A Phase Space Trajectory Proofs Here we present the proofs for the propositions from Section 4, concerning a

Then the time derivative of z (t) is d dt x d x dt ... d Single phase space trajectories can feed into themselves representing periodic motion. Effectively an additional dimension is added to phase space, which is time. This maintains generality and allows NODEs to be used as a component of a larger model. These are the same equations that were derived by Chen et al. The gradients from the positional part and the velocity part are found separately and added.