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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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  • [AO] J.D. Allen and R.E. O'Malley, Jr., Singularly perturbed boundary value problems viewed in the Liénard plane, Asymptotic and Computational Analysis, (R. Wong, ed.), Lecture Notes in Pure and Applied Math., 124, Dekker, 1990, pp. 357-378. MR 1052441 (91h:34085)
  • [AP] D.K. Arrowsmith and C.M. Place, An introduction to dynamical systems, Cambridge University Press, 1990. MR 1069752 (91g:58068)
  • [B] P. Bonckaert, Partially hyperbolic fixed points with constraints, preprint (1993). MR 1321568 (96h:58157)
  • [Ca] J. Carr, Applications of centre manifold theory, Applied Mathematical Sciences 35, Springer, New York, 1981. MR 635782 (83g:34039)
  • [CD] C. Chicone and F. Dumortier, Finiteness for critical periods of planar analytic vector fields, Nonlinear Analysis, Theory, Methods and Applications 20 (1993), 315-335. MR 1206421 (94b:58082)
  • [DeR] Z. Denkowska and R. Roussarie, A method of desingularization for analytic two-dimensional vector field families, Bol. Soc. Brasil. Mat. 22 (1991), 93-126. MR 1159387 (93c:58171)
  • [D] F. Dumortier, Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations (notes written with B. Smits), Bifurcations of Periodic Orbits and Vector Fields (D. Schlomiuk, ed.), Nato ASI-series C408, 1993. MR 1258518 (94j:58123)
  • [DR] F. Dumortier and R. Roussarie, Canard cycles and center manifolds, preprint (1993). MR 1327208 (96k:34113)
  • [T1] F. Takens, Singularities of vector fields, Inst. Hautes Etudes Sci. Publ. Math. 43 (1973), 47-100. MR 0339292 (49:4052)
  • [T2] F. Takens, Partially hyperbolic fixed points, Topology 10 (1971), 133-147. MR 0307279 (46:6399)

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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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