FAQ: What is an example of a system with Re(λ)=0 that is not stable? What if Im(λ) is not zero?
From FBSwiki
Posted by Murray 07:31, 12 November 2005 (PST)
The system
![\left[\begin{matrix} \dot x_1 \\ \dot x_2 \end{matrix}\right] =
\left[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}\right] \left[\begin{matrix} x_1 \\ x_2 \end{matrix}\right]](/~murray/amwiki/images/math/d/e/6/de6c828dc0087dbc8a421a39b82b9d3d.png)
which has two zero eigenvalues, has this solution
which is not stable.
For the second question, we have to go to a larger system, such as
![\left[\begin{matrix} \dot x_1 \\ \dot x_2 \\ \dot x_3 \\ \dot x_4\end{matrix}\right] =
\left[\begin{matrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 2 & 0 & 01 & 0\end{matrix}\right]
\left[\begin{matrix} x_1 \\ x_2 \\ x_3 \\ x_4\end{matrix}\right]](/~murray/amwiki/images/math/b/6/2/b628e53075d51ebc1bd56cf2e76ae4f9.png)
Here the eigenvalues are 
, each with multiplicity 2. The solution is quite complicated (it's on p. 36 and 37 of Perko, Differential Equations and Dynamical Systems) but you can see that you're going to have a similar effect as in the first example, and you'll have terms that grow linearly with time.
