Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T19:21:14.658Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant manifolds in singular perturbation problems for ordinary differential equations

Published online by Cambridge University Press:  14 November 2011

Kunimochi Sakamoto
Kunimochi Sakamoto
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Based on Fenichel's geometric idea, invariant manifold theory is applied to singular perturbation problems. This approach clarifies the nature of outer and inner solutions. A specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds. A method to approximate the connections is also presented.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 116 , Issue 1-2 , 1990 , pp. 45 - 78
Copyright
Copyright © Royal Society of Edinburgh 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Chow, S. N. and Lu, K.. Ck centre unstable manifolds. Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 303–320.CrossRefGoogle Scholar
2Chow, S. N. and Lu, K.. Invariant manifolds for flows in Banach spaces. J. Differential Equations 74 (1988), 285–317.CrossRefGoogle Scholar
3Fenichel, N.. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21 (1971), 193–226.CrossRefGoogle Scholar
4. Fenichel, N.. Geometric singular perturbation theory for ordinary differential equation. J. Differential Equations 31 (1979), 53–89.CrossRefGoogle Scholar
5Hale, J. K. and Sakamoto, K.. Existence and stablility of transition layers. Japan. J. Appl. Math. 5 (1988), 367–405.CrossRefGoogle Scholar
6Henry, D.. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).CrossRefGoogle Scholar
7Hoppensteadt, F. C.. Properties of Solutions of ordinary differential equations with a small parameter. Comm. Pure Appl. Math. 24 (1971), 807–840.CrossRefGoogle Scholar
8Kokubu, H.. Homoclinic and heteroclinic bifuractions of vector fields. Japan J. Appl. Math. 5 (1989), 455–501.CrossRefGoogle Scholar
9Knobloch, K. W. and Aulbach, B.. Singular perturbations and integral manifolds. J. Math. Phys. Sci. 18 (1984), 415–424.Google Scholar
10Knobloch, H. W.. A method for construction of invariant manifolds. In Asymptotic methods in mathematical physics. Proceedings of Conference in honor of J. A. Mitropoliskii on his 70th Birthday, edited by Korolyuk, V. S., pp. 100–18 (Kiev: Naukova Dumka, 1988).Google Scholar
11Levin, J.. Singular perturbations of non-linear systems of differential equations related to conditional stability. Duke J. Math. 24 (1956), 609–620.Google Scholar
12Lin, X. B.. Heteroclinic bifurcations and singularly perturbed boundary value problems (preprint, 1989).CrossRefGoogle Scholar
13. Palmer, K. J.. Exponential dichotomies and transversal homoclinic points: J. Differential Equations 55 (1984), 225–256.CrossRefGoogle Scholar
14Vanderbauwehde, A. and Van Gils, S. A.. Center manifolds and contractions on a scale of Banach spaces. J. Funct. Anal. 72 (1987), 209–224.CrossRefGoogle Scholar