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**Problem:** Let $A$ be a square matrix of size $2$ with eigenvalues $\lambda=a \pm ib$ $(b \neq 0)$.

I know that the general solution of the dynamical system $X_k=AX_{k-1}$ with given $X_0$ is given by $X_k=r^kPR_{k\theta}P^{-1}X_0$, where $R_{k\theta}$ is the rotation matrix counterclockwise $k\theta$ degrees and $r=\sqrt{a^2+b^2}$. I just proved this fact myself.

**Need help:** Let $r=1$ and $\theta=s\pi$, where $s$ is a constant. How can I determine if the system is periodic or chaotic?

Thanks for any help.

I have crossposted this question here: differential equations - Behavior of a dynamical system - Mathematics Stack Exchange