Abstract
We study the existence and smoothness of global center, center-stable, and center-unstable manifolds for skew-product flows. Smooth invariant foliations to the center stable and center unstable manifolds are also discussed.
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References
Carr, J. (1981).Applications of Center Manifold Theory, Springer-Verlag, New York.
Chow, S.-N., and Hale, J. K. (1982).Methods of Bifurcation Theory, Springer-Verlag, New York.
Fenichel, N. (1971). Persistence and smoothness of invariant manifolds for flows.Indiana Univ. Math. J 21, 192–226.
Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations.J. Diff. Eqs. 31, 53–89.
Hale, J. K. (1961). Integral manifolds of perturned differential systems.Ann. Math. 43, 496–531.
Henry, D. (1981). Geometric theory of semilinear parabolic equations.Lect. Notes Math. 840, Springer-Verlag, New York.
Hirsch, M. W., Pugh, C. C., and Shub, M. (1970). Invariant manifolds.Bull. Am. Math. Soc. 76, 1015–1019.
Johnson, R. A. (1987). Concerning a theorem of Sell.J. Diff. Eqs. 30, 324–339.
Johnson, R. A., and Sell, G. R. (1981). Smoothness of spectral subbundles and reducibility of Q-periodic linear differential systems.J. Diff. Eqs. 41, 262–288.
Johnson, R. A., and Yi, Y. (1994). Hopf bifurcations from nonperiodic solutions of differential equations.J. Diff. Eqs. 107, 310–340.
Johnson, R. A., Palmer, K. J., and Sell, G. R. (1987). Ergodic properties of linear dynamical systems.SIAM J. Math. Anal. 18, 1–33.
Kelley, A. (1967). The stable, center-stable, center unstable and unstable manifolds.J. Diff. Eqs. 3, 546–570.
Lu, K. (1991). A Hartmann-Grobman theorem for scalar reaction-diffusion equations.J. Diff. Eqs. 93, 364–194.
Palmer, K. J. (1986). Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system.J. Diff. Eqs. 65, 321–360.
Palmer, K. J. (1987). On the stability of center manifolds.J. Appl. Math. Phys. 38, 273–278.
Pliss, V. (1964). Principal reduction in the theory of stability of motion.Izv. Akad. Nauk. SSSR Mat. Ser. 28, 1297–1324 (Russian).
Rybakowski, K. P. (1990). An implicit function theorem for a scale of Banach spaces and smoothness of invariant manifolds. CDSNS preprint.
Sacker, R. J. (1965). A new approach to the perturbation theopry of invariant surfaces.Comm. Pure Appl. Math. 18, 717–732
Sacker, R. J., and Sell, G. R. (1974/1976). Existence of dichotomies and invariant splitting for linear differential systems, I and II.J. Diff. Eqs. 15/22, 429–458.
Sacker, R. J., and Sell, G. R. (1978). A spectral theory for linear differential systems.J. Diff. Eqs. 27, 320–358.
Sacker, R. J., and Sell, G. R. (1980). The spectrum of an invariant submanifold.J. Diff. Eqs. 38, 135–160.
Sell, G. R. (1971).Topological Dynamical Systems and ODE, Van Nostrand, New York.
Sell, G. R. (1978). The structure of a flow in the vicinity of an almost periodic motion.J. Diff. Eqs. 27(3), 359–393.
Vanderbauwhede, A. (1989). Center manifolds, normal forms and elementary bifurcations.Dynam. Report. 2, 89–170.
Vanderbauwhede, A., and Van Gils, S. A. (1987). Center manifolds and contractions on a scale of B-spaces.J. Funct. Anal. 72, 209–224.
Yi, Y. (1993a). Generalized integral manifold theorem.J. Diff. Eqs. 102(1), 153–187.
Yi, Y. (1993b). Stability of integral manifold and orbital attraction of quasi-periodic motion.J. Diff. Eqs. 103(2), 278–322.
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Chow, SN., Yi, Y. Center manifold and stability for skew-product flows. J Dyn Diff Equat 6, 543–582 (1994). https://doi.org/10.1007/BF02218847
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DOI: https://doi.org/10.1007/BF02218847