Differential embedding of the Lorenz attractor

Phys. Rev. E 81, 066220 – Published 25 June 2010
Daniel J. Cross and R. Gilmore

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.

DOI: http://dx.doi.org/10.1103/PhysRevE.81.066220

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  • Received 21 January 2010
  • Published 25 June 2010

© 2010 The American Physical Society

Authors & Affiliations

Daniel J. Cross and R. Gilmore

  • Physics Department, Drexel University, Philadelphia, Pennsylvania 19104, USA

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