Transition time analysis in singularly perturbed boundary value problems

Abstract:

The paper deals with the boundary value problem $\varepsilon \ddot x + x\dot x - {x^2} = 0$, with $x(0) = A,x(T) = B$ for $A,B,T > 0$ and $\varepsilon > 0$ 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 $T \geqslant T(\varepsilon ) > 0$ and we show that asymptotically for $\varepsilon \to 0,\quad T(\varepsilon ) \sim - {\text {ln}}(\varepsilon )$, ${\text {i}}{\text {.e}}{\text {.}}\quad {\text {li}}{{\text {m}}_{\varepsilon \to 0}} - \frac {{T(\varepsilon )}} {{{\text {ln(}}\varepsilon {\text {)}}}} = 1$. The main tools are transit time analysis in the Liénard plane and normal form techniques. As such the methods are rather qualitative and useful in other similar problems.