Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-12T19:39:40.302Z Has data issue: false hasContentIssue false
  • English
  • Français

LOWER BOUNDS ALONG STABLE MANIFOLDS

Part of: Stability theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  02 October 2017

LUIS BARREIRA  and
CLAUDIA VALLS
LUIS BARREIRA
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal e-mail: barreira@math.tecnico.ulisboa.pt, cvalls@math.tecnico.ulisboa.pt
CLAUDIA VALLS
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal e-mail: barreira@math.tecnico.ulisboa.pt, cvalls@math.tecnico.ulisboa.pt
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is well known that along any stable manifold the dynamics travels with an exponential rate. Moreover, this rate is close to the slowest exponential rate along the stable direction of the linearization, provided that the nonlinear part is sufficiently small. In this note, we show that whenever there is also a fastest finite exponential rate along the stable direction of the linearization, similarly we can establish a lower bound for the speed of the nonlinear dynamics along the stable manifold. We consider both cases of discrete and continuous time, as well as a nonuniform exponential behaviour.

MSC classification

Primary: 34D99: None of the above, but in this section
Secondary: 37C75: Stability theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 3 , September 2018 , pp. 527 - 537
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Barreira, L. and Pesin, Ya., Lyapunov exponents and smooth ergodic theory, University Lecture Series, vol. 23 (American Mathematical Society, Providence, RI, 2002).Google Scholar
2. Barreira, L. and Valls, C., Characterization of stable manifolds for nonuniform exponential dichotomies, Discrete Contin. Dyn. Syst. 21 (2008), 1025–1046.CrossRefGoogle Scholar
3. Barreira, L. and Valls, C., Nonuniform exponential contractions and Lyapunov sequences, J. Differ. Equ. 246 (2009), 4743–4771.CrossRefGoogle Scholar
4. Coppel, W., Dichotomies in stability theory, Lecture Notes in Mathematics, vol. 629 (Springer-Verlag, Berlin-New York, 1978).CrossRefGoogle Scholar
5. Hale, J., Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25 (American Mathematical Society, Providence, RI, 1988).Google Scholar
6. Henry, D., Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840 (Springer-Verlag, Berlin-New York, 1981).CrossRefGoogle Scholar
7. Mañé, R., Lyapounov exponents and stable manifolds for compact transformations, in Geometric dynamics (Rio de Janeiro, 1981) (Palis J., Editor), Lecture Notes in Mathematics, vol. 1007 (Springer, Berlin, 1983), 522–577.CrossRefGoogle Scholar
8. Pesin, Ya., Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv. 10 (1976), 1261–1305.CrossRefGoogle Scholar
9. Ruelle, D., Characteristic exponents and invariant manifolds in Hilbert space, Ann. Math. 115 (2) (1982), 243–290.CrossRefGoogle Scholar
10. Sell, G. and You, Y., Dynamics of evolutionary equations, Applied Mathematical Sciences, vol. 143 (Springer-Verlag, New York, 2002).Google Scholar