Abstract
We present a set of advanced techniques for the visualization of 2D Poincaré maps. Since 2D Poincaré maps are a mathematical abstraction of periodic or quasi-periodic 3D flows, we propose to embed the 2D visualization with standard 3D techniques to improve the understanding of the Poincaré maps. Methods to enhance the representation of the relation x ↔ P(x), e.g., the use of spot noise, are presented as well as techniques to visualize the repeated application of P, e.g., the approximation of P as a warp function. It is shown that animation can be very useful to further improve the visualization. For example, the animation of the construction of Poincaré map P is a very intuitive visualization. During the paper we present a set of examples which demonstrate the usefulness of our techniques.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. H. Abraham and C. D. Shaw, Dynamics — the geometry of behavior, 2nd ed., Addison-Wesley, 1992.
Advanced Visual System Inc, Avs developers guide,release 4, May 1992.
V. I. Arnold, Ordinary differential equations, M.I.T. Press, Cambridge, MA, 1973.
D. K. Arrowsmith and C. A. Place, An introduction to dynamical systems, Cambridge University Press, 1990.
T. Beier and S. Neely, Feature-based image metamorphosis, Computer Graphics (SIGGRAPH ‘82 Proceedings) (E. E. CATMULL, ed.), vol. 26, July 1992, pp. 35–42.
E. Doedel, Software for continuation and bifurcation problems in ordinary differential equations (auto 86 user manual), 2nd ed., February 1986, See also URL http://ute.usi.utah.edu/software/math/automan.html.
J. H. HUBBARD AND B. H. WEST, Differential equations: A dynamical systems approach: Ordinary differential equations, Texts in Applied Mathematics, vol. 5, Springer, 1991.
J. P. M. Hultquist, Constructing stream surfaces in steady 3d vector fields, Proceedings Visualization ‘82, October 1992, pp. 171–177.
H. Löffelmann and E. Gröller, DynSys3D: A workbench for developing advanced visualization techniques in the field of three-dimensional dynamical systems, Proceedings of The Fifth International Conference in Central Europe on Computer Graphics and Visualization ‘87 (WSCG ‘87) (N. Thalmann and V. Skala, eds.), Plzen, Czech Republic, February 1997, pp. 301–310.
H. Löffelmann, L. Mroz, and E. Gröller, Hierarchical streamarrows for the visualization of dynamical systems, Accepted for publication at the 8th EG Workshop on Visualization in Scientific Computing, Boulogne sur Mer, France, April 1997.
H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and fractals - new frontiers of science, Springer, 1992.
H. Poincarf, Les méthodes nouvelles de la mécanique céleste (3 vols.), Gauthier-Villars, Paris, 1899.
S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry,and engineering, Addison-Wesley, 1994.
A. A. Tsonis, Chaos - from theory to applications, Plenum Press, 1992.
S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, 3rd ed., Texts in Applied Mathematics, vol. 2, Springer, 1996.
J. J. van Wijk, Spot noise, Proceedings SIGGRAPH ‘81, July 1991, pp. 309–318.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Löffelmann, H., Kučera, T., Gröller, E. (1998). Visualizing Poincaré Maps together with the Underlying Flow. In: Hege, HC., Polthier, K. (eds) Mathematical Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03567-2_23
Download citation
DOI: https://doi.org/10.1007/978-3-662-03567-2_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08373-0
Online ISBN: 978-3-662-03567-2
eBook Packages: Springer Book Archive