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Transactions of the American Mathematical Society

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Properties of center manifolds


Author: Jan Sijbrand
Journal: Trans. Amer. Math. Soc. 289 (1985), 431-469
MSC: Primary 58F14; Secondary 34C30
MathSciNet review: 783998
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Abstract: The center manifold has a number of puzzling properties associated with the basic questions of existence, uniqueness, differentiability and analyticity which may cloud its profitable application in e.g. bifurcation theory. This paper aims to deal with some of these subtle properties.

Regarding existence and uniqueness, it is shown that the cut-off function appearing in the usual existence proof is responsible for the selection of a single center manifold, thereby hiding the inherent nonuniqueness. Conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection. This intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium. In the case of nonuniqueness the difference between any two center manifolds will be less than $ {c_1}\exp ({c_2}{x^{ - 1}})$ with $ {c_1}$ and $ {c_2}$ constants, which explains why the formal Taylor expansions of different center manifolds are the same, while the expansions do not converge.

The differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost.

Center manifolds are usually not analytic; however, an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant. Using this manifold, a new and very simple proof of the Lyapunov subcenter theorem is given.


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  • [1] Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141
  • [2] V. I. Arnol′d, Loss of stability of self-induced oscillations near resonance, and versal deformations of equivariant vector fields, Funkcional. Anal. i Priložen. 11 (1977), no. 2, 1–10, 95 (Russian). MR 0442987
  • [3] Yuri N. Bibikov, Local theory of nonlinear analytic ordinary differential equations, Lecture Notes in Mathematics, vol. 702, Springer-Verlag, Berlin-New York, 1979. MR 547669
  • [4] J. Carr, Applications of centre manifold theory, Lecture Notes, Brown University, Providence, R.I., 1979.
  • [5] Nathaniel Chafee, A bifurcation problem for a functional differential equation of finitely retarded type, J. Math. Anal. Appl. 35 (1971), 312–348. MR 0277854
  • [6] Émile Cotton, Sur les solutions asymptotiques des équations différentielles, Ann. Sci. École Norm. Sup. (3) 28 (1911), 473–521 (French). MR 1509144
  • [7] J. J. Duistermaat, On periodic solutions near equilibrium points of conservative systems, Arch. Rational Mech. Anal. 45 (1972), 143–160. MR 0377196
  • [8] -, Stable manifolds, Preprint, University of Utrecht, Utrecht, The Netherlands, 1976.
  • [9] -, Periodic solutions near equilibrium points of Hamiltonian systems. Preprint, University of Utrecht, Utrecht, The Netherlands, 1980.
  • [10] J. J. Duistermaat and J. A. Vogel, The Moser-Weinstein method (work in progress).
  • [11] Neil Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), no. 1, 53–98. MR 524817, 10.1016/0022-0396(79)90152-9
  • [12] -, Center manifolds in bifurcation theory and singular perturbation theory, Preprint, University of British Columbia. Vancouver, B.C., Canada, 1978.
  • [13] Jack K. Hale, Ordinary differential equations, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. Pure and Applied Mathematics, Vol. XXI. MR 0419901
  • [14] Philip Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc. 11 (1960), 610–620. MR 0121542, 10.1090/S0002-9939-1960-0121542-7
  • [15] Brian Hassard, Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon, J. Theoret. Biol. 71 (1978), no. 3, 401–420. MR 0479443
  • [16] B. Hassard and Y. H. Wan, Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl. 63 (1978), no. 1, 297–312. MR 0488152
  • [17] Brian D. Hassard, Nicholas D. Kazarinoff, and Yieh Hei Wan, Theory and applications of Hopf bifurcation, London Mathematical Society Lecture Note Series, vol. 41, Cambridge University Press, Cambridge-New York, 1981. MR 603442
  • [18] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes, University of Kentucky,
  • [19] 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
  • [20] Philip Holmes and Jerrold Marsden, Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis, Automatica—J. IFAC 14 (1978), no. 4, 367–384. MR 0495662
  • [21] Philip J. Holmes (ed.), New approaches to nonlinear problems in dynamics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1980. MR 584582
  • [22] G. Iooss, Bifurcation of maps and applications, North-Holland Mathematics Studies, vol. 36, North-Holland Publishing Co., Amsterdam-New York, 1979. MR 531030
  • [23] N. D. Kazarinoff, Y.-H. Wan, and P. van den Driessche, Hopf bifurcation and stability of periodic solutions of differential-difference and integro-differential equations, J. Inst. Math. Appl. 21 (1978), no. 4, 461–477. MR 0492724
  • [24] A. Kelley, The center manifold and integral manifolds for Hamiltonian systems, Notices Amer. Math. Soc. 12 (1965), 143-144.
  • [25] Al Kelley, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations 3 (1967), 546–570. MR 0221044
  • [26] Al Kelley, On the Liapounov subcenter manifold, J. Math. Anal. Appl. 18 (1967), 472–478. MR 0216114
  • [27] N. Kopell and L. N. Howard, Bifurcations and trajectories joining critical points, Advances in Math. 18 (1975), no. 3, 306–358. MR 0397078
  • [28] Ivar Stakgold, Daniel D. Joseph, and David H. Sattinger (eds.), Nonlinear problems in the physical sciences and biology, Lecture Notes in Mathematics, Vol. 322, Springer-Verlag, Berlin-New York, 1973. MR 0371548
  • [29] A. Lyapunov, Problème général de la stabilité du mouvement, Ann. of Math. Studies, No. 17, Princeton Univ. Press, Princeton, N.J., 1949. (Russian original Kharkow, 1892.)
  • [30] J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale; Applied Mathematical Sciences, Vol. 19. MR 0494309
  • [31] J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math. 29 (1976), no. 6, 724–747. MR 0426052
  • [32] J. Moser, Addendum to: “Periodic orbits near an equilibrium and a theorem by Alan Weinstein” (Comm. Pure Appl. Math. 29 (1976), no. 6, 727–747), Comm. Pure Appl. Math. 31 (1978), no. 4, 529–530. MR 0467821
  • [33] P. Negrini and A. Tesei, Attractivity and Hopf bifurcation in Banach spaces, J. Math. Anal. Appl. 78 (1980), no. 1, 204–221. MR 595777, 10.1016/0022-247X(80)90223-1
  • [34] J. Palis and F. Takens, Topological equivalence of normally hyperbolic dynamical systems, Topology 16 (1977), no. 4, 335–345. MR 0474409
  • [35] Oskar Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z. 29 (1929), no. 1, 129–160 (German). MR 1544998, 10.1007/BF01180524
  • [36] V. A. Pliss, The reduction principle in the stability of motion, Dokl. Akad. Nauk SSSR 154 (1964), 1044–1046 (Russian). MR 0173056
  • [37] Paul H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), no. 2, 157–184. MR 0467823
  • [38] -, Periodic solutions of Hamiltonian systems: a survey, Preprint, University of Wisconsin, 1980.
  • [39] R. D. Richtmyer, Invariant manifolds and bifurcations in the Taylor problem (work in progress).
  • [40] David Ruelle and Floris Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971), 167–192. MR 0284067
  • [41] David H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, Vol. 309, Springer-Verlag, Berlin-New York, 1973. MR 0463624
  • [42] Dieter S. Schmidt, Hopf’s bifurcation theorem and the center theorem of Liapunov with resonance cases, J. Math. Anal. Appl. 63 (1978), no. 2, 354–370. MR 0477298
  • [43] P. R. Sethna, Bifurcation theory and averaging in mechanical systems, in [21].
  • [44] Carl Ludwig Siegel and Jürgen K. Moser, Lectures on celestial mechanics, Springer-Verlag, New York-Heidelberg, 1971. Translation by Charles I. Kalme; Die Grundlehren der mathematischen Wissenschaften, Band 187. MR 0502448
  • [45] J. Sijbrand, Studies in nonlinear stability and bifurcation theory, Chapter III, Ph.D. Thesis, 1981.
  • [46] Sebastian J. van Strien, Center manifolds are not 𝐶^{∞}, Math. Z. 166 (1979), no. 2, 143–145. MR 525618, 10.1007/BF01214040
  • [47] Floris Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47–100. MR 0339292
  • [48] Applications of global analysis. I, Mathematisch Instituut, Rijksuniversiteit Utrecht, Utrecht, 1974. Symposium held at Utrecht State University, Utrecht, February 8, 1973; Communications of the Mathematical Institute, Rijksuniversiteit, Utrecht, No. 3 – 1974. MR 0431273
  • [49] M. M. Vaĭnberg and V. A. Trenogin, Teoriya vetvleniya reshenii nilineinykh uravnenii, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0261416
  • [50] F. Verhulst, Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies, Philos. Trans. Roy. Soc. London Ser. A 290 (1979), 435-465.
  • [51] Yieh Hei Wan, On the uniqueness of invariant manifolds, J. Differential Equations 24 (1977), no. 2, 268–273. MR 0455047

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DOI: https://doi.org/10.1090/S0002-9947-1985-0783998-8
Article copyright: © Copyright 1985 American Mathematical Society