Globalizing two-dimensional unstable manifolds of maps

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 8, Issue 3, 1998, Pages 483-503

  Krauskopf, Bernd ^a   Osinga, Hinke ^b  

^a VU UNIVERSITY AMSTERDAM   (Netherlands)

^b UNIVERSITY OF MINNESOTA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032016858     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127498000310     Document Type: Article

References (33)

1. Arnol'd, V. I. [1965] "Small denominators I," Trans. Amer. Math. Soc. 2nd. Series 46, 213-284.
2. Back, A., Guckenheimer, J., Myers, M. R., Wicklin, F. J. & Worfolk, P. A. [1992] "DsTool: Computer assisted exploration of dynamical systems," Notices Amer. Math. Soc. 39(4), 303-309.
3. Bai, F., Spence, A. & Stuart, A. M. [1993] "The numerical computation of heteroclinic connections in systems of gradient partial differential equations," SIAM J. Appl. Math. 53(3), 743-769.
4. Beyn, W.-J. [1990] "The numerical computation of connecting orbits in dynamical systems," IMA J. Num. Anal. 9, 379-405.
5. Broer, H. W., Osinga, H. M. & Vegter, G. [1996] "On the computation of normally hyperbolic invariant manifolds," in Progress in Nonlinear Differential Equations and Their Applications 19, eds. Broer, H. W., Van Gils, S. A., Hoveijn, I. & Takens, F. (Birkhäuser Verlag, Basel), pp. 423-447.
6. Broer, H. W., Osinga, H. M. & Vegter, G. [1997] "Algorithms for computing normally hyperbolic invariant manifolds," Z. Angew. Math. Phys. 48, 480-524.
7. Chow, S.-N., Li, C. & Wang, D. [1994] Normal Forms and Bifurcation of Planar Vector Fields (Cambridge University Press).
8. Dellnitz, M. & Hohmann, A. [1996] "The computation of unstable manifolds using subdivision and continuation," in Progress in Nonlinear Differential Equations and Their Applications 19, eds. Broer, H. W., Van Gils, S. A., Hoveijn, I. & Takens, F. (Birkhäuser Verlag, Basel), pp. 449-459.
9. Friedman, M. J. & Doedel, E. J. [1991] "Numerical computation and continuation of invariant manifolds connecting fixed points," SIAM J. Numer. Anal. 28(3), 789-808.
10. Guckenheimer, J. & Holmes, P. [1986] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer Verlag, Berlin), second printing.
11. Guckenheimer, J. & Worfolk, P. [1993] "Dynamical systems: Some computational problems," in Bifurcations and Periodic Orbits of Vector Fields, ed. Schlomiuk, D. (Kluwer Academic Publishers), pp. 241-277.
12. Haller, G. & Wiggins, S. [1995] "Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forced nonlinear Schrödinger equation," Physica D85, 311-347.
13. Hirsch, M. W., Pugh, C. & Shub, M. [1977] Invariant Manifolds, Lecture Notes in Mathematics 583 (Springer Verlag, Berlin).
14. Hobson, D. [1993] "An efficient method for computing invariant manifolds of planar maps," J. Comput. Phys. 104(1), 14-22.
15. Homburg, A. J., Sands, D. & De Vilder, R. [1996] "Computing invariant sets," Technical Report 57/96, Schwerpunktprogramm Danse der DFG.
16. Johnson, M. E., Jolly, M. S. & Kevrekidis, I. G. [1997] "Two-dimensional invariant manifolds and global bifurcations: Some approximation and visualization studies," Numer. Algorithms 14, 125-140.
17. Krauskopf, B. & Osinga, H. M. [1997] "Growing unstable manifolds of planar maps," IMA preprint 1517, http://www.ima.umn.edu/preprints/OCT97/1517.ps.gz.
18. Krauskopf, B. & Osinga, H. M. [1988] "Growing ID and quasi 2D unstable manifolds of maps," Math. Phys. Archive preprint 98-56, http://www.ma.utexas.edu/mp_arc/mp_arc-home.html.
19. Lomelí, H. E. & Meiss, J. D. [1997] "Quadratic volume preserving maps," Nonlinearity 11(3), 557-574.
20. Malhotra, N. & Wiggins, S. [1998] "Geometric structures, lobe dynamics, and Lagrangian transport in flows with aperiodic time-dependence, with applications to Rossby wave flow," J. Nonlin. Sci., to appear.
21. Moore, G. [1995] "Computation and parametrization of periodic and connecting orbits," IMA J. Numer. Anal. 15, 245-263.
22. Osinga, H. M. [1996] "Computing invariant manifolds: An implementation of the graph transform," PhD thesis, Groningen University.
23. Palis, J. & De Melo, W. [1982] Geometric Theory of Dynamical Systems (Springer Verlag, Berlin).
24. Parker, T. S. & Chua, L. O. [1989] Practical Numerical Algorithms for Chaotic Systems (Springer Verlag, Berlin).
25. Phillips, M., Levy, S. & Munzner, T. [1993] "Geomview: An interactive geometry viewer," Notices of the Amer. Math. Soc. 40, 985-988. This software and the accompanying manual are available via http://www.geom.umn.edu/software/download/geomview.html
26. Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. [1992] Numerical Recipes in C: the Art of Scientific Computing, 2nd edition (Cambridge University Press).
27. Simó, C. [1989] "On the analytical and numerical approximation of invariant manifolds," in Les Méthodes Modernes de la Mécanique Céleste, eds. Benest, D. & Froeschlé, C. (Goutelas), pp. 285-329.
28. Sosnovtseva, O., Feudel, U., Kurths, J. & Pikovsky, A. [1996] "Multiband strange nonchaotic attractors in quasiperiodically forced systems," Phys. Lett. A218, 255-267.
29. Tabacman, E. [1993] "Visualization of stable and unstable manifolds of diffeomorphisms," Technical Report GCG60, The Geometry Center, Minneapolis.
30. Taylor, M. A., Jolly, M. S. & Kevrekidis, I. G. [1989] "SCIGMA: Stability computations and interactive graphics for invariant manifold analysis," Technical Report, Department of Chemical Engineering Princeton University.
31. Thurman, R. & Worfolk, P. [1996] "The geometry of halo orbits in the circular restricted three-body problem," Technical Report GCG95, The Geometry Center, Minneapolis.
32. Worfolk, P. [1997] private communication.
33. You, Z., Kostelich, E. J. & Yorke, J. A. [1991] "Calculating stable and unstable manifolds," Int. J. Bifurcation and Chaos 1(3), 605-623.