Invariant foliations near normally hyperbolic invariant manifolds for semiflows

Bates, Peter; Lu, Kening; Zeng, Chongchun

doi:10.1090/S0002-9947-00-02503-4

Invariant foliations near normally hyperbolic invariant manifolds for semiflows
HTML articles powered by AMS MathViewer

by Peter W. Bates, Kening Lu and Chongchun Zeng PDF

Trans. Amer. Math. Soc. 352 (2000), 4641-4676 Request permission

Abstract:

Let $M$ be a compact $C^1$ manifold which is invariant and normally hyperbolic with respect to a $C^1$ semiflow in a Banach space. Then in an $\epsilon$-neighborhood of $M$ there exist local $C^1$ center-stable and center-unstable manifolds $W^{cs}(\epsilon )$ and $W^{cu}(\epsilon )$, respectively. Here we show that $W^{cs}(\epsilon )$ and $W^{cu}(\epsilon )$ may each be decomposed into the disjoint union of $C^1$ submanifolds (leaves) in such a way that the semiflow takes leaves into leaves of the same collection. Furthermore, each leaf intersects $M$ in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time.

References

D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. , Proceedings of the Steklov Institute of Mathematics, No. 90 (1967), American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by S. Feder. MR 0242194
Bernd Aulbach and Barnabas M. Garay, Partial linearization for noninvertible mappings, Z. Angew. Math. Phys. 45 (1994), no. 4, 505–542. MR 1289659, DOI 10.1007/BF00991895
Peter W. Bates and Christopher K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 1–38. MR 1000974
Peter W. Bates and Kening Lu, A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations, J. Dynam. Differential Equations 6 (1994), no. 1, 101–145. MR 1262725, DOI 10.1007/BF02219190
Peter W. Bates, Kening Lu, and Chongchun Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc. 135 (1998), no. 645, viii+129. MR 1445489, DOI 10.1090/memo/0645
P. W. Bates, K. Lu and C. Zeng, Invariant Manifolds and Invariant Foliations for Semiflows, book, in preparation.
Almut Burchard, Bo Deng, and Kening Lu, Smooth conjugacy of centre manifolds, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), no. 1-2, 61–77. MR 1149984, DOI 10.1017/S0308210500014980
Xu-Yan Chen, Jack K. Hale, and Bin Tan, Invariant foliations for $C^1$ semigroups in Banach spaces, J. Differential Equations 139 (1997), no. 2, 283–318. MR 1472350, DOI 10.1006/jdeq.1997.3255
Shui-Nee Chow and Xiao-Biao Lin, Bifurcation of a homoclinic orbit with a saddle-node equilibrium, Differential Integral Equations 3 (1990), no. 3, 435–466. MR 1047746
Shui-Nee Chow, Xiao-Biao Lin, and Kening Lu, Smooth invariant foliations in infinite-dimensional spaces, J. Differential Equations 94 (1991), no. 2, 266–291. MR 1137616, DOI 10.1016/0022-0396(91)90093-O
Shui-Nee Chow and Kening Lu, Invariant manifolds and foliations for quasiperiodic systems, J. Differential Equations 117 (1995), no. 1, 1–27. MR 1320181, DOI 10.1006/jdeq.1995.1046
Shui-Nee Chow, Kening Lu, and John Mallet-Paret, Floquet theory for parabolic differential equations, J. Differential Equations 109 (1994), no. 1, 147–200. MR 1272403, DOI 10.1006/jdeq.1994.1047
Bo Deng, The Šil′nikov problem, exponential expansion, strong $\lambda$-lemma, $C^1$-linearization, and homoclinic bifurcation, J. Differential Equations 79 (1989), no. 2, 189–231. MR 1000687, DOI 10.1016/0022-0396(89)90100-9
Bo Deng, The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations, SIAM J. Math. Anal. 22 (1991), no. 6, 1631–1650. MR 1129402, DOI 10.1137/0522102
Neil Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23 (1973/74), 1109–1137. MR 339276, DOI 10.1512/iumj.1974.23.23090
Neil Fenichel, Asymptotic stability with rate conditions. II, Indiana Univ. Math. J. 26 (1977), no. 1, 81–93. MR 426056, DOI 10.1512/iumj.1977.26.26006
Neil Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), no. 1, 53–98. MR 524817, DOI 10.1016/0022-0396(79)90152-9
Robert A. Gardner, An invariant-manifold analysis of electrophoretic traveling waves, J. Dynam. Differential Equations 5 (1993), no. 4, 599–606. MR 1250262, DOI 10.1007/BF01049140
I. Gasser and P. Szmolyan, A geometric singular perturbation analysis of detonation and deflagration waves, SIAM J. Math. Anal. 24 (1993), no. 4, 968–986. MR 1226859, DOI 10.1137/0524058
J. Hadamard, Sur l’iteration et les solutions asymptotiques des equations differentielles, Bull. Soc. Math. France 29 (1901), 224–228.
Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR 0271991
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173, DOI 10.1007/BFb0092042
G. Haller and S. Wiggins, $N$-pulse homoclinic orbits in perturbations of resonant Hamiltonian systems, Arch. Rational Mech. Anal. 130 (1995), no. 1, 25–101. MR 1350402, DOI 10.1007/BF00375655
C. K. R. T. Jones, Geometric singular perturbation theory, C.I.M.E. Lectures (1994).
C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations 108 (1994), no. 1, 64–88. MR 1268351, DOI 10.1006/jdeq.1994.1025
Gregor Kovačič and Stephen Wiggins, Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation, Phys. D 57 (1992), no. 1-2, 185–225. MR 1169620, DOI 10.1016/0167-2789(92)90092-2
U. Kirchgraber and K. J. Palmer, Geometry in the neighborhood of invariant manifolds of maps and flows and linearization, Pitman Research Notes in Mathematics Series, vol. 233, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1990. MR 1068954
Xiao-Biao Lin, Shadowing lemma and singularly perturbed boundary value problems, SIAM J. Appl. Math. 49 (1989), no. 1, 26–54. MR 978824, DOI 10.1137/0149002
Y. Li, David W. McLaughlin, Jalal Shatah, and S. Wiggins, Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation, Comm. Pure Appl. Math. 49 (1996), no. 11, 1175–1255. MR 1406663, DOI 10.1002/(SICI)1097-0312(199611)49:11<1175::AID-CPA2>3.3.CO;2-B
Charles Li and Stephen Wiggins, Invariant manifolds and fibrations for perturbed nonlinear Schrödinger equations, Applied Mathematical Sciences, vol. 128, Springer-Verlag, New York, 1997. MR 1475929, DOI 10.1007/978-1-4612-1838-8
Rafael de la Llave, Invariant manifolds associated to nonresonant spectral subspaces, J. Statist. Phys. 87 (1997), no. 1-2, 211–249. MR 1453740, DOI 10.1007/BF02181486
Kening Lu, A Hartman-Grobman theorem for scalar reaction-diffusion equations, J. Differential Equations 93 (1991), no. 2, 364–394. MR 1125224, DOI 10.1016/0022-0396(91)90017-4
K. Lu, Structural stability for time periodic parabolic equations, US-Chinese Conference on Differential Equations and their Applications, edited by P. Bates, et al. (1997), 207-214.
A. M. Liapunov, Problème géneral de la stabilité du mouvement, Annals Math. Studies 17 (1947).
Ricardo Mañé, Lyapounov exponents and stable manifolds for compact transformations, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 522–577. MR 730286, DOI 10.1007/BFb0061433
David Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2) 115 (1982), no. 2, 243–290. MR 647807, DOI 10.2307/1971392
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817.
P. Szmolyan, Analysis of a singularly perturbed traveling wave problem, SIAM J. Appl. Math. 52 (1992), no. 2, 485–493. MR 1154784, DOI 10.1137/0152027
D. Terman, The transition from bursting to continuous spiking in excitable membrane models, J. Nonlinear Sci. 2 (1992), no. 2, 135–182. MR 1169590, DOI 10.1007/BF02429854
Stephen Wiggins, Normally hyperbolic invariant manifolds in dynamical systems, Applied Mathematical Sciences, vol. 105, Springer-Verlag, New York, 1994. With the assistance of György Haller and Igor Mezić. MR 1278264, DOI 10.1007/978-1-4612-4312-0