= Three-state cyclic conversion model
{title2=$\dot A=-aA+cC,\quad\dot B=aA-bB,\quad\dot C=bB-cC$}
Three nonnegative amounts convert in the directed cycle $A\to B\to C\to A$ at positive first-order rates $a,b,c$. The <linear system of ordinary differential equations> conserves $A+B+C=N$. Its <characteristic polynomial> is $\lambda[\lambda^2+(a+b+c)\lambda+ab+bc+ca]$. The two nonzero <eigenvalues> have negative <real part>, so every trajectory converges to the unique <steady state> with that total:
$$
(A_\infty,B_\infty,C_\infty)=\frac{N}{ab+bc+ca}(bc,ca,ab).
$$
This is also a column-convention <continuous-time Markov chain> after normalizing by $N>0$: every component loses to one successor and receives from one predecessor. A complex-conjugate pair produces damped oscillations about equilibrium even though the amounts remain nonnegative. A repeated nonzero <eigenvalue> is defective and requires a <Jordan chain>.
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