TRANSITION TIME ANALYSIS IN SINGULARLY PERTURBED BOUNDARY-VALUE-PROBLEMS

Abstract

The payer deals with the boundary value problem epsilon x + xx-x(2) = 0, with x(0) = A, x(T) = B for A, B, T > 0 and epsilon > 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 greater than or equal to T(epsilon) > 0 and we show that asymptotically for epsilon --> 0, T(epsilon) similar to ln (epsilon), i.e. lim(epsilon-->0) -T(epsilon)/ln(epsilon) = 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.