Abstract
The geometric approach to singular perturbation problems is based on powerful methods from dynamical systems theory. These techniques have been very successful in the case of normally hyperbolic critical manifolds. However, at points where normal hyperbolicity fails, e.g. fold points or points of self-intersection of the critical manifold, the well developed geometric theory does not apply. We present a method based on blow-up techniques which leads to a rigorous geometric analysis of these problems. The blow-up method leads to problems which can be analysed by standard methods from the theory of invariant manifolds and global bifurcations. The presentation is in the context of a planar singularly perturbed fold. The blow-up used in the analysis is closely related to the rescalings used in the classical analysis based on matched asymptotic expansions. The relationship between these classical results and our geometric analysis is discussed.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Research supported by the Austrian Science Foundation under grant Y 42-MAT.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
V.I. Arnold (Ed.), Dynamical Systems 5, Encyclopedia of Mathematical Sciences, Springer (1989).
B. Braaksma, Critical phenomena in dynamical systems of van der Pol type, Thesis, University of Utrecht (1993).
C.M. Bender and S.A. Orszag, Advanced mathematical methods for scientists and engineers, McGraw-Hill (1978).
E. Benoit (Ed.), Dynamic Bifurcations, Lecture Notes in Mathematics 1493, Springer, Berlin (1991).
M. Diener, Regularizing microscopes and rivers, SIAM J. Appl. Math. 25, pp. 148–173 (1994).
F. Dumortier, Techniques in the Theory of Local Bifurcations: Blow-Up, Normal Forms, Nilpotent Bifurcations, Singular Perturbations, in Bifurcations and Periodic Orbits of Vector Fields (ed. D. Szlomiuk), Kluwer C408, Dordrecht (1993).
F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Memoirs of the AMS 557 (1996).
W. Eckhaus, Relaxation oscillations including a standard chase on French ducks, in Asymptotic Analysis II, Springer Lecture Notes Math. 985, pp. 449–494 (1983).
N. Fenichel, Geometric singular perturbation theory, J. Diff. Eq. 31, pp. 53–98 (1979).
J. Grasman, Asymptotic methods for relaxation oscillations and applications, Springer, New York (1987).
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York (1983).
C.K.R.T. Jones, Geometric singular perturbation theory, in Dynamical Systems, Springer Lecture Notes Math. 1609, pp. 44–120 (1995).
J. Kevorkian J.D. Cole, Perturbation methods in applied mathematics, Springer, New York (1981).
M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points, in preparation.
P.A. Lagerstrom, Matched Asymptotic Expansions, Springer (1988).
X.B. Lin, Heteroclinic bifurcation and singularly perturbed boundary value problems, J. Diff. Eq. 84, pp. 319–382 (1990).
E.F. Mishchenko and N.Kh. Rozov, Differential equations with small parameters and relaxation oscillations, Plenum, New York (1980).
E.F. Mishchenko, Yu.S. Kolesov, A.Yu. Kolesov and N. Kh. Rozov, Asymptotic methods in singularly perturbed systems, Consultants Bureau, New York and London (1994).
A.I. Neishtadt, Persistence of stability loss for dynamical bifurcations I, Differential Equations 23, pp. 1385–1391 (1987). stability loss for dynamical bifurcations II
K. Nipp, Breakdown of stability in singularly perturbed autonomous systems I. Orbit equations, SIAM J. Math. Anal. 17, pp. 512–532 (1986).
R.E. O’Malley, Introduction to singular perturbations, Academic Press, Inc., New York (1974).
L.S. Pontryagin, Asymptotic properties of solutions of differential equations with small parameter multiplying leading derivatives, Izv. AN SSSR, Ser. Matem. 21, 5, pp. 605–626 (1957).
S. Sternberg, On the nature of local homeomorphisms of Euclidean n-space II, Am. J. Math. 80 pp. 623–631 (1958).
P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Diff. Eq. 92, pp. 252–281 (1991).
P. Szmolyan, Geometry of singular perturbations: A case study, in preparation.
M. Wechselberger, Singularly perturbed folds and canards in ℝ3, Thesis, TU-Wien (1998).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media New York
About this paper
Cite this paper
Krupa, M., Szmolyan, P. (2001). Geometric Analysis of the Singularly Perturbed Planar Fold. In: Jones, C.K.R.T., Khibnik, A.I. (eds) Multiple-Time-Scale Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol 122. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0117-2_4
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0117-2_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6529-0
Online ISBN: 978-1-4613-0117-2
eBook Packages: Springer Book Archive