Assume you have a complicated nonlinear ODE system with some parameter $p$. Numerical simulations of the system show, that for any initial conditions and $p$, the solution tends to a periodic function $g_p$. (Possible a limit cycle, as suggested in the comments) A full analytical solution of the system does not seem possible, at this point. However, is it possible to study at least some properties of the periodic solution for a given $p$? More precisely, if I were interested in the average value of the attractor $\left( \bar g = \lim_{t \rightarrow \infty} \frac1t \int_0^t g(s)ds \text{, such that } g = \bar g + \hat g \text{, and } \hat g \text{ is "osillating around zero"} \right)$, or its frequency, could something be found?
Please note, I am not currently interested in proving existence of periodic solution (I only mention because the majority of content I have as of yet found on Google seems to relate to this aspect of the question).
I know I do not provide much technical information. For that reason I am asking for general ideas of what could be tried, similar cases from literature, related papers, other pieces of advice.
EDIT: Also, this is a system of equations. Only some of the components behave in the described way. Others may behave otherwise (for instance, they can grow)