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:

$\begin{cases}\begin{array}[]{l}\dot{z_{1}}(t)=z_{1}(t)+z_{2}(t-1)\\ \dot{z_{2}}(t)=z_{2}(t)+z_{3}(t-1)\\ \dot{z_{3}}(t)=z_{3}(t)-z_{1}(t-1)\end{array}\end{cases}$ |

The characteristic matrix is: $\Delta(\sigma)=\sigma\cdot Id-(Id+Je^{{-\sigma}})$, where $Id$ is the $3\times 3$ identity matrix, and

$J=\left(\begin{array}[]{rrr}0&1&0\\ 0&0&1\\ -1&0&0\end{array}\right)$ |

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

$p(0)=(-1)^{3}+e^{0}=-1+1=0,$ |

$p^{{\prime}}(0)=3(\sigma-1)^{2}-3e^{{-3\sigma}}|_{{\sigma=0}}=3-3=0,$ |

and

$p"(0)=6(\sigma-1)+9e^{{-3\sigma}}|_{{\sigma=0}}=-6+9=3\not=0$ |

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

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

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## Re: Generalized Eigenvector in Dynamical System in Infinite ...

It looks like you are trying to use Latex, but it doesn't seem to be working. It is almost unreadable.