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

Question

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 +- ib, with b not equal to 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^k*P*R_(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.

Explanation / Answer

Cosine and sine are periodic functions with period 2?/k.

If ?=s? where s=p/q for p,q?0 and p,q are integers then the period =(2?/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.

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