Repelling or attracting spiral phase portrait in canonical basis

Question

If matrix $\bf{A}$ of a system $\bf{x}'=\bf{A}x$ (*) has only complex eigenvalues and eigenvectors with non-zero real parts, and we make the substitution $\bf{y}'=\bf{B}x$ (**), where $\bf{B=P^{-1}AP}$, then will the phase portrait of the spiral be the same in case (*) and case (**)?

Answer 1

If you extend the first system to $\mathbb C^n$, the phase portraits will be the "same" topologically and differentiably. The fact that the eigenvectors have nonzero real parts is irrelevant, although of course unavoidable. Note: This answer assumes that your first matrix has only real entries.