The new chaotic pendulum

A more detailed study of the chaotic pendulum
Nondimensionalization

The chaotic pendulum's equations (1) and (2) being quite complex (they involve six parameters), we will first try to simplify them by nondimentionalization ^[ref].

First of all, we can recognize that the term $\frac{g}{l}$ is the equivalent of a squared frequency. We can thus write : $$\sqrt{\frac{g}{l}}=\omega$$ The equations $(1)$ and $(2)$ then become :
$$ \begin{eqnarray} \ddot{\theta}&=&\dot{\varphi}^2\sin\theta\cos\theta+\omega ^2(\cos\theta\sin\varphi\sin\alpha-\sin\theta\cos\alpha) \\ % \ddot{\varphi}&=&\frac{\sin\theta(\omega ^2\cos\varphi\sin\alpha-2\dot{\theta}\dot{\varphi}\cos\theta)}{\sin^2\theta+\frac{MR^2}{ml^2}} \end{eqnarray} $$ As we did for $\omega$, we will consider that $\beta = \frac{MR^2}{ml^2}$ is a control parameter. It represents the report between the disc's and the mass $m$'s moments of inertia

Finally, we will also redefine the scale of time. To achieve that, all we have to do is to pose $\tilde{t}=\frac{t}{\tau}$ , where $\tau = \frac{1}{\omega}$ is the chaotic pendulum's characteristic time.
We get the nondimensionalized equations :

$$ \begin{eqnarray} % \ddot{\theta}&=&\dot{\varphi}^2\sin\theta\cos\theta+\cos\theta\sin\varphi\sin\alpha-\sin\theta\cos\alpha \qquad \quad (3) \\ % \ddot{\varphi}&=&\frac{\sin\theta(\cos\varphi\sin\alpha-2\dot{\theta}\dot{\varphi}\cos\theta)}{\sin^2\theta+\beta} \qquad \qquad \qquad \qquad \;\;\, (4) \end{eqnarray} $$
The system is now nondimensionalized and greatly simplified : we started with six parameters $( l,m,R,M,\alpha,g )$ and there are now only two left $( \alpha$ and $\beta )$