Lets start by taking any system of ordinary differential equations. We can of course convert this to a first order system by creating stand-in variables for derivatives, (e.g. if $y'' = y$ we can create the equivalent system $y' = w$ and $w' = y$).
We define a full model:
A full model for a system should exactly determine the evolution of a system based on the state of the system
This means that there are unique solutions that exist. This seems fair to us. This means that when we take a solution curve in the phase plane, it must not intersect another curve unless at a limit point.
We ask that our derivatives all be finite. It is ok if our paths are jagged, but we cannot jump between points in our system. This is the most controversial thing I'll ask be assumed. There are many reasons why you might not want this to be the case, but all the ones I can think of rely on us being at a molecular level.
We can say another thing. There is energy associated with our system, and it is lost over time. This is more controversial, but then only to a small subset of schizophrenics. For the rest of us we can happily claim that there are no periodic orbits in our phase planes.
If we aim to create a model that fully describes the evolution of a system
- Bounded variation: $ \|y'\| < \inf $
- Unique solutions that exist: Picard–Lindelöf theorem satisfied
- Non time-dependent: Phase‐portrait stationarity
- No perpetual motion: a-periodic
This is a very strong condition, and is aided by phase lines being continuous. Small pertubations in one place create small pertubations everywhere (in delta-epsilon style).
Limited hidden variables
Full hidden variables
Constraints on those
Neural Network Analogues
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