Keep 'er steady

A dynamical system is described by the equation \(\dot{r} = f(r)\) where \(f(r)\) is an \(n^\text{th}\) order polynomial with roots \(\{r_1,\ldots,r_n\}\). In other words, the roots of \(f\) are the steady states of the system (\(\dot{r}(r_i) = 0\)).

Suppose the system is placed into one of the steady states \(r_i\) and is perturbed very slightly away to \(r_i+\Delta r\). How does the perturbation change over time?

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