Figure 9

Example canard orbits near the type-II excitable regime. (a) Bifurcation diagram for \(C_{3} = 80\), \(C_{5}=450\) and \(B \in [16, 18]\) (see Fig. 5a for the whole diagram). Stable and unstable solutions are represented by continuous and dashed curves, respectively. Hopf bifurcation (\(H_{4}\), red dot) is marked on the diagram. Numbered orbits on the lower branch of periodic solutions are presented in panels (c–d). The orange curve traces the frequency of the oscillations emerging at \(H_{4}\). (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). (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) are shown with respective color codes. Period is normalized to 1 (\(\tilde{t}/ \tilde{T} =1\), where T̃ represents period of a cycle)