I have an autonomous system of nonlinear equations of the form:
Mx'' + C(\omega)x' + K(\omega)x + F_{nl}(x) = 0
where $M$ is the mass, $C$ the damping and $K$ the stiffness matrix. The nonlinear force term $F_{nl}$ depends on the solution and is of a piecewise linear shape, where the nonsmoothness occurs at $x=0$
$F_{nl}(x) = \begin{cases} 0 & \text{if }x < 0 \\ 10x & \text{if }x \ge 0 \end{cases}$
The system results from two rings rotating relative to each other at $\omega$. For certain rotational speeds, energy can be transferred from the rotating ring to the stationary ring, resulting in a divergence of the zero solution.
I am trying to study the stability of the zeros solution fixed point ($x=0$). Floquet multipliers won't work as the linearization at the fixed point will only take into account the left or right sided derivative.
Except for doing time simulation with a small perturbation around $x=0$, I am lost as to how to study stability in a more academically sound way. Any help is appreciated!