Physical Review E
Ideally an embedding of an N-dimensional dynamical system is N-dimensional. Ideally, an embedding of a dynamical system with symmetry is symmetric. Ideally, the symmetry of the embedding is the same as the symmetry of the original system. This ideal often cannot be achieved. Differential embeddings of the Lorenz system, which possesses a twofold rotation symmetry, are not ideal. While the differential embedding technique happens to yield an embedding of the Lorenz attractor in three dimensions, it does not yield an embedding of the entire flow. An embedding of the flow requires at least four dimensions. The four dimensional embedding produces a flow restricted to a twisted three dimensional manifold in R4. This inversion symmetric three-manifold cannot be projected into any three dimensional Euclidean subspace without singularities.
“Differential Embedding of the Lorenz Attractor,” D. J. Cross and R. Gilmore, Physical Review E 81, 066220 (2010).