Integral manifolds generalize invariant manifolds to nonautonomous ordinary differential equations. In this paper, we develop a method to calculate their Taylor approximation with respect to the state space variables. This is of decisive importance, e.g., in nonautonomous bifurcation theory or for an application of the reduction principle in a time-dependent setting.
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Abraham R.H., Marsden J.E., and Ratiu T. (1988). Manifolds, Tensor Analysis, and Applications. Springer, New York
Arnold L. (1998). Random Dynamical Systems. Springer, Berlin, Heidelberg, New York
Aulbach B. (1982). A reduction principle for nonautonomous differential equations. Arch. Math. 39:217–232
Aulbach B., and Wanner T. (1996). Integral manifolds for carathéodory type differential equations in banach spaces. In: Aulbach B., and Colonius F. (eds). Six Lectures on Dynamical Systems. World Scientific, Singapore, pp 45–119
Aulbach B., Rasmussen M., and Siegmund S. (2005). Approximation of attractors of nonautonomous dynamical systems. Discrete Contin. Dyn. Syst. (Series B) 5(2):215–238
Aulbach B., Rasmussen M., and Siegmund S. (2006). Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete Contin. Dyn. Syst. 15(2):579–596
Beyn W.-J., and Kleß W. (1998). Numerical Taylor expansion of invariant manifolds in large dynamical systems. Numer. Math. 80:1–38
Carr, J. (1981). Applications of centre manifold theory, Appl. Math. Sci. Vol. 35, Springer-Verlag, Berlin
Cheban D.N., Kloeden P.E., and Schmalfuß B. (2001). Pullback attractors in dissipative nonautonomous differential equations under discretization. J. Dyn. Differ. Equations 13(1):185–213
Chicone C., and Latushkin Y. (1997). Center manifolds for infinite-dimensional nonautonomous differential equations. J. Diff. Eq. 141(2):356–399
Chow S.-N., and Hale J.K. (1982). Methods of Bifurcation Theory. Springer, Berlin
Chow S.-N., Li C., and Wang D. (1994). Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press.
Coppel W.A. (1978). Dichotomies in Stability Theory. Lecture Notes in Mathematics. Vol. 629, Springer, Berlin
Daleckii, J. L., and Krein, M. G. (1974). Stability of solutions of differential equations in banach space. In Translations of Mathematical Monographs, Vol. 43, AMS, Providence, RI.
Dellnitz M., and Hohmann A. (1997). A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75:293–317
Eirola T., and von Pfaler J. (2004). Taylor xpansions for invariant manifolds. Numer. Math. 99:25–46
Flandoli F., and Schmalfuß B. (1996). Random attractors for the 3-D stochastic Navier–Stokes equation with mulitiplicative white noise. Stoch. Stoch. Rep. 59:21–45
Foias C., and Jolly M.S. (1995). On the numerical algebraic approximations of global attractors. Nonlinearity 8(3):295–319
Fuming M., and Küpper T. (1994). A numerical method to calculate center manifolds of ODE’s. Appl. Anal. 54:1–15
Guckenheimer J., and Vladimirsky A. (2004). A fast method for approximating invariant manifolds. SIAM J. Appl. Dyna. Syst. 3(3):232–260
Hassard B.D. (1980). Computation of invariant manifolds. In: Holmes P.J. (eds) New Approaches to Nonlinear Problems in Dynamics. SIAM, Philadelphia
Homburg A.J., Osinga H.M., and Vegter G. (1995). On the Computation of invariant manifolds of fixed points. J. Appl. Math. Phys. (ZAMP) 46(2):171–187
Johnson R.A. (1989). Hopf bifurcation from nonperiodic solutions of differential equations. I. Linear Theory. J. Dyn. Diff. Eq. 1:179–198
Johnson R.A., and Mantellini F. (2003). A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete Contin. Dyn. Syst. (Series A) 9:209–224
Johnson R.A., and Yi Y. (1994). Hopf bifurcation from non-periodic solutions of differential equations II. J. Differ. Equations 107:310–340
Jolly M.S., and Rosa R. (2005). Computation of non-smooth local centre manifolds. IMA J. Numer. Anal. 25:698–725
Kriegl A., and Michor P.W. (1997). The Convenient Setting of Global Analysis. AMS, Providence
Kuznetsow Y.A. (2004). Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York
Lorenz E.N. (1963). Deterministic Nonperiodic Flow. J. Atmos. Sci. 20:130–141
Langa J.A., Robinson J.C., and Suárez A. (2002). Stability, instability and bifurcation phenomena in non-autonomous differential equations. Nonlinearity 15:887–903
Langa J.A., Robinson J.C., and Suárez A. (2006). Bifurcations in non-autonomous scalar equations. J. Diff. Eq. 221(1):1–35
Pliss V.A. (1964). The reduction principle in the theory of stability of motion. Sov. Math. Dokl. 5:247–250
Pötzsche C., and Rasmussen M. (2005). Taylor approximation of invariant fiber bundles for nonautonomous difference equations. Nonlinear Anal. (TMA) 60:1303–1330
Rasmussen, M. (2006). Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Ph.D. Thesis, University of Augsburg.
Rybakowski K.P. (1991). Formulas for higher-order finite expansions of composite maps. In: Gauß C.F., Rassias G.M. (eds) The Mathematical Heritage of. World Scientific, Singapore, pp 652–669
Sell G.R. (1978). The structure of a flow in the vicinity of an almost periodic motion. J. Diff. Eq. 27(3):359–393
Sell G.R. (1979). Bifurcation of higher-dimensional tori. Arch. Rational Mech. Anal. 69(3):199–230
Sell G.R. (1985). Smooth linearization near a fixed point. Amer. J. Math. 107(5):1035–1091
Siegmund, S. (1999). Spektraltheorie, glatte Faserungen und Normalformen für Differentialgleichungen vom Carathéodory-Typ (in german). Ph.D. Thesis, University of Augsburg.
Siegmund S. (2002). Normal forms for nonautonomous differential equations. J. Diff. Eq. 178(2):541–573
Vanderbauwhede, A. (1989). Centre manifolds, normal forms and elementary bifurcations. Dynamics Reported. In Dynamic Report Series Dynamic Systems Applies, Vol. 2, Wiley, Chichester, 89–169.
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Pötzsche, C., Rasmussen, M. Taylor Approximation of Integral Manifolds. J Dyn Diff Equat 18, 427–460 (2006). https://doi.org/10.1007/s10884-006-9011-8
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DOI: https://doi.org/10.1007/s10884-006-9011-8