Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse

SIAM Journal on Applied Dynamical Systems

Volumn 3, Issue 2, 2004, Pages 161-190

  England, J P ^a   Krauskopf, B ^a   Osinga, H M ^a  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Discrete time system; Noninvertible map; Piecewise smooth map; Planar map; Stable manifold; Stable set

Indexed keywords

EID: 3042531966     PISSN: 15360040     EISSN: None     Source Type: Journal    
DOI: 10.1137/030600131     Document Type: Article

References (35)

1. R. H. ABRAHAM, L. GARDINI, AND C. MIRA, Chaos in Discrete Dynamical Systems, A visual introduction in 2 Dimensions, Springer-Verlag, New York, 1997.
2. V. I. ARNOL'D, Catastrophe Theory, 3rd ed., Springer-Verlag, Berlin, 1992.
3. A. BACK, J. GUCKENHEIMER, M. R. MYERS, F. J. WICKLIN, AND P. A. WORFOLK, DsTool: Computer assisted exploration of dynamical systems, Notices Amer. Math. Soc., 39 (1992), pp. 303-309.
4. M. A. DAVIS, J. R. PRATT, B. DUTTERER, AND T. J. BURNS, The stability of low radial immersion, Ann. CIRP, 49 (2000), pp. 37-40.
5. M. A. DAVIS, J. R. PRATT, B. DUTTERER, AND T. J. BURNS, Stability prediction for low radial immersion milling, J. Manufacturing Sci. Engrg., 124 (2002), pp. 217-225.
6. M. DELLNITZ AND A. HOHMANN, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math., 75 (1997), pp. 293-317.
7. C. E. FROUZAKIS, L. GARDINI, I. G. KEVREKIDIS, G. MILLERIOUX, AND C. MIRA, On some properties of invariant sets of two-dimensional noninvertible maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 7 (1997), pp. 1167-1194.
8. C. E. FROUZAKIS, I. G. KEVREKIDIS, AND B. PECKHAM, A route to computational chaos revisited: Noninvertibility and the breakup of an invariant circle, Phys. D 177 (2003), pp. 101-121.
9. J. GUCKENHEIMER AND P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, Berlin, 1983.
10. I. GUMOWSKI AND C. MIRA, Dynamique Chaotique, Cepadues Éditions, Toulouse, 1980.
11. I. GUMOWSKI AND C. MIRA, Recurrences and Discrete Dynamic Systems, Springer-Verlag, New York,1980.
12. S. M. HAMMEL, C. K. R. T. JONES, AND J. V. MALONEY, Global dynamical behaviour of the optical field in a ring cavity, J. Opt. Soc. Amer. Ser. B, 2 (1985), pp. 552-564.
13. D. HOBSON, An efficient method for computing invariant manifolds, J. Comput. Phys., 104 (1991), pp. 14-22.
14. A. J. HOMBURG, H. M. OSINGA, AND G. VEGTER, On the computation of invariant manifolds of fixed points, Z. Angew. Math. Phys., 46 (1995), pp. 171-187.
15. K. IKEDA, Multiple-valued stationary state and its instability of the transmitted light from a ring cavity system, Optics Communications, 30 (1979), pp. 257-261.
16. E. J. KOSTELICH, J. A. YORKE, AND Z. YOU, Plotting stable manifolds: Error estimates and noninvertible maps, Phys. D, 93 (1996), pp. 210-222.
17. B. KRAUSKOPF AND H. M. OSINGA, Growing 1D and quasi-2D unstable manifolds of maps, J. Comput. Phys., 146 (1998), pp. 406-419.
18. 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.
19. B. KRAUSKOPF AND H. M. OSINGA, Investigating torus bifurcations in the forced Van der Pol oscillator, in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, E. J. Doedel and L. S. Tuckerman, eds., IMA Vol. Math. Appl. 119, Springer-Verlag, New York, 2000, pp. 199-208.
20. Maplesoft Product Suite, http://www.maplesoft.com/ (2004).
21. C. MIRA, C. JEAN-PIERRE, G. MILLÉRIOUX, AND L. GARDINI, Plane foliation of two-dimensional non-invertible maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), pp. 1439-1462.
22. C. MIRA, L. GARDINI, A. BARUGOLA, AND J. C. CATHALA, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, World Scientific Ser. Nonlinear Sci. Ser A: Monographs and Treaties 20 A, World Scientific, River Edge, NJ, 1996.
23. P. MORESCO AND S. P. DAWSON, The PIM-simplex method: An extension of the PIM-triple method to saddles with an arbitrary number of expanding directions, Phys. D, 126 (1999), pp. 38-48.
24. C.-H. NIEN AND F. J. WICKLIN, An algorithm for the computation of preimages in noninvertible mappings, 8 (1998), pp. 415-422.
25. H. E. NUSSE AND J. A. YORKE, A procedure for finding numerical trajectories in chaotic saddles, Phys. D, 36 (1989), pp. 137-156.
26. H. M. OSINGA AND J. P. ENGLAND, Global Manifold 1D Code, Version 2, Software for Use with DsTool, http://www.dynamicalsystems.org/sw/sw/detail?item= 27 (2003).
27. J. PALIS AND W. DE MELO, Geometric Theory of Dynamical Systems, Springer-Verlag, New York, Berlin, 1982.
28. J. PALIS AND F. TAKENS, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, Cambridge, UK, 1993.
29. T. S. PARKER AND L. O. CHUA, Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, Berlin, 1989.
30. C. SIMÓ, On the analytical and numerical approximation of invariant manifolds, in Les Méthodes Modernes de la Mécanique Céleste, D. Benest and C. Froeschlé, eds., Goutelas, 1989, pp. 285-330.
31. S. H. STROGATZ, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Perseus Books, Reading, MA, Cambridge, MA, 1994.
32. D. SWEET, H. E. NUSSE, AND J. A. YORKE, Stagger-and-step method: Detecting and computing chaotic saddles in higher dimensions, Phys. Rev. Lett., 86 (2001), pp. 2261-2264.
33. R. SZALAI, Nonlinear Vibrations of Interrupted Cutting Processes, M.Sc. thesis, Budapest University of Technology and Economics, Budapest, Hungary, 2002.
34. F. J. WICKLIN, Pisces: A Platform for Implicit Surfaces and Curves and the Exploration of Singularities, Tech. report GCG 89, The Geometry Center, University of Minnesota, Minneapolis, MN, 1995. Available online at http://www.geom.uiuc.edu/~fjw/pisces/.
35. Z. YOU, E. J. KOSTELICH, AND J. A. YORKE, Calculating stable and unstable manifolds, 1 (1991), pp. 605-623.