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Generalized Eigenvector in Dynamical System in Infinite Dime...

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Generalized Eigenvector in Dynamical System in Infinite Dime...

Consider a system of linear delay differential equations:

{z1˙(t)=z1(t)+z2(t-1)z2˙(t)=z2(t)+z3(t-1)z3˙(t)=z3(t)-z1(t-1)=⁢˙z1t+⁢z1t⁢z2-t1=⁢˙z2t+⁢z2t⁢z3-t1=⁢˙z3t-⁢z3t⁢z1-t1\begin{cases}\begin{array}[]{l}\dot{z_{1}}(t)=z_{1}(t)+z_{2}(t-1)\\ \dot{z_{2}}(t)=z_{2}(t)+z_{3}(t-1)\\ \dot{z_{3}}(t)=z_{3}(t)-z_{1}(t-1)\end{array}\end{cases}

The characteristic matrix is: Δ(σ)=σId-(Id+Je-σ)normal-Δσnormal-⋅σIdIdJsuperscripteσ\Delta(\sigma)=\sigma\cdot Id-(Id+Je^{{-\sigma}}), where IdIdId is the 3×3333\times 3 identity matrix, and

J=(010001-100)J010001-100J=\left(\begin{array}[]{rrr}0&1&0\\ 0&0&1\\ -1&0&0\end{array}\right)

Clearly, the characteristic equation is: p(σ)=det(Δ(σ))=0pσnormal-Δσ0p(\sigma)=\det(\Delta(\sigma))=0, i.e. p(σ)=(σ-1)3+e-3σ=0pσsuperscriptσ13superscripte3σ0p(\sigma)=(\sigma-1)^{3}+e^{{-3\sigma}}=0. It is easy to see that σ=0σ0\sigma=0 is a characteristic root of algebraic multiplicity 2, as

p(0)=(-1)3+e0=-1+1=0,p0superscript13superscripte0110p(0)=(-1)^{3}+e^{0}=-1+1=0,
p(0)=3(σ-1)2-3e-3σ|σ=0=3-3=0,superscriptpnormal-′03superscriptσ12evaluated-at3superscripte3σσ0330p^{{\prime}}(0)=3(\sigma-1)^{2}-3e^{{-3\sigma}}|_{{\sigma=0}}=3-3=0,

and

p"(0)=6(σ-1)+9e-3σ|σ=0=-6+9=30pnormal-"06σ1evaluated-at9superscripte3σσ06930p"(0)=6(\sigma-1)+9e^{{-3\sigma}}|_{{\sigma=0}}=-6+9=3\not=0

However, when I tried to find the two generalized eigenvectors by solving Δ(0)ϕ2=ϕ1normal-Δ0subscriptϕ2subscriptϕ1\Delta(0)\phi_{2}=\phi_{1}, where ϕ1=(1-11)Tsubscriptϕ1superscript111T\phi_{1}=(1-11)^{T}, and ϕ1subscriptϕ1\phi_{1} is derived by solving Δ(0)ϕ1=0normal-Δ0subscriptϕ10\Delta(0)\phi_{1}=0, I found that the equation Δ(0)ϕ2=ϕ1normal-Δ0subscriptϕ2subscriptϕ1\Delta(0)\phi_{2}=\phi_{1} is inconsistent, i.e., there is no solution!

I did realize that Δ(0)normal-Δ0\Delta(0) is a matrix of rank 2, that is, the null space of Δ(0)normal-Δ0\Delta(0) is only one dimensional. But unfortunately, the null space of (Δ(0))2superscriptnormal-Δ02(\Delta(0))^{2} is one dimensional too! This makes me unable to find ϕ2subscriptϕ2\phi_{2}. I believe I must have missed something, or have misunderstood something. Any comment or suggestion would be highly appreciated!


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