# Non-Analytic Equations and Chaos

Physics Asked by Evariste on May 8, 2021

Could anyone please tell me an example of an equation with no analytic solution(s) that is not a chaotic one? And what is the physical meaning of having analytic solution? For instance, the three body problem does not have in general analytic solution and it leads to chaos. But I don’t know if this is a general statement. I have absolutely no idea. Could anyone explain me, please?

Could anyone please tell me an example of an equation with no analytic solution(s) that is not a chaotic one ??

A simple fifth order polynomial ($$k_5 x^5 + k_4 x^4 + k_3 x^3 + k_2 x^2 + k_1 x + k_0 = 0$$) has no analytic solution, but is not chaotic.

And what is the physicall meaning of having analytic solution ??

There is no physical meaning. Nature doesn’t care if we have nice functions to describe something.

Answered by Dale on May 8, 2021

Like the following?

$$sin (x) = lambda x tag{1}$$ for $$lambda < 1$$

Or do you want an ODE? You did not specify in the question.

Answered by John Alexiou on May 8, 2021

As a real everyday physical model, consider a non-damping pendulum: ($$ddot{theta} + frac{g}{l}sintheta = 0$$). It does not have a general analytical solution (Wikipedia) and yet it is not chaotic.

Answered by MKMS on May 8, 2021