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)

Abstract

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.

References

  1. 1.
    D. Asimov, Notes on the topology of vector fields and flows. Technical Report RNR-93-003, NASA Ames Research Center, 1993Google Scholar
  2. 2.
    U. Bauer, M. Kerber, J. Reininghaus, PHAT: persistent homology algorithm toolbox. http://phat.googlecode.com/
  3. 3.
    C. Chen, M. Kerber, Persistent homology computation with a twist, in 27th European Workshop on Computational Geometry (EuroCG 2011), 2011. Extended abstractGoogle Scholar
  4. 4.
    G. Chen, K. Mischakow, R.S. Laramee, E. Zhang, Efficient morse decompositions of vector fields. IEEE Trans. Vis. Comput. Graph. 14, 848–862 (2008)CrossRefGoogle Scholar
  5. 5.
    M. Dellnitz, O. Junge, On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36, 491–515 (1999)CrossRefMathSciNetGoogle Scholar
  6. 6.
    H. Edelsbrunner, D. Letscher, A. Zomorodian, Topological persistence and simplification. Discret. Comput. Geom. 28, 511–533 (2002)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    H. Edelsbrunner, A. Zomorodian, Computing linking numbers of a filtration, in Algorithms in Bioinformatics. LNCS, vol. 2149 (Springer, Berlin/Heidelberg, 2001), pp. 112–127Google Scholar
  8. 8.
    C. Garth, G.-S. Li, X. Tricoche, C.D. Hansen, H. Hagen, Visualization of coherent structures in transient 2D flows. In Topology-Based Methods in Visualization II (Springer, Berlin/Heidelberg, 2007), pp. 1–13Google Scholar
  9. 9.
    G. Haller, Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149, 248–277 (2001)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    J. L. Helman, L. Hesselink, Visualizing vector field topology in fluid flows. IEEE Comput. Graph. Appl. 11(3), 36–46 (1991)CrossRefGoogle Scholar
  11. 11.
    M. Hirsch, S. Smale, R. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, 2nd edn. (Academic, 2004)Google Scholar
  12. 12.
    J. Kasten, C. Petz, I. Hotz, B. Noack, H.-C. Hege, Localized finite-time lyapunov exponent for unsteady flow analysis, in Proceedings Vision, Modeling and Visualization 2008, Braunschweig, 2009, pp. 265–274Google Scholar
  13. 13.
    A. Kuhn, C. Rössl, T. Weinkauf, H. Theisel, A benchmark for evaluating FTLE computations, in Proceedings IEEE Pacific Visualization 2012, Songdo, 2012, pp. 121–128Google Scholar
  14. 14.
    R. Peikert, F. Sadlo, Topologically relevant stream surfaces for flow visualization, in Proceedings of Spring Conference on Computer Graphics, Budmerice, 2009, pp. 171–178Google Scholar
  15. 15.
    F.H. Post, The state of the art in flow visualization: feature extraction and tracking. Comput. Graph. Forum 22(4), 775–792 (2003)CrossRefMathSciNetGoogle Scholar
  16. 16.
    W. Reich, D. Schneider, C. Heine, A. Wiebel, G. Chen, G. Scheuermann, Combinatorial vector field topology in three dimensions, in Topological Methods in Data Analysis and Visualization II (Springer, Berlin/Heidelberg, 2012)Google Scholar
  17. 17.
    F. Sadlo, R. Peikert, Visualizing lagrangian coherent structures and comparison to vector field topology, in Topology-Based Methods in Visualization II (Springer, Berlin/Heidelberg, 2009)Google Scholar
  18. 18.
    A. Sanderson, G. Chen, X. Tricoche, E. Cohen, Understanding quasi-periodic fieldlines and their topology in toroidal magnetic fields, in Topological Methods in Data Analysis and Visualization II, Mathematics and Visualization, ed. by R. Peikert, H. Hauser, H. Carr, R. Fuchs, (Springer, Berlin/Heidelberg, 2012) pp. 125–140.CrossRefGoogle Scholar
  19. 19.
    O.K. Smith, Eigenvalues of a symmetric 3 × 3 matrix. Commun. ACM 4(4), 168 (1961)Google Scholar
  20. 20.
    H. Theisel, T. Weinkauf, H.-C. Hege, H.-P. Seidel, Saddle connectors – an approach to visualizing the topological skeleton of complex 3D vector fields, in Proceedings of IEEE Visualization 2003, ed. by G. Turk, J.J. van Wijk, R. Moorhead, Seattle, Oct 2003, pp. 225–232Google Scholar
  21. 21.
    H. Theisel, T. Weinkauf, H.-C. Hege, H.-P. Seidel, Grid-independent detection of closed stream lines in 2D vector fields, in Proceedings Vision, Modeling and Visualization 2008, Konstanz, 2004, pp. 421–428Google Scholar
  22. 22.
    D. Weiskopf, B. Erlebacher, Overview of flow visualization, in The Visualization Handbook, (Academic, 2005), pp. 261–278Google Scholar
  23. 23.
    T. Wischgoll, G. Scheuermann, Detection and visualization of closed streamlines in planar flows. IEEE Trans. Vis. Comput. Graph. 7(2), 165–172 (2001)CrossRefGoogle Scholar

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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