Complex Eigenvalue Periodic

Question

This is probably a simple question, but I'm having trouble finding a clear answer. Let's say we have a system, with the complex eigenvalue:

$$\lambda = \alpha + i\beta$$

I know that $\alpha < 0$ means the system is stable, that $\alpha > 0$ means the system is unstable and that $\alpha = 0$ means the system is neutrally stable. The thing is, I need to find out whether the system is periodic. How can I determine this from a given complex eigenvalue?

Answer 1

Assuming that you're facing with a linear time invariant dynamical system, then periodic behaviour occurs when you have a couple of purely imaginary conjugate eigenvalues. That is:

$$\lambda = \pm i\beta, ~\text{with} ~\beta > 0$$

Anyway, when you have

$$\lambda = \alpha \pm i\beta, ~\text{with} ~\beta > 0$$

there are oscillation that can diverge ($\alpha > 0$) or converge ($\alpha < 0$), but in this case we can't speak of a periodic behavior.