Extending geometric singular perturbation theory to nonhyperbolic points – fold and canard points in two dimensions.

*(English)*Zbl 1002.34046The 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.

Reviewer: E.V.Shchetinina (Berlin)

##### 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 |