Keep 'er steady

A dynamical system is described by the equation r˙=f(r)\dot{r} = f(r) where f(r)f(r) is an nthn^\text{th} order polynomial with roots {r1,,rn}\{r_1,\ldots,r_n\}. In other words, the roots of ff are the steady states of the system (r˙(ri)=0\dot{r}(r_i) = 0).

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

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