Stability of roots of polynomial

Question

I'm given the time-dependent polynomial $$p(x_t,x^*_{t-1},t)=x_t^3+ax_t^2+b(t)x_t+c(x^*_{t-1},t),$$ where $a$ is constant, $b(t)$ is a time-dependent parameter and $c(x^*_{t-1},t)$ also depends on the previous root $x^*_{t-1}$. I want to study the stability of its time-dependent roots $x_t^*$, i.e.the solution to $p(x^*_t,x^*_{t-1},t)=0$ as a sequence in time. From numerical simulations I know that the sequence stabilizes if perturbed and converges back to the same value. But I want to prove that this is the case and ideally for what size of perturbations it becomes unstable. Naively, I tried plugging in $x_t\to x_t^*+\varepsilon$ and look at when the first order coefficient of $\varepsilon$ is negative, as this usually does imply linear stability (for ODEs), however I don't know if this is correct. I'm not sure what approach to use, as this isn't a map, i.e. $x_{t+1}=f(x_t)$, or an ODE.