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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Transition time analysis in singularly perturbed boundary value problems

Authors: Freddy Dumortier and Bert Smits
Journal: Trans. Amer. Math. Soc. 347 (1995), 4129-4145
MSC: Primary 34E15; Secondary 34B15
MathSciNet review: 1308009
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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.

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Keywords: Global blowing up, transition time, asymptotic analysis, center manifold
Article copyright: © Copyright 1995 American Mathematical Society

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