Is the Dynamical System Periodic or Chaotic?

[kalish1]

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.

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

 

[jvicens]

Hello,

Thank you for reaching out for help with understanding the behavior of a dynamical system. Based on the information provided, it is difficult to determine the behavior of the system without knowing the specific values of $a$ and $b$. However, there are a few general principles that can help determine if the system is periodic or chaotic.

First, let's define what we mean by periodic and chaotic behavior. A periodic system is one in which the state of the system repeats itself after a certain period of time. In other words, the system will return to a previous state after a certain number of iterations. On the other hand, a chaotic system is one in which small changes in initial conditions can lead to drastically different outcomes. This means that the state of the system is highly sensitive to initial conditions and it is difficult to predict its behavior over time.

Now, let's look at the specific values given in the problem. We know that $r=1$ and $\theta=s\pi$. If $r=1$, this means that the eigenvalues are equal to $1\pm ib$. In this case, the system will have periodic behavior if $b$ is a rational number, and chaotic behavior if $b$ is an irrational number. This is because a rational number can be expressed as a ratio of two integers, and therefore the state of the system will eventually repeat itself after a certain number of iterations. On the other hand, an irrational number cannot be expressed as a ratio of two integers, and therefore the system will exhibit chaotic behavior.

To determine the specific behavior of the system, you can plug in different values for $b$ and observe the behavior over time. Additionally, you can also look at the values of $a$ and $b$ in relation to each other to see if there are any patterns or relationships that can help determine the behavior of the system.

I hope this helps and good luck with your research!