Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T14:33:17.457Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of L1 contractions†

Published online by Cambridge University Press:  01 April 2015

LUIS BARREIRA  and
CLAUDIA VALLS
LUIS BARREIRA
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal. e-mail: [email protected]; [email protected]
CLAUDIA VALLS
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal. e-mail: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The notion of an exponential contraction is only one among many possible rates of contraction of a nonautonomous system, while for an autonomous system all contractions are exponential. We consider the notion of an L1 contraction that includes exponential contractions as a very particular case and that is naturally adapted to the variation-of-parameters formula. Both for discrete and continuous time, we show that under very general assumptions the notion of an L1 contraction persists under sufficiently small linear and nonlinear perturbations, also maintaining the type of stability. As a natural development, we establish a version of the Grobman–Hartman theorem for nonlinear perturbations of an L1 contraction.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 1 , July 2015 , pp. 23 - 46
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Supported by Portuguese funds through FCT - Fundação para a Ciência e a Tecnologia: project PEst-OE/EEI/LA0009/2013 (CAMGSD)

References

REFERENCES

[1] Grobman, D. Homeomorphism of systems of differential equations. Dokl. Akad. Nauk SSSR 128 (1959), 880–881.Google Scholar
[2] Grobman, D. Topological classification of neighbourhoods of a singularity in n-space Mat. Sb. (N.S.) 56 (98) (1962), 77–94.Google Scholar
[3] Hartman, P. A lemma in the theory of structural stability of differential equations. Proc. Amer. Math. Soc. 11 (1960), 610–620.Google Scholar
[4] Hartman, P. On the local linearization of differential equations. Proc. Amer. Math. Soc. 14 (1963), 568–573.Google Scholar
[5] Moser, J. On a theorem of Anosov. J. Differential Equations 5 (1969), 411–440.Google Scholar
[6] Palis, J. On the local structure of hyperbolic points in Banach spaces. An. Acad. Brasil. Ci. 40 (1968), 263–266.Google Scholar
[7] Palmer, K. A generalisation of Hartman's linearisation theorem. J. Math. Anal. Appl. 41 (1973), 753–758.Google Scholar
[8] Pugh, C. On a theorem of P. Hartman. Amer. J. Math. 91 (1969), 363–367.Google Scholar