Transition time analysis in singularly perturbed boundary value problems
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- by Freddy Dumortier and Bert Smits PDF
- Trans. Amer. Math. Soc. 347 (1995), 4129-4145 Request permission
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.References
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Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 4129-4145
- MSC: Primary 34E15; Secondary 34B15
- DOI: https://doi.org/10.1090/S0002-9947-1995-1308009-3
- MathSciNet review: 1308009