Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-14T22:26:53.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Partial and complete linearisations at stationary points of infinite-dimensional dynamical systems with foliations and applications*

Published online by Cambridge University Press:  14 November 2011

W. M. Rivera
W. M. Rivera
Affiliation:
CDSNSf†, Georgia Tech, Atlanta, GA 30332; West Georgia College, Carrollton, GA 30118, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we discuss C1-linearisations of diffeomorphisms and flows on Banach spaces. Strong foliations of the neighbourhood of the fixed point composed of leaves based on successively larger subspaces (similar to those in [14]) are constructed. Generalised gap conditions which involve the width and separation of vertical bands containing the spectrum of a linear operator are imposed to achieve maximal smoothness. The method of proof generalises that of Hartman and of Mora and Solá-Morales. Our theorems apply to weakly coupled systems of damped wave and beam equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 5 , 1996 , pp. 1067 - 1085
Copyright
Copyright © Royal Society of Edinburgh 1996

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.)

References

1Afraimovich, V.. On smooth changes in variables. Selecta Math. Soviet. 9, No. 3 (1990).Google Scholar
2Burchard, A., Deng, B. and Lu, K.. Smooth conjugacy of center manifolds (CDSNS Preprint #91-46).Google Scholar
3Chow, S. N.. Lin, X.-B. and Lu, K.. Smooth invariant foliations in infinite dimensional space (IMA Preprint Series #685).Google Scholar
4Chow, S. N.. L, K.. and Sell, G.. Smoothness of inertial manifolds (Preprint, 1989).Google Scholar
5Deng, B.. The Silnikov problem, exponential expansion, λ-lemma, C1 linearisation and homoclinic bifurcation. J. Differential Equations 79 (1989), 189–231.CrossRefGoogle Scholar
6Fenichel, N.. Asymptotic stability with rate conditions II. Indiana Univ. Math. J. 26 (1977), 81–93.CrossRefGoogle Scholar
7Hale, J. K.. Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25 (Providence, RI: American Mathematical Society, 1989).Google Scholar
8Hanse, S.. and Zuazua, E.. Exact controllability and stabilization of a vibrating string with an interior point mass (IMA Preprint Series #1140, June 1993).Google Scholar
9Hartman, P.. On local homeomorphisms of Euclidean spaces. Sociedad Mathematica Mexicana 5 (1960), 220–41.Google Scholar
10Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds, Lecture Notes in Mathematics 583 (Berlin: Springer, 1977).CrossRefGoogle Scholar
11Holmes, P. and Marsden, J.. Chaotic oscillations of a forced beam. Arch. Rational Mech. Anal. 76 (1981), 135–66xs.CrossRefGoogle Scholar
12, Krein and , Dalekii. Stability of Solutions of Differential Equations in Banach Space (Providence, RI: American Mathematical Society, 1974).Google Scholar
13Llave, R. de la. Some new invariant manifold theorems and applications to the study of cohomology equations and smooth conjugacy (Preprint, 1992).Google Scholar
14Lu, K.. A Hartman-Grobman theorem for scalar reaction diffusion equations. J. Differential Equations 93 (1991), 364–94.CrossRefGoogle Scholar
15Mora, X. and Sola-Morales, J.. Existence and non-existence of finite dimensional globally attracting invariant manifolds in semilinear damped wave equation. In Dynamics of Infinite Dimensional Systems, 187–120 (New York: Springer, 1987).Google Scholar
16Palis, J. and Melo, W. de. Geometric Theory of Dynamical Systems (New York: Springer, 1982).CrossRefGoogle Scholar
17Pugh, C.. On a theorem of P. Hartman. Amer. J. Math. 19 (1969), 363–7.CrossRefGoogle Scholar
18Rivera, W.. A new invariant manifold with an application to a smooth conjugacy at a node. J. Dynam. Systems Applications (to appear).Google Scholar
19Ruelle, D.. Elements of Differentiable Dynamics and Bifurcation Theory (New York: Academic Press, 1989).Google Scholar
20Shub, M.. Global Stability of Dynamical Systems (New York: Springer, 1987).CrossRefGoogle Scholar
21Sternberg, S.. Local contractions and a theorem of Poincare. Amer. J. Math. 79 (1957), 809–24.CrossRefGoogle Scholar