For a given n dimensional chaotic system such as a chaotic attractor or really a time dependent chaotic dynamic system does there exist a higher dimensional representation where the behavior is in fact deterministic?
This question comes from a data science problem in multivariate time series forecasting. Most real world behavior is chaotic in nature, such as the stock market or Earth climate cycles (milankovitch cycles etc.). Lets say I could build a really big model that takes into account everything that might impact the system of interest, would there be some set where all the chaos would just go away?
Take the TV show foundation based on the books by Isaac Asimov. In there they postulate the Abraxis conjecture which is solved by folding space and is critical to the success of psyco-history (which is really just data science). One of the steepest challenges with time series forecasting is the ease at which it is possible to leave conditions where your inital state no longer applies. If chaos could be eliminated then this problem could be pushed out. Effectively with infinite information we could forecast for infinite time (in the theoretical limit only).
Chaotic systems are sensitive to initial conditions, which means that a tiny perturbation of the system’s state can result in a completely different state after some time. If we want to predict a chaotic system in reality, one of the problems we have is that we cannot possibly know all these tiny influences and thus we can only predict the system’s state for a short time interval (the prediction horizon).
Your question seems to be based on the misunderstanding that these tiny influences cause chaos. This is false. The effect of tiny changes is merely what characterises chaos. You can have chaos in idealised mathematical models, e.g., a simple ODE model of the double pendulum. This model per construction does not feature any details on the molecular level or similar. Such models are also deterministic: Identical initial conditions will yield identical results; it’s only similar initial conditions that will yield completely different results. (Contrast with a stochastic system, where also identical conditions can yield completely different results.)
For a real-world chaotic system, having better models or more detailed knowledge of the system’s state only increases your prediction horizon; it does not remove the chaos. In most practical applications, the effect of perturbations is so quick that there is a small upper limit to your prediction horizon corresponding to changes on the atomic scale.
Note that Foundation was written before the “discovery” that chaos is ubiquitous and there are inherent limits to predictability. Given our current knowledge about these kind of things, psychohistory is extremely unrealistic. You might as well break the laws of thermodynamics.