intersection between to curves (the invariant manifolds of a dynamical system). I do this with a numerical algorithm, but I would like to fit a function with this data.

Apparently the angle of intersection reduces from $\pi$ and then increases as I change a parameter (which is the energy of the system). For example, I get the following data:

$$ \begin{array}{|l|r|}\hline h &\alpha\\\hline 0.0524 &3.14152323\approx \pi\,\text{rad} \\ 0.053 & 3.141592636<\pi\,\text{rad} \\ 0.05556& 1.9204623818\,\text{rad} \\ 0.056008& 1.7608587162\,\text{rad} \\ 0.0571 & 1.8637600112\,\text{rad}\\ \hline\end{array}$$ where $h$ is the parameter (energy of the system) and $\alpha$ the angle of intersection between the curves.

I wonder if anybody can give me an advise as to what type of function use to fit this data. For example a function of four parameters which I have to determine with the data, using four data points.

Thanks.

Answer 1

So you have 2D data [x,y] and want to fit a curve to it. Using zunzun.com I get :

  y = f(x) =  a( x0.5) + b( ln(x)-1) + c( cosh(x) ) + Offset

Note that in input data there is a big difference between $\alpha$ and $h$ range :

$$d_\alpha = 3.141592636-1.7608587162 = 1.3807339198$$

$$d_h = 0.0571-0.0524 = 0.004699999999999996$$

so

$$ d_\alpha/d_h = 293.7731744255321$$

HTH