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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Properties of center manifolds
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by Jan Sijbrand PDF
Trans. Amer. Math. Soc. 289 (1985), 431-469 Request permission

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.
References
  • 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
  • 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
  • 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
    • J. Carr, Applications of centre manifold theory, Lecture Notes, Brown University, Providence, R.I., 1979.
  • Nathaniel Chafee, A bifurcation problem for a functional differential equation of finitely retarded type, J. Math. Anal. Appl. 35 (1971), 312–348. MR 277854, DOI 10.1016/0022-247X(71)90221-6
  • Émile Cotton, Sur les solutions asymptotiques des équations différentielles, Ann. Sci. École Norm. Sup. (3) 28 (1911), 473–521 (French). MR 1509144
  • J. J. Duistermaat, On periodic solutions near equilibrium points of conservative systems, Arch. Rational Mech. Anal. 45 (1972), 143–160. MR 377196, DOI 10.1007/BF00253043
    • —, Stable manifolds, Preprint, University of Utrecht, Utrecht, The Netherlands, 1976. —, Periodic solutions near equilibrium points of Hamiltonian systems. Preprint, University of Utrecht, Utrecht, The Netherlands, 1980. J. J. Duistermaat and J. A. Vogel, The Moser-Weinstein method (work in progress).
  • 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
    • —, Center manifolds in bifurcation theory and singular perturbation theory, Preprint, University of British Columbia. Vancouver, B.C., Canada, 1978.
  • Jack K. Hale, Ordinary differential equations, Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. MR 0419901
  • Philip Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc. 11 (1960), 610–620. MR 121542, DOI 10.1090/S0002-9939-1960-0121542-7
  • 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 479443, DOI 10.1016/0022-5193(78)90168-6
  • B. Hassard and Y. H. Wan, Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl. 63 (1978), no. 1, 297–312. MR 488152, DOI 10.1016/0022-247X(78)90120-8
  • 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
    • D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes, University of Kentucky,
  • 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
  • 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 495662, DOI 10.1016/0005-1098(78)90036-5
  • Philip J. Holmes (ed.), New approaches to nonlinear problems in dynamics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1980. MR 584582
  • G. Iooss, Bifurcation of maps and applications, North-Holland Mathematics Studies, vol. 36, North-Holland Publishing Co., Amsterdam-New York, 1979. MR 531030
  • 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 492724
    • A. Kelley, The center manifold and integral manifolds for Hamiltonian systems, Notices Amer. Math. Soc. 12 (1965), 143-144.
  • Al Kelley, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations 3 (1967), 546–570. MR 221044, DOI 10.1016/0022-0396(67)90016-2
  • Al Kelley, On the Liapounov subcenter manifold, J. Math. Anal. Appl. 18 (1967), 472–478. MR 216114, DOI 10.1016/0022-247X(67)90039-X
  • N. Kopell and L. N. Howard, Bifurcations and trajectories joining critical points, Advances in Math. 18 (1975), no. 3, 306–358. MR 397078, DOI 10.1016/0001-8708(75)90048-1
  • 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
    • 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.)
  • J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Applied Mathematical Sciences, Vol. 19, 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. MR 0494309
  • J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math. 29 (1976), no. 6, 724–747. MR 426052, DOI 10.1002/cpa.3160290613
  • 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 467821, DOI 10.1002/cpa.3160310408
  • P. Negrini and A. Tesei, Attractivity and Hopf bifurcation in Banach spaces, J. Math. Anal. Appl. 78 (1980), no. 1, 204–221. MR 595777, DOI 10.1016/0022-247X(80)90223-1
  • J. Palis and F. Takens, Topological equivalence of normally hyperbolic dynamical systems, Topology 16 (1977), no. 4, 335–345. MR 474409, DOI 10.1016/0040-9383(77)90040-4
  • Oskar Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z. 29 (1929), no. 1, 129–160 (German). MR 1544998, DOI 10.1007/BF01180524
  • V. A. Pliss, The reduction principle in the stability of motion, Dokl. Akad. Nauk SSSR 154 (1964), 1044–1046 (Russian). MR 0173056
  • Paul H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), no. 2, 157–184. MR 467823, DOI 10.1002/cpa.3160310203
    • —, Periodic solutions of Hamiltonian systems: a survey, Preprint, University of Wisconsin, 1980. R. D. Richtmyer, Invariant manifolds and bifurcations in the Taylor problem (work in progress).
  • David Ruelle and Floris Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971), 167–192. MR 284067
  • David H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, Vol. 309, Springer-Verlag, Berlin-New York, 1973. MR 0463624
  • 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 477298, DOI 10.1016/0022-247X(78)90081-1
    • P. R. Sethna, Bifurcation theory and averaging in mechanical systems, in [21].
  • Carl Ludwig Siegel and Jürgen K. Moser, Lectures on celestial mechanics, Die Grundlehren der mathematischen Wissenschaften, Band 187, Springer-Verlag, New York-Heidelberg, 1971. Translation by Charles I. Kalme. MR 0502448
    • J. Sijbrand, Studies in nonlinear stability and bifurcation theory, Chapter III, Ph.D. Thesis, 1981.
  • Sebastian J. van Strien, Center manifolds are not $C^{\infty }$, Math. Z. 166 (1979), no. 2, 143–145. MR 525618, DOI 10.1007/BF01214040
  • Floris Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47–100. MR 339292
  • Applications of global analysis. I, Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, No. 3 – 1974, Rijksuniversiteit Utrecht, Mathematisch Instituut, Utrecht, 1974. Symposium held at Utrecht State University, Utrecht, February 8, 1973. MR 0431273
  • M. M. Vaĭnberg and V. A. Trenogin, Teoriya vetvleniya resheniĭ nilineĭnykh uravneniĭ, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0261416
    • 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.
  • Yieh Hei Wan, On the uniqueness of invariant manifolds, J. Differential Equations 24 (1977), no. 2, 268–273. MR 455047, DOI 10.1016/0022-0396(77)90150-4
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 289 (1985), 431-469
  • MSC: Primary 58F14; Secondary 34C30
  • DOI: https://doi.org/10.1090/S0002-9947-1985-0783998-8
  • MathSciNet review: 783998