Numerically solve the differential equation found in Problem 4.3: $$l\ddot{\theta} + (g+\ddot{z}) \sin \theta = 0 \tag{8.34}$$

Take the motion of the platform to be periodic, and interactively explore the dynamics of the pendulum as a function of the amplitude and frequency of the excitation.

Let's assume an excitation of the form:

$$z = A\sin{\omega t} \tag{1}$$

We can play with the two parameters $A$ and $\omega$. This system is chaotic because of the unstable point(s) of the pendulum; one at the top if swings all the way, and another at the bottom if the platform is going down, causing the pendulum to swing one left or right. Let's combine equations (1) and (8.34):

$$l\ddot{\theta} + \left(g-A\omega^2 \sin(\omega t)\right) \sin \theta = 0 \tag{2}$$

To solve this numerically, we'll need a 2 DOF state vector:

$$y = \begin{bmatrix}\theta \\ \dot{\theta}\end{bmatrix} \tag{3}$$

We can formulate the problem as:

$$\begin{bmatrix}\dot{\theta} \\ \ddot{\theta}\end{bmatrix} = \begin{bmatrix} \dot{\theta} \\ \left(\frac{A\omega^2}{l} \sin(\omega t) - \frac{g}{l}\right) \sin \theta \end{bmatrix} \tag{4} $$