I am studying a system of the form
$$Q\ddot{x}+F = 0$$
where $Q$ is an $n\times n$ matrix, with entries depending on the dependent variables $x_1,x_2,..,x_n$, while F is an $n\times 1$ vector, and has entries that depend on $x_1,...,x_n,\dot{x}_1,...,\dot{x}_n$. (All variables are a function of time, and dot represents a derivative with respect to time.) Note, solutions to this system exist provided that $Q$ is invertible, i.e. $det(Q)\neq 0$.
Now, I can find equilibrium points, where $F=0$, and these solutions can all be defined by one parameter, which we denote as $\alpha$.
I am examining the behavior of these stationary points under perturbations. Numerically, I observe that for a critical value of $\alpha$, which we denote $\alpha_*$, perturbations with a $\textbf{positive}$ amplitude lead to $det(Q)\to 0$, that is the solutions "blow up" in a certain heuristic sense. For $\textbf{negative}$ perturbation amplitudes, the solutions exist for all integration times I've examined.
My questions are:
1) Is there any way to analytically predict what the value of $\alpha_*$ is? I.e., why does the system bifurcate here?
2) I'd like to better understand the sensitivity to the signature of the perturbation. I can define metrics in phase space, between the different perturbations and the unperturbed state, and use this to quantify what I'm observing, but this doesn't really get at why it's happening. Alternatively, I can try to deduce a nonlinear asymptotic model for the behavior of the perturbation. Is this the way forward here?
I have seen the Krein signature used to deduce the point $\alpha_*$ for linear perturbation analysis (which occurs because of critical points in the energy ( as a function of $\alpha$)), but do not see how that would help in this case.
If this is confusing, or needs more information, let me know and I can clarify.