Abstract
Here we present simple examples, showing that solutions of singularly perturbed differential equations naturally have composite asymptotic expansions (ca se s) near turning points. A theory of ca se s might thus help to understand them.All examples are linear equations of first order. The first example is among the simplest ones having a turning point. The second one contains a control parameter for “duck hunting” or “canard hunting”. The third example also contains a control parameter, but the turning point is no longer simple; this implies that the canard solutions are no longer overstable solutions in the sense of Guy Wallet. Finally the fourth example relates to “fake ducks” or “fake canard solutions”: the slow curve is first repelling and then attracting. In this situation, any solution with bounded initial condition at the turning point is defined and bounded on an interval containing this turning point, but this solution can have a ca se only if the initial condition has an asymptotic expansion. We will see that this necessary condition is also sufficient.
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Notes
- 1.
In this memoir, we will use the notation \({y}^{-}\sim \widehat{y}\) for the Poincaré asymptotic.
- 2.
Here we are interested only in the case p even, but the case p odd also has its interest, cf. Eq. (6.15) in Sect. 6.2.
- 3.
Recall that “bounded” means uniformly with respect to \(\epsilon \) in some interval \(]0,{\epsilon }_{0}]\) . In the present context, it turns out that, for all fixed \(\epsilon \) , there is a unique value \(\alpha = \alpha (\epsilon )\) for which (1.13) has a solution \(y = y(x,\epsilon )\) bounded on \(\mathbb{R}\) in the classical sense, and that the function y so defined is also bounded on \(\mathbb{R}\times \,]0,{\epsilon }_{0}]\).
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Fruchard, A., Schäfke, R. (2013). Four Introductory Examples. In: Composite Asymptotic Expansions. Lecture Notes in Mathematics, vol 2066. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34035-2_1
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