wmwv
commented
8 years ago I've been thinking about this as representing reduced dimensionality. I.e., what fraction of possible phase space is occupied by your system. You could look at any of the physical parameters: position, velocity, acceleration (and in either linear or angular space).
For a Poincaré map, you could take some slice in phase space that seemed to have some meaning and look at the points that are filled in that slice. Run the system for a given set of initial conditions until it reaches its steady-state chaotic behavior. This general 6-D phase space will be a mess, but perhaps there are interesting slices to take through it.
So I'm not quite sure how to incorporate the idea of fractal dimensions into our project; the concept of finding for and solving the differential equations of the swinging atwood machine seems straightforward enough, but I'm having some difficulty figuring out how to take it further.
My understanding of fractal dimensions is that you might have a curve that occupies more than two dimensions due to its bendiness, or that you might have a surface that is so wrinkly it oughtn't be considered simply two dimensions. But I'm having difficulty seeing what curves/surfaces we would analyze from our system.
Originally I had assumed that this would pertain to some analysis of the phase space, perhaps with poincare maps or something to do with phase space, but I'm having difficulty transferring this initial thought to an actual plan of action.
On a somewhat related note, I was reading about this system, and wikipedia was saying that although there are four dimensions to the phase space ($r, \theta, p{r}, p{\theta}$) (p = momentum), but also that because of energy conservation there were really only three dimensions. I was hoping you could clarify this point, since wikipedia did not go into detail on this topic. Perhaps the three dimensions are r, \theta, and the sum of the momenta?