Toward the Extraction of Saddle Periodic Orbits

  • Jens Kasten
  • Jan Reininghaus
  • Wieland Reich
  • Gerik Scheuermann
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


Saddle periodic orbits are an essential and stable part of the topological skeleton of a 3D vector field. Nevertheless, there is currently no efficient algorithm to robustly extract these features. In this chapter, we present a novel technique to extract saddle periodic orbits. Exploiting the analytic properties of such an orbit, we propose a scalar measure based on the finite-time Lyapunov exponent (FTLE) that indicates its presence. Using persistent homology, we can then extract the robust cycles of this field. These cycles thereby represent the saddle periodic orbits of the given vector field. We discuss the different existing FTLE approximation schemes regarding their applicability to this specific problem and propose an adapted version of FTLE called Normalized Velocity Separation. Finally, we evaluate our method using simple analytic vector field data.


Vector Field Periodic Orbit Stream Line Lagrangian Coherent Structure Separation Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



First, we thank the reviewers of this paper for their ideas and critical comments. In addition, we thank Ronny Peikert and Filip Sadlo for a fruitful discussions. This research is supported by the European Commission under the TOPOSYS project FP7-ICT-318493-STREP, the European Social Fund (ESF App. No. 100098251), and the European Science Foundation under the ACAT Research Network Program.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Jens Kasten
    • 1
  • Jan Reininghaus
    • 2
  • Wieland Reich
    • 1
  • Gerik Scheuermann
    • 1
  1. 1.Leipzig UniversityLeipzigGermany
  2. 2.Institute of Science and Technology AustriaKlosterneuburgAustria

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