Existence and Bifurcation of Canards in ³ in the Case of a Folded Node

We give a geometric analysis of canards of folded node type in singularly perturbed systems with two-dimensional (2D) folded critical manifold using the blow-up technique. The existence of two primary canards is known provided a nonresonance condition μ is satisfied, where μ=λ₁/λ₂ denotes the ratio of the eigenvalues of the associated folded singularity of the reduced flow. We show that, due to resonances, bifurcation of secondary canards occurs. We give a detailed geometric explanation of this phenomenon using an extension of Melnikov theory to prove a transcritical bifurcation of canards for odd μ. Furthermore, we show numerically the existence of a pitchfork bifurcation for even μ and a novel turning point bifurcation close to μ. We conclude the existence of [(μ-1)/2] secondary canards away from the resonances. Finally, we apply our results to a network of Hodgkin--Huxley neurons with excitatory synaptic coupling and explain the observed slowing of the firing rate of the synchronized network due to the existence of canards of folded node type.