Three-state cyclic conversion model ($\dot{A}=-aA+cC,\dot{B}=aA-bB,\dot{C}=bB-cC$)

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: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.

When the displayed equality holds, the two nonzero eigenvalues of the three-state cyclic conversion model coincide at $\lambda =-(a+b+c)/2$. The positive cyclic rates force each eigenspace to have dimension at most one: if its $A$ coordinate vanished, the first eigenvector equation would force $C=0$, and then the third would force $B=0$. Thus the repeated eigenvalue has only one eigenvector direction and the matrix is not diagonalizable. A length-two Jordan chain produces both $e^{\lambda t}$ and $t{e}^{\lambda t}$ transients. The rates $(a,b,c)=(1,4,1)$ give an explicit example with $\lambda =-3$.