Abstract
Area-preserving maps arise in the study of conservative dynamical systems describing a wide variety of physical phenomena, from the rotation of planets to the dynamics of a fluid. The visual inspection of these maps reveals a remarkable topological picture in which invariant manifolds form the fractal geometric scaffold of both quasi-periodic and chaotic regions. We discuss in this paper the visualization of such maps built upon these invariant manifolds. This approach is in stark contrast with the discrete Poincare plots that are typically used for the visual inspection of maps. We propose to that end several modified definitions of the finite-time Lyapunov exponents that we apply to reveal the underlying structure of the dynamics. We examine the impact of various parameters and the numerical aspects that pertain to the implementation of this method. We apply our technique to a standard analytical example and to a numerical simulation of magnetic confinement in a fusion reactor. In both cases our simple method is able to reveal salient structures across spatial scales and to yield expressive images across application domains.
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
Arnold, V.I.: Proof of A. N. Kolmogorov’s thereom on the preservation of quasiperiodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18(5), 9 (1963)
Bagherjeiran, A., Kamath, C.: Graph-based methods for orbit classification. In: Proc. of 6th SIAM International Conference on Data Mining, April 2006
England, J., Krauskopf, B., Osinga, H.: Computing one-dimensional global manifolds of poincaré maps by continuation. SIAM J. Appl. Dyn. Syst. 4(4), 1008–1041 (2005)
Garth, C., Gerhardt, F., Tricoche, X., Hagen, H.: Efficient computation and visualization of coherent structures in fluid flow applications. IEEE Trans. Visual. Comput. Graph. 13(6), 1464–1471 (2007)
Garth, C., Wiebel, A., Tricoche, X., Hagen, H., Joy, K.: Lagrangian visualization of flow embedded structures. Comput. Graph. Forum 27(3), 1007–1014 (2008)
Grasso, D., Borgogno, D., Pegoraro, F., Schep,T.: Barriers to field line transport in 3d magnetic configurations. J. Phys. Conference Series 260(1), 012012 (2010). http://stacks.iop.org/1742-6596/260/i=1/a=012012
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer, Berlin (2006)
Haller, G.: Distinguished material surfaces and coherent structures in three-dimensional flows. Physica D 149, 248–277 (2001)
Hauser, H., Hagen, H., Theisel, H.: Topology Based Methods in Visualization. Springer, Berlin (2007)
Hege, H.-C., Polthier, K., Scheuermann, G.: Topology Based Methods in Visualization II. Springer, Berlin (2009)
Helman, J.L., Hesselink, L.: Visualizing vector field topology in fluid flows. IEEE Comput. Graph. Appl. 11(3), 36–46 (1991)
Kolmogorov, A.N.: On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Dokl. Akad. Nauk. SSR 98, 469 (1954)
Levnajic, Z., Mezic, I.: Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partitions and invariant sets. Chaos 20(3), 033114 (2010)
Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics, 2nd edn. Springer, New York (1992)
Löffelmann, H., Gröller, E.: Enhancing the visualization of characteristic structures in dynamical systems. In: Visualization in Scientific Computing ’98, pp. 59–68 (1998)
Löffelmann, H., Kucera, T., Gröller, E.: Visualizing poincaré maps together with the underlying flow. In: Hege, H.-C., Polthier, K. (eds.) Mathematical Visualization: Proceedings of the International Workshop on Visualization and Mathematics ’97, pp. 315–328. Springer, Berlin (1997)
Löffelmann, H., Doleisch, H., Gröller, E.: Visualizing dynamical systems near critical points. In: 14th Spring Conference on Computer Graphics, pp. 175–184 (1998)
Marsden, J., West, M.: Discrete mechanics and variational integrators. Acta Numerica 10, 357–514 (2001). doi:10.1017/S096249290100006X
Marsden, J.E., Ratiu, T.: Introduction to mechanics and symmetry. Texts in Applied Mathe-matics, vol. 17. Springer, Berlin (2003)
Mathur, M., Haller, G., Peacock, T., Ruppert-Felsot, J., Swinney, H.: Uncovering the lagrangian skeleton of turbulence. Phys. Rev. Lett. 98, 144502 (2007)
Morrison, P.: Magnetic field lines, hamiltonian dynamics, and nontwist systems. Phys. Plasma 7(6), 2279–2289 (2000)
Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. II 1,1 Kl(1), 1 (1962)
Peikert, R., Sadlo, F.: Visualization methods for vortex rings and vortex breakdown bubbles. In: Museth, A.Y.K., Möller, T. (eds.) Proceedings of the 9th Eurographics/IEEE VGTC Symposium on Visualization (EuroVis’07), pp. 211–218, May 2007
Peikert, R., Sadlo, F.: Flow topology beyond skeletons: Visualization of features in recirculating flow. In: Hege, H.-C., Polthier, K., Scheuermann, G. (eds.) Topology-Based Methods in Visualization II, pp. 145–160. Springer, Berlin (2008)
Prince, P.J., Dormand, J.R.: High order embedded runge-kutta formulae. J. Comput. Appl. Math. 7(1) 67–75 (1981)
Sadlo, F., Peikert, R.: Efficient visualization of Lagrangian coherent structures by filtered AMR ridge extraction. IEEE Trans. Visual. Comput. Graph. 13(5), 1456–1463 (2007)
Sadlo, F., Peikert, R.: Visualizing Lagrangian coherent structures and comparison to vector field topology. In: Hege, H.-C., Polthier, K., Scheuermann, G., Farin, G., Hege, H.-C., Hoffman, D., Johnson, C.R., Polthier, K. (eds.) Topology-Based Methods in Visualization II. Mathematics and Visualization, pp. 15–29. Springer, Berlin (2009)
Sadlo, F., Weiskopf, D.: Time-dependent 2-D vector field topology: An approach inspired by Lagrangian coherent structures. Comput. Graph. Forum 29(1), 88–100 (2010)
Sanderson, A., Tricoche, X., Garth, C., Kruger, S., Sovinec, C., Held, E., Breslau, J.: Poster: A geometric approach to visualizing patterns in the poincaré plot of a magnetic field. In: Proc. of IEEE Visualization 06 Conference, 2006
Sanderson, A., Cheng, G., Tricoche, X., Pugmire, D., Kruger, S., Breslau, J.: Analysis of recurrent patterns in toroidal magnetic fields. IEEE Trans. Visual. Comput. Graph. 16(6), 1431–1440 (2010)
Scheuermann, G., Tricoche, X.: Topological methods in flow visualization. In: Johnson, C., Hansen, C. (eds.) Visualization Handbook, pp. 341–356. Academic, NY (2004)
Soni, B., Thompson, D., Machiraju, R.: Visualizing particle/flow structure interactions in the small bronchial tubes. IEEE Trans. Visual. Comput. Graph. 14(6), 1412–1427 (2008)
Yip, K.M.-K.: KAM: A System for Intelligently Guiding Numerical Experimentation by Computer. MIT, MA (1992)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Tricoche, X., Garth, C., Sanderson, A., Joy, K. (2012). Visualizing Invariant Manifolds in Area-Preserving Maps. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds) Topological Methods in Data Analysis and Visualization II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23175-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-23175-9_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23174-2
Online ISBN: 978-3-642-23175-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)