Asymptotic expansions for a degenerate canard explosion

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.