FAQ: What is an example of a system with Re(λ)=0 that is not stable? What if Im(λ) is not zero?
Posted by Murray 07:31, 12 November 2005 (PST)
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
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.