Journal of Statistical Physics

, Volume 65, Issue 3–4, pp 579–616


  • Tim Sauer
  • James A. Yorke
  • Martin Casdagli

DOI: 10.1007/BF01053745

Cite this article as:
Sauer, T., Yorke, J.A. & Casdagli, M. J Stat Phys (1991) 65: 579. doi:10.1007/BF01053745


Mathematical formulations of the embedding methods commonly used for the reconstruction of attractors from data series are discussed. Embedding theorems, based on previous work by H. Whitney and F. Takens, are established for compact subsetsA of Euclidean space Rk. Ifn is an integer larger than twice the box-counting dimension ofA, then almost every map fromRk toRn, in the sense of prevalence, is one-to-one onA, and moreover is an embedding on smooth manifolds contained withinA. IfA is a chaotic attractor of a typical dynamical system, then the same is true for almost everydelay-coordinate map fromRk toRn. These results are extended in two other directions. Similar results are proved in the more general case of reconstructions which use moving averages of delay coordinates. Second, information is given on the self-intersection set that exists whenn is less than or equal to twice the box-counting dimension ofA.

Key words

Embedding chaotic attractor attractor reconstruction probability one prevalence box-counting dimension delay coordinates 

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Tim Sauer
    • 1
  • James A. Yorke
    • 2
  • Martin Casdagli
    • 3
  1. 1.Department of Mathematical SciencesGeorge Mason UniversityFairfax
  2. 2.Institute of Physical Science and TechnologyUniversity of MarylandCollege Park
  3. 3.Santa Fe InstituteSanta Fe
  4. 4.Tech PartnersStamford

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