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Transition time analysis in singularly perturbed boundary value problems

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