Computing geodesic level sets on global (un)stable manifolds of vector fields

SIAM Journal on Applied Dynamical Systems

Volumn 2, Issue 4, 2003, Pages 546-569

  Krauskopf, Bernd ^a   Osinga, Hinke M ^a  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Geodesic parametrization; Global stable and unstable manifolds; Numerical method

Indexed keywords

EID: 0348091154     PISSN: 15360040     EISSN: None     Source Type: Journal    
DOI: 10.1137/030600180     Document Type: Article

References (34)

1. R. H. Abraham and C. D. Shaw, Dynamics - the Geometry of Behavior. Part 3: Global Behavior, Aerial Press, Santa Cruz, CA, 1985.
2. A. Arneodo, P. Coullet, E. Spiegel, and C. Tresser, Asymptotic chaos, Phys. D, 14 (1985), pp. 327-347.
3. H. W. Broer, H. M. Osinga, and G. Vegter, Algorithms for computing normally hyperbolic invariant manifolds, Z. Angew. Math. Phys., 48 (1997), pp. 480-524.
4. M. Dellnitz and A. Hohmann, The computation of unstable manifolds using subdivision and continuation, in Nonlinear Dynamical Systems and Chaos, H. W. Broer, S. A. Van Gils, I. Hoveijn, and F. Takens, eds., Progr. Nonlinear Differential Equations Appl. 19, Birkhäuser, Basel, 1996, pp. 449-459.
5. M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math., 75 (1997), pp. 293-317.
6. E. J. Doedel, A. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Sandstede, and X. Wang, AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations, http://indy.cs.concordia.ca/auto/main.html, 1997.
7. E. J. Doedel, private communication, IMA, Minneapolis, MN, 1997.
8. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 2nd ed., Springer-Verlag, New York, 1986.
9. J. Guckenheimer and A. Vladimirsky, A Fast Method for Approximating Invariant Manifolds, preprint, Cornell University, Ithaca, NY, 2003. Available online from http://www.math.cornell.edu/~vlad/papers/InvMfolds/InvMfolds.pdf.
10. J. Guckenheimer and P. Worfolk, Dynamical systems: Some computational problems, in Bifurcations and Periodic Orbits of Vector Fields, D. Schlomiuk, ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993, pp. 241-277.
11. J. Hauser and H. M. Osinga, On the geometry of optimal control: The inverted pendulum example, in Proceedings of the American Control Conference Vol. 2, Arlington, VA, 2001, pp. 1721-1726.
12. M. E. Henderson, Computing Invariant Manifolds by Integrating Fat Trajectories, IBM Research Report RC22944, 2003.
13. D. Hobson, An efficient method for computing invariant manifolds of planar maps, J. Comput. Phys., 104 (1993), pp. 14-22.
14. J. G. Hocking and G. S. Young, Topology, 2nd ed., Dover Publications, New York, 1988.
15. A. Jadbabaie, J. Yu, and J. Hauser, Unconstrained receding-horizon control of nonlinear systems, IEEE Trans. Automat. Control, 46 (2001), pp. 776-783.
16. M. E. Johnson, M. S. Jolly, and I. G. Kevrekidis, Two-dimensional invariant manifolds and global bifurcations: Some approximation and visualization studies, Numer. Algorithms, 14 (1997), pp. 125-140.
17. M. E. Johnson, M. S. Jolly, and I. G. Kevrekidis, The Oseberg transition: Visualization of global bifurcations for the Kuramoto-Sivashinsky equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 1-18.
18. O. Junge, Rigorous discretization of subdivision techniques, in Proceedings of the International Conference on Differential Equations, B. Fiedler, K. Gröger, and J. Sprekels, eds., World Scientific, Singapore, 2000, pp. 916-918.
19. E. N. Lorenz, Deterministic nonperiodic flows, J. Atmospheric Sci., 20 (1963), pp. 130-141.
20. B. Krauskopf and H. M. Osinga, Globalizing two-dimensional unstable manifolds of maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), pp. 483-503.
21. B. Krauskopf and H. M. Osinga, Growing 1D and quasi 2D unstable manifolds of maps, J. Comput. Phys., 146 (1998), pp. 404-419.
22. B. Krauskopf and H. M. Osinga, Two-dimensional global manifolds of vector fields, Chaos, 9 (1999), pp. 768-774.
23. B. Krauskopf and H. M. Osinga, Visualizing the structure of chaos in the Lorenz system, Computers and Graphics, 26 (2002), pp. 815-823.
24. Yu. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, Berlin, 1995.
25. H. M. Osinga, Non-orientable manifolds of periodic orbits, in Proceedings of the International Conference on Differential Equations, B. Fiedler, K. Gröger, and J. Sprekels, eds., World Scientific, Singapore, 2000, pp. 922-924.
26. H. M. Osinga, Non-orientable manifolds in three-dimensional vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), pp. 553-570.
27. J. Palis and W. De Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, Berlin, 1982.
28. M. Phillips, S. Levy, and T. Munzner, Geomview: An interactive geometry viewer, Notices Amer. Math. Soc., 40 (1993), pp. 985-988; software and the accompanying manual are available at http://www.geom.uiuc.edu/.
29. W. C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations, 2nd ed., CBMS-NSF Regional Conf. Ser. in Appl. Math. 70, SIAM, Philadelphia, 1998.
30. S. H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA, 1994.
31. A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, UK, 1996.
32. M. Spivak, A Comprehensive Introduction to Differential Geometry, Volume 1, 2nd ed., Publish or Perish, Houston, TX, 1979.
33. 2-Gain and Passivity Techniques in Nonlinear Control, Lecture Notes in Control and Inform. Sci. 218, Springer-Verlag, Berlin, 1994.
34. P. Worfolk, Private communication, Geometry Center, Minneapolis, MN, 1997.