Advertisement

Extended hierarchies of invariant fiber bundles for dynamic equations on measure chains

  • Christian PötzscheEmail author
Original Research
  • 40 Downloads

Abstract

If a linear autonomous ordinary differential of difference equation possesses a coefficient operator, which is (pseudo-) hyperbolic or allows a more specific splitting of its spectrum into appropriate spectral sets, then this gives rise to a so-called hierarchy of invariant linear subspaces of X related to the ranges to the corresponding spectral projections. Together with the intersections of these invariant subspaces, we get an extended hierarchy. Each member of the hierarchy can be characterized dynamically as set of initial points for orbits with a certain asymptotic growth rate in forward or backward time.

In this paper we show that such a scenario persists under perturbations w.r.t. two points of view: In the first instance, the invariant linear spaces become an “extended hierarchy” of invariant manifolds, if the linear part is perturbed by a globally Lipschitzian (or smooth) mapping on X. This will be done in the nonautonomous context of dynamic equations on measure chains or time scales, where the time-varying invariant manifolds are called invariant fiber bundles. Secondly, we derive perturbation results well-suited for up-coming applications in analytical discretization theory.

Keywords

Invariant fiber bundles Extended hierarchy Exponential splitting Time scale Dynamic equation 

Mathematics Subject Classification (2000)

Primary 37D10, 39A11 Secondary 37C60, 37B55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aulbach B., Hierarchies of invariant manifolds, J. Nigerian Mathematical Society, 6, 71–89, (1987)Google Scholar
  2. 2.
    Aulbach B., Hierarchies of invariant fiber bundles, Southeast Asian Bulletin of Math., 19, 91–98, (1995)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Aulbach B., The fundamental existence theorem on invariant fiber bundles, Journal of Difference Equations and Applications, 3, 501–537, (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Aulbach B. and Wanner T., Integral manifolds for Carathéodory type differential equations in Banach spaces, Six Lectures on Dynamical Systems (Aulbach B. and Colonius F., eds.), World Scientific, Singapore, 45–119, (1996)Google Scholar
  5. 5.
    Aulbach B. and Pötzsche C., Reducibility of linear dynamic equations on measure chains, Journal of Computational and Applied Mathematics, 141, 101–115, (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bohner M. and Peterson A., Dynamic Equations on Time Scales - An Introduction with Applications, Birkhäuser, Boston, (2001)zbMATHGoogle Scholar
  7. 7.
    Chow S.-N. and Hale J. K., Methods of Bifurcation Theory, Springer-Verlag, Grundlehren der mathematischen Wissenschaften, 251, Berlin-Heidelberg-New York, (1996)Google Scholar
  8. 8.
    Hartman P., Ordinary Differential Equations, John Wiley & Sons, Chichester, (1964)zbMATHGoogle Scholar
  9. 9.
    Hilger S., Analysis on measure chains - A unified approach to continuous and discrete calculus, Results in Mathematics, 18, 18–56, (1990)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Hirsch M. W., Pugh C. C. and Shub M., Invariant Manifolds, Lecture Notes in Mathematics 583, Springer-Verlag, Berlin, (1977)Google Scholar
  11. 11.
    Kelley A., The stable, center-stable, center, center-unstable, unstable manifolds, Journal of Differential Equations, 3, 546–570, (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Keller S., Asymptotisches Verhalten invarianter Faserbündel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen (in german), Ph.D. Thesis, Univ. Augsburg, (1999)Google Scholar
  13. 13.
    Keller S. and Pötzsche C., Integral manifolds under explicit variable time-step discretization, Journal of Difference Equations and Applications, 12(3–4), 321–342, (2005)Google Scholar
  14. 14.
    Kriegl A. and Michor P. W., The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, Providence, (1997)Google Scholar
  15. 15.
    Neidhart L., Integration im Rahmen des Maßkettenkalküls (in german), Thesis, Univ. Augsburg, (2001)Google Scholar
  16. 16.
    Pötzsche C., Langsame Faserbündel dynamischer Gleichungen auf Maßketten (in german), Ph.D. Thesis, Univ. Augsburg, (2002)Google Scholar
  17. 17.
    Pötzsche C., Pseudo-stable and pseudo-unstable fiber bundles for dynamic equations on measure chains, Journal of Difference Equations and Applications, 9(10), 947–968, (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pötzsche C., Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients, Journal of Mathematical Analysis and Applications, 289, 317–335, (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Pötzsche C., Invariant foliations and stability in critical cases, Advances in Difference Equations, 2006, 19, (2006)CrossRefGoogle Scholar
  20. 20.
    Pötzsche C., Topological decoupling, linearization and perturbation on inhomogenous time scales, Journal of Differential Equations, 245, 1210–1242, (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Pötzsche C., Topological linearization under explicit variable time-step discretization, in preparationGoogle Scholar
  22. 22.
    Pötzsche C. and Siegmund S., C m-smoothness of invariant fiber bundles for dynamic equations on measure chains, Advances in Difference Equations, 2, 141–182, (2004)CrossRefGoogle Scholar
  23. 23.
    Siegmund S., Spektraltheorie, glatte Faserungen und Normalformen für Differentialgleichungen vom Carathéodory-Typ (in german), Ph.D. thesis, Universität Augsburg, (1999)Google Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2010

Authors and Affiliations

  1. 1.Centre forMathematical SciencesMunich University of TechnologyGarchingGermany

Personalised recommendations