Abstract
When studying the behaviour of a dynamical system in the neighbourhood of an equilibrium point the first step is to construct the stable, unstable and centre manifolds. These are manifolds that are invariant under the flow relative to a neighbourhood of the equilibrium point and carry the solutions that decay or grow (or neither) at certain rates. These ideas have a long history, see for instance Poincaré [32] and Hadamard [11]. Sophisticated recent results can be found in Fenichel [7], Hirsch, Pugh and Shub [17] and Kelley [22].
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References
J. Ball, Saddle point analysis for an ordinary differential equation in a Banach space, and an application to dynamic buckling of a beam, in Nonlinear Elasticity ( R. W. Dickey, ed.), Academic Press, New York, (1973), pp. 93–160.
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II. Existence of infinitely many solutions. Arch. Rat. Mech. Anal., 82 (1983), 347–76.
G. Carpenter, A geometric approach to singular perturbation problems with applications to nerve impulse equations. J. Differential Equations 23 (1977), 335–67.
J. Carr, Applications of Centre Manifold Theory. Springer-Verlag, New York, 1981.
S. N. Chow and J. K. Hale, Methods of Bifurcation Theory. Springer-Verlag, New York, 1982.
J. W. Evans, Nerve axon equations, III: stability of the nerve impulse. Indiana Univ. Math. 22 (1972), 577–94.
N. Fenichel, Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J., 21 (1971), 193–226.
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membranes. Biophys. J. 1 (1961), 445–66.
C. Foias, B. Nicolaenko, G. Sell and R. Temam, Varietes inertielles pour l’ equation de Kuramoto—Sivashinski. C.R. Acad. Sc. Paris, 301, Serie I (1985), 285–8.
C. Foias, G. Sell and R. Temam, Varietes inertielles des equations differentielles dissipatives, C.R. Acad. Sc. Paris, 301, Serie I (1985), 139–41.
J. Hadamard, Sur l’iteration et les solutions asymptotiques des equations differentielles. Bull. Soc. Math. France, 29 (1901), 224–8.
J. K. Hale, Ordinary Differential Equations: Wiley—Interscience, New York, 1969.
P. Hartman, Ordinary Differential Equations. Wiley, New York, 1964.
S. P. Hastings, On the existence of homoclinic and periodic orbits for the FitzHugh—Nagumo equations. Quarterly J. Math. Oxford, 27 (1976), 123–34.
D. Henry, Geometric theory of semilinear parabolic equations. Lecture Notes in Math., 840, Springer-Verlag, New York, 1981.
E. Hille and R. S. Phillips, Functional Analysis and Semi-groups. Amer. Math. Soc., Providence, 1957.
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds. Bull. Amer. Math. Soc., 76 (1970), 1015–19. Invariant manifolds, Lecture Notes in Math. 583, Springer-Verlag, New York, 1977.
C. K. R. T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system. Trans. Amer. Math. Soc., 286 (1984), 431–69.
C. K. R. T. Jones and T. Kupper, On the infinitely many solutions of a semilinear elliptic equation, to appear SIAM J. Math. Anal.
T. Kato, Perturbation theory for linear operators. Springer-Verlag, New York, 1966.
T. Kato, A spectral mapping theorem for the exponential function, and some counterexamples. MR Report no. 2316, Univ. of Wisc., Jan. 1982.
A. Kelley, The stable, center-stable, center, center-unstable, and unstable manifolds. J. Differential Equations, 3 (1967), 546–70 or an appendix to: Transversal Mappings and Flows by R. Abraham and J. Robbin, Benjamin, New York, 1967.
C. Keller, Stable and unstable manifolds for the nonlinear wave equation with dissipation. J. Differential Equations, 50 (1983), 330–47.
M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, Oxford, 1964.
R. Langer, Existence of homoclinic travelling wave solutions to the FitzHughNagumo equations, PhD thesis, Northeastern Univ., 1980.
A. M. Liapunov, Probleme general de la stabilite du movement, Princeton Univ. Press, Princeton, N.J., 1947.
J. Mallet-Paret and G. Sell, On the theory of inertial manifolds for reaction diffusion equations in higher space dimension. J. Amer. Math. Soc., 1 (1988).
R. McGehee, The stable manifold theorem via an isolating block, Symposium on Ordinary Differential Equations (W. Harris and Y. Sibuya ed.), Springer-Verlag, Berlin, (1973), 135–44.
J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axons. Proc. IRI, 50 (1960), 2061–70.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
O. Perron, Die Stabilitatsfrage bei Differentialgleichungssysteme, Math. Zeit., 32 (1930), 703–728.
H. Poincaré, Memoire sur les courbes definie par une equation differentielle, I—IV, J. Math. Pures Appl., 3 (1881), 375–422; 3 (1882), 251–80; 4 (1885), 167–244; 4 (1886), pp. 151–217.
I. E. Segal, Non-linear semi-groups. Amer. J. of Math., 78 (1963), 339–64.
J. Shatah, Unstable Ground State of Nonlinear Klein-Gordon Equations. Trans. Amer. Math. Soc., 290 (1985), 701–710.
M. Slemrod, Asymptotic behaviour of Co semi-groups as determined by the spectrum of the generator. Indiana Univ. Math. J., 25 (1976), 783–92.
W. A. Strauss, Existence of solitary waves in higher dimensions. Comm. Math. Phys., 55 (1977), 149–62.
W. A. Strauss, Stable and unstable states of nonlinear wave equations, in Nonlinear Partial Differential Equations, Contemporary Math., 17, Amer. Math. Soc., Providence (1983), pp. 429–441.
A. E. Taylor, Introduction to Functional Analysis. Wiley, New York, 1961.
I. Vidav, Spectra of perturbed semigroups with applications to transport theory. J. Math. Anal. Appl., 30 (1970), 264–79.
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© 1989 John Wiley & Sons Ltd and B. G. Teubner
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Bates, P.W., Jones, C.K.R.T. (1989). Invariant Manifolds for Semilinear Partial Differential Equations. In: Kirchgraber, U., Walther, H.O. (eds) Dynamics Reported. Dynamics Reported, vol 2. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-96657-5_1
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DOI: https://doi.org/10.1007/978-3-322-96657-5_1
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