Can someone explain to me the meaning of "real parts" in the conext of eigenvalues?

Question

While I'm busy trying to understand how to evaluate dynamical systems with help of the Jacobian matrix I got stuck on some terminology.

I'm struggling with the meaning of this paragraph in found in Wikipedia

Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point, if any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, the Jacobian matrix does not allow for an evaluation of the stability.

What are "real parts"? My first language is neither English, nor Math ;-) so I'd appreciate it if you could explain it without an avalanche of new terms and maybe with an example.

Thanks in advance.

Answer 1

If the eigenvalues are expressed as $e_i=a_i+ib_i$ with $a_i, b_i$ real, the real parts are $Re(e_i)=a_i$. The point is that the eigenfunctions are of the form $e^{e_i}=e^{(a_i+ib_i)t}$ If $a_i$ is negative, the function decays away to zero. $b_i$ represents an oscillatory behavior, but if $a_i \lt 0$ at long times you won't see it because of the decay.