The application of centre manifolds to amplitude expansions. I. Ordinary differential equations. (English) Zbl 0544.34038
Some basic facts of center manifold theory for ordinary differential equations are summarized and their relations to amplitude expansions is discussed by means of examples. Emphasis is placed upon the practical computational aspects of applying center manifold theory to parameter depending systems which are close to critical ones. The calculations are illustrated by a detailed analysis of a five-dimensional system arising in fluid mechanics.
Reviewer: B.Aulbach
MSC:
| 37-XX | Dynamical systems and ergodic theory |
| 34C30 | Manifolds of solutions of ODE (MSC2000) |
Citations:
Zbl 0544.34039References:
| [1] | Arnold, V. I., Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields, Functional Anal. Appl., 11, 1-10 (1977) · Zbl 0411.58013 |
| [2] | Ball, J. M., Saddle point analysis for an ordinary differential equation in a Banach space, and an application to dynamic buckling of a beam, (Dickey, R. W., Nonlinear Elasticity (1973), Academic Press: Academic Press New York), 93-160 · Zbl 0288.34057 |
| [3] | Carr, J., Applications of Centre Manifold Theory, (Applied Mathematical Sciences, Vol. 35 (1981), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0464.58001 |
| [4] | Carr, J.; Al-amood, N., Rate of decay in critical cases. I: Finite dimensional problems, J. Math. Anal. Appl., 75, 242-250 (1980) · Zbl 0457.34041 |
| [5] | Carr, J.; Al-amood, N., Rate of decay in critical cases. II: Infinite dimensional problems, J. Math. Anal. Appl., 75, 293-300 (1980) · Zbl 0457.34042 |
| [6] | Hale, J. K., Critical cases for neutral functional differential equations, J. Differential Equations, 10, 59-82 (1970) · Zbl 0223.34057 |
| [7] | Hassard, B. D.; Kazarinoff, N. D.; Wan, Y.-H, Theory and Applications of Hopf Bifurcation (1981), Cambridge Univ. Press: Cambridge Univ. Press Cambridge/New York · Zbl 0474.34002 |
| [8] | Henry, D., Geometric Theory of Semilinear Parabolic Equations, (Lecture Notes in Mathematics No. 840 (1981), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0456.35001 |
| [9] | Holmes, P. J., Bifurcations to divergence and flutter in flow-induced oscillations: A finite dimensional analysis, J. Sound Vibration, 53, 471-503 (1977) · Zbl 0363.73059 |
| [10] | Holmes, P. J.; Marsden, J., Bifurcations to divergence and flutter in flow-induced oscillations: An infinite dimensional analysis, Automatica, 14, 367-384 (1978) · Zbl 0385.93028 |
| [11] | Holmes, P. J.; Rand, D. A., Identification of Vibrating Systems by Generic Modeling, with an Application to Flutter, Institute of Sound and Vibration Research, Technical Report No. 79 (1975), Southampton |
| [12] | Hoppensteadt, F.; Gordon, N., Nonlinear stability analysis of static states which arise through bifurcation, Comm. Pure Appl. Math., 28, 355-373 (1975) · Zbl 0293.35023 |
| [13] | Hoppensteadt, F.; Gordon, N., An analysis of transient behavior in the onset of convection, (Lecture Notes in Mathematics No. 594 (1977), Springer-Verlag: Springer-Verlag Berlin/New York), 231-243 · Zbl 0366.76017 |
| [14] | Kelley, A., The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3, 546-570 (1967) · Zbl 0173.11001 |
| [15] | Kelley, A., Stability of the center-stable manifold, J. Math. Anal. Appl., 18, 336-344 (1967) · Zbl 0166.08304 |
| [16] | Kogelman, S.; Keller, J. B., Transient behaviour of unstable nonlinear systems with applications to the Bénard and Taylor problems, SIAM J. Appl. Math., 20, 619-637 (1971) · Zbl 0198.30402 |
| [17] | Rubenfeld, L. A.; Siegmann, W. L., A nonlinear model for double-diffusive convection, SIAM J. Appl. Math., 29, 540-557 (1975) · Zbl 0346.76037 |
| [18] | Rubenfeld, L. A.; Siegmann, W. L., Nonlinear dynamic theory for a double-diffusive convection model, SIAM J. Appl. Math., 32, 871-894 (1977) · Zbl 0362.76089 |
| [19] | Ruelle, D.; Takens, F., On the nature of turbulence, Comm. Math. Phys., 20, 167-192 (1971) · Zbl 0223.76041 |
| [20] | Takens, F., Singularities of Vector Fields, ((1974), Publ. Inst. des Haute Etudes Scientifiques: Publ. Inst. des Haute Etudes Scientifiques Bures-sur-Yvette, Paris), 47-100, 43 · Zbl 0279.58009 |
| [21] | Takens, F., Forced Oscillations and Bifurcations, (Applications of Global Analysis, Comm. 3 of Math. Inst. (1974), Rijksuniversiteit Utrecht) · Zbl 1156.37315 |
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