Differential Embedding of the Lorenz Attractor

Authors

Daniel J. Cross, Haverford CollegeFollow

Document Type

Journal Article

Role

Author

Standard Number

1539-3755

Journal Title

Physical Review E

Volume

81

Issue

6

First Page

066220

Last Page

066229

Publication Date

2010

Abstract

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.

Repository Citation

“Differential Embedding of the Lorenz Attractor,” D. J. Cross and R. Gilmore, Physical Review E 81, 066220 (2010).