Invariant foliations near normally hyperbolic invariant manifolds for semiflows
Authors:
Peter W. Bates, Kening Lu and Chongchun Zeng
Journal:
Trans. Amer. Math. Soc. 352 (2000), 46414676
MSC (2000):
Primary 37D30, 37L45; Secondary 53C12, 37D10, 37K55
Published electronically:
June 14, 2000
MathSciNet review:
1675237
Fulltext PDF Free Access
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Abstract: Let be a compact manifold which is invariant and normally hyperbolic with respect to a semiflow in a Banach space. Then in an neighborhood of there exist local centerstable and centerunstable manifolds and , respectively. Here we show that and may each be decomposed into the disjoint union of submanifolds (leaves) in such a way that the semiflow takes leaves into leaves of the same collection. Furthermore, each leaf intersects in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time.
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 P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach spaces, Mem. Amer. Math. Soc. 135 (1998). MR 99b:58210
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 A. Burchard, B. Deng and K. Lu, Smooth conjugacy of centre manifolds, Proceedings of the Royal Society of Edingburgh 120A (1992), 6177. MR 93c:58197
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 SN. Chow and K. Lu, Invariant manifolds and foliations for quasiperiodic systems, J. Differential Equations 117 (1995), 127. MR 96b:34064
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 SN. Chow, K. Lu and J. MalletParet, Floquet theory for parabolic equations, J. Differential Equations 109 (1994), 147200. MR 95c:35116
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 B. Deng, The Sil'nikov problem, exponential expansion, strong lemma, linearization, and homoclinic bifurcation, J. Differential Equations 82 (1989), 156173. MR 90k:58161
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 [HP]
 M. W. Hirsch and C. C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis, Proc. Sympos. Pure Math. 14 (1970), 133163. MR 42:6872
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 M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, vol. 583, SpringerVerlag, New York, 1977. MR 58:18595
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 Y. Li, D. W. McLaughlin, J. Shatah and S. Wiggins, Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation, Comm. Pure Appl. Math 49 (1996), 11751255. MR 98d:35208
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 Y. Li and S. Wiggins, Invariant manifolds and fibrations for perturbed nonlinear Schrödinger equations, Applied Mathematical Sciences, 128, SpringerVerlag, New York, 1997. MR 99e:58165
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 R. de la Llave, Invariant manifolds associated to nonresonant spectral subspaces, J. Statist. Phys. 87 (1997), 211249. MR 98f:58038
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 K. Lu, A HartmanGrobman theorem for scalar reactiondiffusion equations, J. Differential Equations 93 (1991), 364394. MR 92k:35147
 [Lu2]
 K. Lu, Structural stability for time periodic parabolic equations, USChinese Conference on Differential Equations and their Applications, edited by P. Bates, et al. (1997), 207214. CMP 98:08
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Additional Information
Peter W. Bates
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email:
peter@math.byu.edu
Kening Lu
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email:
klu@math.byu.edu
Chongchun Zeng
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, New York 10012
Email:
zengch@math1.cims.nyu.edu
DOI:
http://dx.doi.org/10.1090/S0002994700025034
PII:
S 00029947(00)025034
Received by editor(s):
December 18, 1996
Received by editor(s) in revised form:
June 5, 1998
Published electronically:
June 14, 2000
Additional Notes:
The first author was partially supported by NSF grant DMS9622785 and the Isaac Newton Institute
The second author was partially supported by NSF grant DMS9622853
The third author was partially supported by the Isaac Newton Institute
Article copyright:
© Copyright 2000 American Mathematical Society
