Does the concept of topological equivalence of systems of ODEs with constant coefficients only make sense when the systems have no eigenvalues with real part equal to zero?
I've been checking Arnold's Ordinary Differential Equations and I haven't found a discussion of that case. Is it because it doesn't make sense?
The necessary and sufficient condition of having the same number of eigenvalues for each sign of the real part seems to exclude the possibility to define topological equivalence for such systems. Is there an similar notion for that case?
Sure it makes sense, and the notion of topological equivalence is the same. But it happens much less and that's why it is rarely discussed.
For example, consider the equations $$ x''+x=0\quad\text{and}\quad x''+2x=0. $$ All the eigenvalues of the associated $2$-dimensional systems have zero real part. But the orbits (all periodic other than the origin) of the two systems have different periods and so there is no topological equivalence.
In such cases one sometimes might add the possibility of a time change, such as in bifurcation theory, so that we would still consider the two equations topologically equivalent up to a time change. This is always done in bifurcation theory although it is rarely made explicit in textbooks.