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Extending geometric singular perturbation theory to nonhyperbolic points – fold and canard points in two dimensions. (English) Zbl 1002.34046
The geometric approach to singular perturbation problems is based on methods from dynamical systems theory. These techniques are very successful in the case of normally hyperbolic critical manifolds. However, at points where normal hyperbolicity fails, the well-developed geometric theory could not be applied. The authors present a method based on blow-up techniques, that leads to a rigorous geometric analysis of these problems. A detailed analysis of fold points and canard points is given.

MSC:
34E15 Singular perturbations for ordinary differential equations
34C30 Manifolds of solutions of ODE (MSC2000)
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C26 Relaxation oscillations for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34C40 Ordinary differential equations and systems on manifolds
37C10 Dynamics induced by flows and semiflows
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