Considering the following Lorentz's attractor,
$$\begin{aligned} &\dot x=\sigma (y-x)\\ &\dot y=x(\rho -z)-y\\ &\dot z=xy-\beta z \end{aligned}$$
I would like to convert it to a position-velocity-acceleration problem (via variable changes) as follows:
$$\begin{aligned} &\dot p=v\\ &\dot v=a\\ &\dot a=??? \end{aligned}$$
The jerk $\dot a$ is a function of acceleration $a$, velocity $v$ and position $p$. I look for a non-trivial solution which do not loose the information of the original problem.
Can anyone please help such a conversion?