Functions from numerical solutions

Question

Here is the context, I have numerically solved a cubic equation which is an irreducible case.

Here is the polynomial I have solved: $P(x)=x^{3}+(1.45-2i\omega)x^{2}+(0.64 \lambda-w^{2}-1.9i\omega+1.06)x + (-0.66 i \lambda \omega + 0.32 \lambda -0.45\omega^{2} + 0.19 i \omega + 0.2425)$

It is the characteristic polynomial of a dynamical system that I am performing stability analysis upon - I am interested when the real part of at least one of the eigenvalues becomes positive.

This has led me to solving for when one of the roots of the system becomes zero, these solutions produce the data in the following figure:

I thought I might be able to find some analytical form to fit this data with. I should note here that I have been able to do this with a slightly different cubic equation, and I thought I could use the solution I found there to inform my fit here. I used that information alongside my knowledge of the system and what we see here (symmetry around 0 with 3 stationary points) to fit the data (blue points) in the above plot with a function (red line).

If it's helpful, here's what I know about the system:

• I have successfully found the analytic form of a solution to a very similar cubic equation, one that this is a variation of, with a quadratic function.
• From the physics of this system, I know that no other stationary points exist and I know it is completely symmetrical around zero.
• The two symmetric minima around 0 are (almost) the positive and negative values of a physical quantity ($\pm\Omega$) that I have set to 1.

My attempt here (one of many) is using a quartic equation which I'm now fairly confident doesn't provide the correct shape despite having all the features.

So my questions are:

• Is it possible for a system such as this to produce smooth behaviour but have no analytical form?
• And if this system should have an analytical form, are there any functions that would produce the features we see here; the stationary points and symmetry, beyond a simple quartic equation?

Answer 1

• Is it possible for a system such as this to produce smooth behaviour but have no analytical form?

Yes! There are plenty of completely smooth functions (in some sense "nearly all" of them), that can not be written down in a nice analytic form. Also, it is not obvious what "nice analytic form" even means. Only basic arithmetic ($+-\times /$)? or also $\sin(x),\cos(x),\exp(x)$? What about infinite series and integrals and all that? where do you draw the line?

• And if this system should have an analytical form, are there any functions that would produce the features we see here; the stationary points and symmetry, beyond a simple quartic equation?

Too many for a brute-force search. So we need some more information about where your data comes from. Some general ideas:

• If you just want to interpolate the numerical data to produce a smooth plot, use splines.
• If you are interested in an approximation that works on some finite domain (such as $[-3,3]$ in your example), use Chebyshev expansion.
• If you are interested in the small neighborhood of a single point, then Taylor expansion is your friend (even a "Taylor expansion around $x=\infty$" can work)

But really, we need more information about your concrete situation. The general problem of going from numerical data to a functional expression is essentially impossible in the general case.