Linear system of differential equations

A linear differential system has vector form $x\prime =A(t)x+b(t)$.

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 $\lambda =-3$.

For $x^{\prime }-Ax=e^{\lambda t}v$, substituting an exponential particular solution reduces the differential equation to $(\lambda I-A)u=v$. The displayed solution exists whenever $\lambda$ is not an eigenvalue of $A$. Resonance can require polynomial factors in time. The general solution adds $e^{tA}c$, expressed using the matrix exponential.

An eigenvector of $A$ with eigenvalue $\lambda$ gives a homogeneous solution $t^{\lambda }\mathbf{v}$ for $t>0$. If $A$ is diagonalizable, these powers form a fundamental system. A constant vector forcing has a particular solution $t(I-A{)}^{-1}\mathbf{b}$ when $1$ is not an eigenvalue. At a resonant eigenvalue or a nontrivial Jordan block, logarithmic factors in $t$ can occur, equivalently by changing time to $\log t$ and solving a constant-coefficient linear system.

For a constant coefficient system, eigenvectors split the homogeneous equation into scalar exponential or power-law modes.

A state transition matrix maps the state of a homogeneous linear system between two times.

Encoding two planar components as one complex variable turns a rotation-dilation system into a scalar complex equation.