I've recently been learning about nonlinear oscillations, and I noticed a seemingly strong connection between how the equations of motion are solved/approximated, and fiber bundles (or vector bundles considering the systems use vectors).
For instance: in general when looking at a nonlinear system, you usually only consider a small localization around a minimum that allows the equations of motion to be approximated as linear differential equations; however, the global system is in general nonlinear, so I figured it's like dealing with a local trivialization.
But looking online I couldn't seem to find anything relating the two or giving a fiber bundle approach.
I thought it might be interesting, or maybe shed light on understanding nonlinear oscillations. Though I've been told that almost all of these systems have no mathematical solutions (only numerical approximations), so is it maybe that this approach has been tried, but it didn't give anything new? Or is there a fiber/vector bundle approach, but I'm just not looking in the right places?