Figure 2

Critical manifold, slow manifold and folded singular points. (a) Critical manifold \(S^{0}\) (green surface), super-slow manifold \(L^{0}\) (red curve) and a bursting orbit in the \((v_{0}, v_{2}, v_{3})\)-space. The curve \(L^{0}\) is divided into three branches at \(\mathcal{F}_{1}\) and \(\mathcal{F}_{2}\) (red dots) where it changes stability. The middle branch of the \(L^{0}\) (\(L^{0}_{m}\)) curve between \(\mathcal{F}_{1}\) and \(\mathcal{F}_{2}\) is unstable (dashed). The stable left-hand side branch (\(L^{0}_{l}\)), \(\mathcal{F}_{1}\), \(L^{0}_{m}\) and \(\mathcal{F}_{2}\) are entirely on the almost horizontal part of \(S^{0}\) (approximately on the \((v_{3} \approx 0)\)-plane). The stable right-hand side branch of \(L^{0}\) (\(L^{0}_{r}\)) on expands both on the horizontal and vertical parts of \(S^{0}\). Arrows indicate the corresponding time-scale (single-headed for super-slow, double-headed for slow dynamics). (b) Super-slow manifold \(L^{0}\) (red surface), fold curves \(\mathcal{F}_{1,2}\) (black lines) and folded singular points \(p_{1,2}\) (red dots) in the \((y_{7}, v_{2}, v_{0})\)-space. Arrows indicate the corresponding time-scale