# Four Introductory Examples

• Augustin Fruchard
• Reinhard Schäfke
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2066)

## 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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