Asymptotic expansions are of great interest and significance in the study of canard explosions in singularly perturbed systems. Several classical methods have been developed to compute such expansions. However, for the non-generic case considered in this letter, those methods fail to do so. There only exists an estimation on the first non-zero term in the literature. Our aim is to propose a new approach to find the asymptotic expansions iteratively. Moreover, the exact value of the first non-zero term for the non-generic case is provided in terms of Airy function. The provided numerical results validate our analytical approximations.

中文翻译：

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使用参数表示形式的非通用Canard系列的渐近展开

渐近展开在奇摄动系统中的Canard爆炸研究中具有极大的兴趣和意义。已经开发了几种经典方法来计算此类展开。但是，对于本信函中考虑的非一般情况，这些方法无法这样做。在文献中仅存在对第一个非零项的估计。我们的目的是提出一种新的方法来迭代地找到渐近展开。此外，根据艾里函数提供了非一般情况的第一个非零项的确切值。提供的数值结果验证了我们的解析近似值。