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Composite Expansions and Singularly Perturbed Differential Equations

Part of the Lecture Notes in Mathematics book series (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.

Keywords

  • Composite Expansion
  • Gevrey Order
  • Canard Solutions
  • Slow Curve
  • Slow Function

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Fig. 5.1
Fig. 5.2
Fig. 5.3
Fig. 5.4
Fig. 5.5
Fig. 5.6

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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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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