A note on the convergence of parametrised non-resonant invariant manifolds

Qualitative Theory of Dynamical Systems

Volumn 10, Issue 1, 2011, Pages 107-121

  Johnson, Tomas ^a   Tucker, Warwick ^a  

^a UPPSALA UNIVERSITY   (Sweden)

Author keywords

Auto validated numerics; Hyperbolic fixed points; Invariant manifolds; Normal forms

Indexed keywords

EID: 84896692987     PISSN: 15755460     EISSN: 16623592     Source Type: Journal    
DOI: 10.1007/s12346-011-0040-2     Document Type: Article

References (27)

1. Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, NewYork (1983)
2. Beyn, W.-J.: The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal. 10(3), 379-405 (1990)
3. Blomquist, F., Hofschuster, W., Krämer, W.: Real and Complex Taylor Arithmetic in C-XSC Preprint 2005/4, Universität Wuppertal (2005). Available from http://www.math.uni-wuppertal.de/xsc
4. Cabré, X., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces. Indiana Univ. Math. J. 52(2), 283-328 (2003)
5. Cabré, X., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds. III. Overview and applications. J. Differ. Equ. 218(2), 444-515 (2005)
6. CXSC-C++ eXtension for Scientific Computation, version 2.2.3. Available from http://www.math. uni-wuppertal.de/xsc
7. Dellnitz, M., Hohmann, A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75(3), 293-317 (1997)
8. Griewank, A.: Evaluating derivatives: principles and techniques of algorithmic differentiation, SIAM Frontiers Appl. Math. 19, SIAM, Philadelphia (2000)
9. Hammer, R., Hocks, M., Kulisch, U., Ratz, D.: C++ toolbox for verified computing. Springer, New York (1995)
10. Hille, E.: Ordinary Differential Equations In The Complex Domain. Pure And Applied Mathematics. Wiley-Interscience, New York (1976)
11. Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Springer, Berlin (1977)
12. Johnson, T., Tucker, W.: Automated computation of robust normal forms of planar analytic vector fields. Discret. Contin. Dyn. Syst.-Series B. 12(4), 769-782 (2009)
13. 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. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15(3), 763-791 (2005)
14. Moore, G., Hubert, E.: Algorithms for constructing stable manifolds of stationary solutions. IMA J. Numer. Anal. 19(3), 375-424 (1999)
15. Moore, R.E.: Interval Analysis. Prentice-Hall, New Jersey (1966)
16. Moore, R.E.: Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics, SIAM, Philadelphia (1979)
17. Neumaier, A.: Interval Methods For Systems Of Equations. Encyclopedia of Mathematics and its Applications, vol. 37. Cambridge University Press, Cambridge (1990)
18. Neumaier, A., Rage, T.: Rigorous chaos verification in discrete dynamical systems. Phys. D 67(4), 327-346 (1993)
19. Ombach, J.: Computation of the local stable and unstable manifolds. Univ. Iagel. Acta Math. 32, 129-136 (1995)
20. Palis, J., de Melo, W.: Geometric theory of dynamical systems. An introduction. Translated from the Portuguese by A. K. Manning. Springer, New York (1982)
21. Siegel, C.L.: Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung. Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt, pp. 21-30 (1952)
22. Siegel, C.L., Moser, J.K.: Lectures on celestial mechanics. Translation by Charles I. Kalme. Die Grundlehren der mathematischen Wissenschaften, Band 187. Springer, New York (1971)
23. Simó, C.: On theAnalytical andNumerical Approximation of InvariantManifolds, Les MéthodesModernes de la Mecánique Céleste, Benest, D., Foeschlé, C. (eds.) Editions Frontièrs, Paris, pp. 285-329 (1990)
24. Tucker, W.: A rigorous ODE solver and Smale's 14th problem. Found. Comput. Math. 2(1), 53-117 (2002)
25. Tucker, W.: Robust normal forms for saddles of analytic vector fields. Nonlinearity 17, 1965-1983 (2004)
26. Zgliczyński, P.: Covering relations, cone conditions and stable manifold theorem. J. Differ. Equ. 246(5), 1774-1819 (2009)
27. Zgliczyński, P.: Private communication (2008)