Geometric Singular Perturbation theory has traditionally dealt only with perturbation problems near normally hyperbolic manifolds of singularities. In this paper we want to show how blow up techniques can permit enlarging the applicability to non-normally hyperbolic points. We will present the method on well chosen examples in the plane and in 3-space.

Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity

Singularly Perturbed Problems in Case of Exchange of Stabilities

The present work is a continuation of the geometric singular perturbation analysis of the Lagerstrom model problem which was commenced in J. Differential Equations (199 (2) (2004) 290–325). We establish the same framework here, reinterpreting Lagerstrom's equation as a dynamical system which is subsequently analyzed by means of methods from dynamical systems theory as well as of the blow-up technique. We show how rigorous asymptotic expansions for the Lagerstrom problem can be obtained using geometric methods, thereby establishing a connection to the method of matched asymptotic expansions. We explain the structure of these expansions and demonstrate that the occurrence of the well-known logarithmic (switchback) terms therein is caused by a resonance phenomenon.

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Rigorous asymptotic expansions for Lagerstrom's model equation—a geometric approach

We apply a method, going back to E Nelson, for simplifying the expression of a vector field to a normal form, for example near a partially hyperbolic fixed point, in the case where some additional properties have to be respected by the change of variables We illustrate the method by solving recent questions encountered in desingularization problems

Conjugacy of vector fields respecting additional properties

The paper deals with the boundary value problem &x+xj*-x2 = 0, with x(0) = A, x(T) = B for A, B, T > 0 and e > 0 close to zero. It is shown that for T sufficiently big, the problem has exactly three solutions, two of which reach negative values. Solutions reaching negative values occur for T > T(e) > 0 and we show that asymptotically for e -+ 0, T(e) -ln(e), i.e. lim6,o T(e) = 1 . The main tools are transit time analysis in the Lienard plane and normal form techniques. As such the methods are rather qualitative and useful in other similar problems. INTRODUCTION In this paper, the authors will apply desingularization techniques to a singularly perturbed boundary value problem. By doing so, we will be able to explain apparent transition phenomena in a geometric way. The equation we study is given by

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Bert Smits

Papers

Transition time analysis in singularly perturbed boundary value problems