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Composite Expansions and Singularly Perturbed Differential Equations

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Composite Asymptotic Expansions

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2066))

Abstract

We assume Φ analytic with respect to x and y in a domain \(\mathcal{D}\subset {\mathbb{C}}^{2}\) and of Gevrey order 1 with respect to \(\epsilon \) in S; this also allows to treat equations containing a control parameter. This is useful for equations where canards solutions might occur for certain values of the parameter.

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Notes

  1. 1.

    Consider the landscape R d given by (5.18) with \(d = p\arg \eta =\arg \epsilon \). We call mountain a connected component of the set R d  > 0. A valley is a connected component of the set R d  < 0. See the lines above Corollary 5.16 for a description and a numbering of these mountains and valleys.

  2. 2.

    Other arguments for η can be reduced to this case by rotations \(x = \xi {e}^{i\varphi }\). If we want a sector of greater opening in η, we cover it by small sectors in η and consider the intersection of the corresponding sectors in x. One can consider an initial condition at a point x = Lη, \(L \in \mathbb{C}\) instead of x = 0, in which case we first study the existence and the asymptotics of the corresponding solution at x = 0 using the inner equation.

  3. 3.

    i.e. the set whose image by F is the triangle with vertices \({x}_{0}^{p},{r}_{1}^{p}\,{e}^{2\delta i}\) and \({r}_{1}^{p}\,{e}^{-2\delta i}\).

  4. 4.

    Instead of integration from 0 to x, however, we must use integration from some point ζη to x.

  5. 5.

    More precisely, consider \(z =\dot{ x} \circ {x}^{-1}\) with the inverse function x  − 1 of x = x(t). For convenience, the independent variable is named x now. We hope the fact that x is a function in the original equation, but the independent variable of the new equation is not too confusing for the reader.

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Informatique et Applications, Université de Haute Alsace, Mulhouse, France

    Augustin Fruchard

  2. Institut de Recherche Mathématique Avancée, Université de Strasbourg, Strasbourg, France

    Reinhard Schäfke

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  1. Augustin Fruchard
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  2. Reinhard Schäfke
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Fruchard, A., Schäfke, R. (2013). Composite Expansions and Singularly Perturbed Differential Equations. In: Composite Asymptotic Expansions. Lecture Notes in Mathematics, vol 2066. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34035-2_5

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