Topological decoupling, linearization and perturbation on inhomogeneous time scales

Journal of Differential Equations

Volumn 245, Issue 5, 2008, Pages 1210-1242

  Pötzsche, Christian ^a  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

Discretization; Invariant fiber bundles; Invariant foliations; Perturbation; Time scales; Topological decoupling; Topological linearization

Indexed keywords

EID: 46449083427     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2008.06.011     Document Type: Article

References (27)

1. Aulbach B., and Garay B.M. Partial linearization for noninvertible mappings. Z. Angew. Math. Phys. 45 4 (1994) 505-542
2. Aulbach B., and Wanner T. The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spaces. Nonlinear Anal. 40 (2000) 91-104
3. Aulbach B., and Wanner T. Topological simplification of nonautonomous difference equations. J. Difference Equ. Appl. 12 3-4 (2006) 283-296
4. Bohner M., and Peterson A.C. Dynamic Equations on Time Scales-An Introduction with Applications (2001), Birkhäuser, Boston/Basel/Berlin
5. Farkas G. A Hartman-Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria. Z. Angew. Math. Phys. 52 (2001) 421-432
6. Grobman D.M. Homeomorphism of systems of differential equations. Dokl. Akad. Nauk SSSR 128 (1959) 880-881
7. Grobman D.M. Topological classification of neighborhoods of a singularity in n-space. Mat. Sb. 56 98 (1962) 77-94
8. Hartman P. A lemma in the theory of structural stability of differential equations. Proc. Amer. Math. Soc. 11 (1960) 610-620
9. Hartman P. On local homeomorphisms of euclidean spaces. Bol. Soc. Mat. Mexicana 2 5 (1960) 220-241
10. Hilger S. Analysis on measure chains-A unified approach to continuous and discrete calculus. Results Math. 18 (1990) 18-56
11. Hilger S. Generalized theorem of Hartman-Grobman on measure chains. J. Aust. Math. Soc. Ser. A 60 (1996) 157-191
12. Kirchgraber U., and Palmer K.J. Geometry in the Neighborhood of Invariant Manifolds of Maps and Flows and Linearization. Pitman Res. Notes Math. vol. 233 (1990), Longman, Burnt Mill
13. Keller S., and Pötzsche C. Integral manifolds under explicit variable time-step discretization. J. Difference Equ. Appl. 12 3-4 (2005) 321-342
14. López-Fenner J., and Pinto M. On a Hartman linearization theorem for a class of ODE with impulse effect. Nonlinear Anal. 38 (1999) 307-325
15. Lu K. A Hartman-Grobman theorem for scalar reaction-diffusion equations. J. Differential Equations 93 (1991) 364-394
16. Muldowney J.S. Dichotomies and asymptotic behaviour for linear differential systems. Trans. Amer. Math. Soc. 283 (1984) 465-484
17. L. Neidhart, Integration im Rahmen des Maßkettenkalküls, Thesis, Universität Augsburg, Germany, 2001
18. Palmer K.J. A generalization of Hartman's linearization theorem. J. Math. Anal. Appl. 41 (1973) 753-758
19. K.J. Palmer, Qualitative behavior of a system of ODE near an equilibrium point-A generalization of the Hartman-Grobman theorem, Preprint 372, Institut für Angewandte Mathematik, Universität Bonn, 1980
20. C. Pötzsche, Langsame Faserbündel dynamischer Gleichungen auf Maßketten, PhD thesis, Universität Augsburg, Germany, 2002
21. C. Pötzsche, Extended hierarchies of invariant fiber bundles for dynamic equations on measure chains, preprint, 2003
22. Pötzsche C. Invariant foliations and stability in critical cases. Adv. Difference Equ. 2006 (2006) 19
23. C. Pötzsche, Topological linearization under explicit variable time-step discretization, in preparation
24. S. Siegmund, Spektraltheorie, glatte Faserungen und Normalformen für Differentialgleichungen vom Carathéodory-typ, PhD thesis, Universität Augsburg, Germany, 1999
25. Šošitai{dotless}̌švili A.N. Bifurcations of topological type at singular points of parametrized vector fields. Funct. Anal. Appl. 5 (1972) 169-170
26. T. Wanner, Invariante Faserbündel und topologische Äquivalenz bei dynamischen Prozessen, Thesis, Universität Augsburg, Germany, 1991
27. Xia Y., Cao J., and Han M. A new analytical method for the linearization of dynamic equation on measure chains. J. Differential Equations 235 (2007) 527-543