Toward the calculation of higher-dimensional stable manifolds and stable sets for noninvertible and piecewise-smooth maps. (English) Zbl 1167.37039

Stable manifolds can be considered as fundamental building blocks of dynamical systems. Together with its sibling, the unstable manifold, each stable manifold might provide important information about a system’s global dynamics. The stable and unstable manifold may indicate, for example, boundaries of basins of attraction or, if a stable and unstable manifold intersect, the occurrence of chaotic dynamics. In this work the author presents a method to compute the stable manifolds and sets of maps in general. Neither the system’s inverse nor its Jacobian is required. It is also not limited to one-dimensional stable manifolds and sets. Theoretically, the method might be capable of computing manifolds independent of their dimension, though in practice it applied it only to get one- and two-dimensional stable manifolds and sets.
For a given map \(f:\mathbb{R}^d \to \mathbb{R}^d\), the proposed method is capable of yielding large parts of stable manifolds and sets within a certain compact region \(M \subset \mathbb{R}^d\). The algorithm divides the region \(M\) in sets and uses an adaptive subdivision technique to approximate an outer covering of the manifolds. The use of suggested method is illustrated by computation of one- and two-dimensional stable manifolds and global stable sets.


37M20 Computational methods for bifurcation problems in dynamical systems
37M99 Approximation methods and numerical treatment of dynamical systems
37-04 Software, source code, etc. for problems pertaining to dynamical systems and ergodic theory
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37E99 Low-dimensional dynamical systems


AnT 4.669
Full Text: DOI


[1] AnT 4.669 project: Home page http://www.AnT4669.de (2005)
[2] Avrutin, V., Lammert, R., Schanz, M., Wackenhut, G., Osipenko, G.S.: On the software package AnT 4.669 for the investigation of dynamical systems. In: Osipenko, G.S. (ed.) Fourth International Conference on Tools for Mathematical Modelling, vol. 9, pp. 24–35. St. Petersburg State Polytechnic University, Russia, June 2003 · Zbl 1236.65157
[3] Dellnitz, M., Hohmann, A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75, 293–317 (1997) · Zbl 0883.65060
[4] Dellnitz, M., Junge, O.: An adaptive subdivision technique for the approximation of attractors and invariant measures. Comput. Vis. Sci. 1, 63–68 (1998) · Zbl 0970.65130
[5] England, J.P., Krauskopf, B., Osinga, H.M.: Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse. SIAM J. Appl. Dyn. Syst. 3(2), 161–190 (2005) · Zbl 1059.37067
[6] Fundinger, D.: Investigating dynamics by multilevel phase space discretization. Ph.D. thesis, University of Stuttgart (2006). Available at http://elib.uni-stuttgart.de/opus/volltexte/2006/2614/
[7] Guckenheimer, J., Worfolk, P.: Dynamical systems: some computational problems. In: Schlomiuk, D. (ed.) Bifurcations and Periodic Orbits of Vector Fields, pp. 241–277. Kluwer Academic, Dordrecht (1993) · Zbl 0795.58034
[8] Henderson, M.E.: Computing invariant manifolds by integrating fat trajectories. SIAM J. Appl. Dyn. Syst. 4(4), 832–882 (2005) · Zbl 1090.37012
[9] Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50, 69–77 (1976) · Zbl 0576.58018
[10] Hobson, D.: An efficient method for computing invariant manifolds. J. Comput. Phys. 104, 14–22 (1991) · Zbl 0768.65025
[11] Homburg, A.J., Osinga, H.M., Vegter, G.: On the computation of invariant manifolds of fixed points. Z. Angew. Math. Phys. 46, 171–187 (1995) · Zbl 0827.58043
[12] Junge, O.: Rigorous discretization of subdivision techniques. In: Proceedings of Equadiff ’99, Berlin (2000) · Zbl 0963.65536
[13] Krauskopf, B., Osinga, H.M.: Globalizing two-dimensional unstable manifolds of maps. Int. J. Bifurc. Chaos 8(3), 483–503 (1998a) · Zbl 0955.37016
[14] Krauskopf, B., Osinga, H.M.: Growing 1D and quasi-2D unstable manifolds of maps. J. Comput. Phys. 146, 406–419 (1998b) · Zbl 0915.65075
[15] Krauskopf, B., Osinga, H.M.: Computing geodesic level sets on global (un)stable manifolds of vector fields. SIAM J. Appl. Dyn. Sys. 4(2), 546–569 (2003) · Zbl 1089.37014
[16] Krauskopf, B., Osinga, H.M., Doedel, E.J., Henderson, M.E., Guckenheimer, J., Vladimirsky, A., Dellnitz, M., Junge, O.: A survey of methods for computing (un)stable manifolds of vector fields. Int. J. Bifurc. Chaos 15(3), 763–791 (2005) · Zbl 1086.34002
[17] Leslie, P.: On the use of matrices in population mathematics. Biometrika 33, 183–212 (1945) · Zbl 0060.31803
[18] Mira, C., Gardini, L., Barugola, A., Cathala, J.C.: Chaotic Dynamics in Two-Dimensional Noninvertible Maps. World Scientific Ser. Nonlinear Sci. A: Monographs and Treaties, vol. 20. World Scientific, River Edge (1996) · Zbl 0906.58027
[19] Nien, C.-H., Wicklin, F.J.: An algorithm for the computation of preimages in noninvertible mappings. Int. J. Bifurc. Chaos 8(2), 415–422 (1998) · Zbl 0982.37036
[20] Nitecki, Z.: Differential Dynamics. MIT Press, Cambridge (1971) · Zbl 0246.58012
[21] Nusse, H.E., Yorke, J.A.: A procedure for finding numerical trajectories in chaotic saddles. Physica D 36, 137–156 (1989) · Zbl 0728.58027
[22] Osipenko, G.S.: Lectures on Symbolic Analysis of Dynamical Systems. St. Petersburg State Polytechnic University (2004) · Zbl 1113.35313
[23] Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, Berlin (1989) · Zbl 0692.58001
[24] Shub, M.: Global Stability of Dynamical Systems. Springer, Berlin (1987) · Zbl 0606.58003
[25] Simó, C.: On the analytical and numerical approximation of invariant manifolds. In: Benest, D., Froeschlé, C. (eds.) Les Méthodes Modernes de la Mécanique Céleste, pp. 285–330. Goutelas (1989)
[26] Ugarcovici, I., Weiss, H.: Chaotic systems of a nonlinearity density dependent population model. Nonlinearity 17, 1689–1711 (2004) · Zbl 1066.37020
[27] You, Z., Kostelich, E.J., Yorke, J.A.: Calculating stable and unstable manifolds. Int. J. Bifurc. Chaos Appl. Sci. Eng. 1(3), 605–623 (1991) · Zbl 0874.58053
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