Figure 8

Example canard orbits near the type-I excitable regime. (a) Bifurcation diagram for \(C_{3} = 50\), \(C_{5}=450\) and \(B\in [18, 22.5]\) (see Fig. 4c for the whole diagram). Stable and unstable solutions are represented by continuous and dashed curves, respectively. Limit point (\(LP_{1,2}\), black squares), Hopf (\(H_{4}\), red dot), homoclinic (\(HOM_{2,3}\), blue stars) bifurcations and saddle-node bifurcation of periodic orbits (SNOP, orange purple square) are marked on the diagram. Numbered solutions are presented in panels (b–d). The orange curves trace the frequency of the oscillations. (b) Periodic orbits marked in panel (a), \(L^{0}\) (red curve), fold curves \(\mathcal{F}_{1,2}\) (red points) and the critical surface \(S^{0}\) (green surface) are projected on the (\(v_{0}\), \(v_{2}\), \(v_{3}\))-space. Arrows indicate the flow direction and its time-scale (single-headed for super-slow, double-headed for slow dynamics. The homoclinic points \(HOM_{2}\) and \(HOM_{3}\) are marked by cyan and dark blue stars. (c) Periodic orbits marked in panel (a), \(L^{0}\) (red surface), fold curves \(\mathcal{F}_{1,2}\) (black curves) and folded singular points \(p_{1,2}\) (red dots) are projected on the \((y_{7},v_{2},v_{0})\)-space. Arrows indicate the corresponding time-scale. (d) Time series of the periodic orbits on panels (b, c) with respective color codes. Period is normalized to 1 (\(\tilde{t}/ \tilde{T} =1\), where T̃ represents period of a cycle)