The canard explosion is a significant phenomenon in singularly perturbed system which has attracted lots of attentions in the literature. Such a periodic behavior often appears near a Hopf bifurcation and variety of methods have been developed for studying it. In the present work, we introduce a degenerate canard explosion of which the canard cycle does not arise from a Hopf bifurcation (a linear center perturbation) but from a nonlinear nilpotent center perturbation. Moreover, we demonstrate an algorithm to find the asymptotic expansions for this type of canard explosion, whereas some classical iterative methods fail to do so. Specifically, our approach provides the exact expressions of the first three terms of the critical value as well as the explicit analytical approximation of the slow manifold in the blow-up coordinates (but not in the original ones) up to the second-order. In fact, the presence of the error function in the involved expressions prevents obtaining best approximations. As far as we know, it is possibly the first time that a high-order analytical approximation of the critical value of the parameter is obtained for this degenerate canard explosion. Numerical results are also given for illustration and they are compared with the analytical predictions.

卡纳德爆炸是奇异摄动系统中的重要现象，引起了文学界的广泛关注。这样的周期性行为经常出现在霍夫夫分叉附近，并且已经开发出多种方法来研究它。在当前的工作中，我们介绍了简并的卡纳爆炸，其卡纳循环不是由霍普夫分叉（线性中心摄动）引起的，而是由非线性幂幂中心摄动引起的。此外，我们演示了一种算法，可以找到这种类型的Canard爆炸的渐近展开，而某些经典的迭代方法无法做到这一点。特别，我们的方法提供了临界值的前三个项的精确表达式，以及直到第二阶的爆炸坐标（而不是原始坐标）中慢流形的显式解析近似。实际上，所涉及的表达式中存在误差函数会妨碍获得最佳近似值。据我们所知，可能是首次针对这种退化的卡纳德爆炸获得参数临界值的高阶分析近似值。还给出了数值结果用于说明，并将其与分析预测进行比较。对于这种退化的Canard爆炸，可能是首次获得参数临界值的高阶分析近似值。还给出了数值结果用于说明，并将其与分析预测进行比较。对于这种退化的Canard爆炸，可能是首次获得参数临界值的高阶分析近似值。还给出了数值结果用于说明，并将其与分析预测进行比较。