Figure 7

Example canard orbits along the transition from sinusoidal oscillations to bursting oscillations (a) Bifurcation diagram for \(C_{3} = 80\), \(C_{5}=450\) and \(B \in [4.7, 4.9]\) (see Fig. 5a for the whole diagram). Stable and unstable solutions are represented by continuous and dashed curves, respectively. Hopf (\(H_{3}\), red dot) is marked on the diagram. The rectangular region is zoomed in panel (b). Numbered orbits from 1-7 are given in panels (c–d). The orange curve traces the frequency of the oscillations emerging at \(H_{3}\). (c) Periodic orbits marked in panels (a, b), \(L^{0}\) (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-head for super-slow, double-head for slow, triple-head for fast). (d) 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. (e) Time series of the periodic orbits on panels are (b, c) shown with respective color codes. Period is normalized to 1 (\(\tilde{t}/ \tilde{T} =1\), where T̃ represents period of a cycle)