Approximation Scheme of a Center Manifold for ... - CiteSeerX

The center manifold theory is one of the most effective tools in the study of the dynamics of differential systems and especially in the infinite- dimensional case.
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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

213, 554]572 Ž1997.

AY975558

Approximation Scheme of a Center Manifold for Functional Differential Equations M. Ait Babram and M. L. Hbid Departement de Mathematiques, L.P.S.S.D., Faculte´ des Sciences Semlalia, Uni¨ ersite´ ´ ´ Cadi Ayyad, B.P. S15, Marrakech, Morocco

and O. Arino Departement de Mathematiques, I.P.R.A, URA 1204, Uni¨ ersite´ de Pau, 1 A¨ enue de ´ ´ l’Uni¨ ersite, ´ 64000, Pau, France Submitted by Jack K. Hale Received February 20, 1997

In this paper, we consider a general autonomous functional differential equation having a local center manifold. Then, we give an algorithmic procedure to compute the terms in the Taylor expansion of this manifold up to any order. Q 1997 Academic Press

1. INTRODUCTION The center manifold theory is one of the most effective tools in the study of the dynamics of differential systems and especially in the infinitedimensional case. When an equilibrium point is non-hyperbolic, this theory allows us to reduce the study of a system near this point to that of an ordinary differential equation. The approach is particularly interesting when the reduced system is finite dimensional. The effectiveness of this theory has been illustrated in various set-ups and by many authors, see for instance the work of Diekmann and van Gils w4x, Vanderbauwhede and Iooss w10x, and Henry w9x. In this paper, we are dealing with application to functional differential equations. Let us briefly fix the framework. For r G 0, we denote by 554 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

FUNCTIONAL DIFFERENTIAL EQUATIONS

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C s C Žwyr, 0x, R n . the Banach space of continuous functions from wyr, 0x to R n endowed with the supremum norm 5 f 5 s supu g wyr, 0x < f Ž u .