# Behavior of a dynamical system

I need some help understanding the behavior of a dynamical system. Here is the problem:

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.

Assuming that $\theta=s\pi$ is expressed in radians (not in degrees), the system is periodic if and only if $s$ is rational. If $s=p/q$ with $p$ and $q$ integers then the period divides $q$.

• Is my following answer okay? Cosine and sine are periodic functions with period $2\pi/k$. If $\theta=s\pi$ where $s=p/q$ for $p,q \neq 0$ and $p,q \in \mathbb{Z}$, then the period $=(2\pi/k)q/p$. Because each ccw rotation is by some radian measure and number of rotations is always a positive integer, exactly $kp$ such rotations are needed to bring a point back to where it was. – user85362 Feb 24 '14 at 6:47
• By the way, thanks for answering my question. Could you take a stab at this one too? math.stackexchange.com/questions/685654/… – user85362 Feb 24 '14 at 6:49