Computing connecting orbits to infinity associated with a homoclinic flip bifurcation

Figure 1.  Bifurcation diagram of system (2) showing: the curve of primary homoclinic bifurcation (brown), along which the homoclinic orbit changes at $ \mathbf{C}_{\rm in} $ from being orientable along $ \mathbf{H_o} $ to being non-orientable along $ \mathbf{H_t} $; curves $ \mathbf{SNP} $ and $ \mathbf{SNP^3} $ (green) of saddle-node bifurcation of periodic orbits; the first two curves $ \mathbf{PD} $ and $ \mathbf{PD^2} $ (red) of a cascade of period-doubling bifurcations; and the curves $ \mathbf{H^n} $ (increasingly darker shades of cyan) of $ n $-homoclinic bifurcations for $ n = 2, 3, 4, 5 $, and $ 6 $. On $ \mathbf{H^n} $ there are points $ \mathbf{C^n_{\rm O}} $ of orbit flip bifurcations (blue dots) and on $ \mathbf{H^2} $ there is a point $ \mathbf{C^2_{\rm I}} $ of inclination flip bifurcation (open dot). Panel (a) shows the $ (\alpha, \beta) $-plane, while panel (b) shows the $ (\alpha, \hat{\beta}) $-plane, where $ \hat{\beta} $ is the distance in the $ \beta $-coordinate from the curve $ \mathbf{H_{o/t}} $ of primary homoclinic bifurcation, which is now at $ \hat{\beta} = 0 $ (brown horizontal line). Panel (c) is an enlargement of the $ (\alpha, \hat{\beta}) $-plane near $ \mathbf{C}_{\rm in} $

Figure 2.  Phase portraits of system (2) along $ \mathbf{H_{t}} $, at $ \mathbf{C}_{\rm in} $ and along $ \mathbf{H_{o}} $ with enlargements near the saddle $ \mathbf{0} $ (top row). Shown are the saddle $ \mathbf{0} $, the homoclinic orbit $ \mathbf{\Gamma_{\rm HOM}} $ (brown curve) formed by one branch of $ W^s(\mathbf{0}) $, the other branch of $ W^s(\mathbf{0}) $ (cyan curve), a first part of $ W^{u}(\mathbf{0}) $ (red surface), and $ W^{uu}(\mathbf{0}) $ (magenta curve). Here $ (\alpha, \beta) = (5.8, 1.7010) $ in panel $ \mathbf{H_{o}} $, $ (\alpha, \beta) = (5.3573, 2.1917) $ in panel $ \mathbf{C}_{\rm in} $ and $ (\alpha, \beta) = (5.1, 2.717) $ in panel $ \mathbf{H_{t}} $

Figure 3.  The primary homoclinic orbit on $ \mathbf{H_t} $ and the $ n $-homoclinic orbits $ \mathbf{H^2} $ to $ \mathbf{H^6} $ of system (2) for $ \alpha = 5.3 $, shown in $ \mathbb{R}^3 $ in brown and increasingly darker shades of cyan to match the colors of the corresponding bifurcation curves in Fig. 1

Figure 4.  Dynamics at infinity for system (5), or system (4) with $ \bar{w} = 0 $, shown in the $ (\bar{x}, \bar{z}) $-plane in panel (a). Panel (b) shows the projection of panel (a) onto the corresponding Poincaré half-sphere with $ y_{\rm s} > 0 $ in the compactified $ (x_{\rm s}, y_{\rm s}, z_{\rm s}) $-cordinates

Figure 5.  Dynamics near the equilibrium $ (\bar{x}, \bar{z}, \bar{w}) = (0, 0, 0) $ of system (4). The behavior in the $ (x_{\rm B}, z_{\rm B}) $-plane, that is, the blow-up chart (6) with $ w_{\rm B} = 0 $, is shown in panel (a). It corresponds to the dynamics on a half-sphere around the origin in the $ (\bar{x}, \bar{z}, \bar{w}) $-space, as is illustrated in panel (b); compare also with Fig. 4(a)

Figure 6.  Numerical simulations suggest the existence of a cylinder-shaped separatrix $ S_{\rm c} $ of system (6) between trajectories that converge to the equilibrium $ (x_{\rm B}, z_{\rm B}, w_{\rm B}) = (0, -\alpha, 0) $, such as the orange trajectory, and those that do not, such as the blue trajectory. Panel (a) shows the $ (x_{\rm B}, z_{\rm B}, w_{\rm B}) $-space near $ (0, -\alpha, 0) $ and panel (b) the associated intersection sets with the plane defined by $ z_{\rm B} = -\alpha $

Figure 7.  The separatrix $ S_{\rm c} $ (purple surface) as represented locally by the cylinder $ C_{r^*} $, shown in the $ (\bar{x}, \bar{z}, \bar{w}) $-space of system (4). Panel (a) shows $ S_{\rm c} $ emerging from the blown-up half-sphere, while in panel (b), $ S_{\rm c} $ is a cone that emerges from the origin

Figure 8.  Set-up with Lin's method to compute a connecting orbit from $ \mathbf{q}_\infty $ to $ \mathbf{0} $ with two orbit segments that meet in the common Lin section $ \Sigma $ (green plane), illustrated in compactified Poincaré coordinates. Panel (a) shows the initially chosen orbit segments $ \mathbf{u} $ (cyan) to $ \mathbf{0} $ and $ \mathbf{u}_{\rm B} $ (magenta) from $ \mathbf{q}_\infty $ for $ \beta = 1.8 $ that define the Lin space $ Z $ (which appears curved in this representation); note that the Lin gap $ \eta $ is initially nonzero. Panel (b) shows the situation for $ \beta = 2.08874 $ where $ \eta = 0 $ and $ \mathbf{u} $ and $ \mathbf{u}_{\rm B} $ connect in $ \Sigma $ to form the heteroclinic connection; here, $ \alpha = 5.3 $

Figure 9.  Bifurcation diagram of system (2) with the additional curve $ \mathbf{Het^\infty} $ (magenta) of heteroclinic bifurcation involving the point $ \mathbf{q}_\infty $ at infinity. Panel (a) shows how $ W^s(\mathbf{0}) $ spirals towards infinity in the $ (x, y, z) $-space to form the heteroclinic connection on $ \mathbf{Het^\infty} $ for $ \alpha = 5.3 $ and $ \beta = 2.08874 $; see Fig. 3 for comparison. Panel (b) shows the overall bifurcation diagram in the $ (\alpha, \hat{\beta}) $-plane and panel (c) is an enlargement near the point $ \mathbf{C}_{\rm in} $; see Fig. 1 for details on the other bifurcation curves

Figure 10.  To the left of the curve $ \mathbf{Het^\infty} $ in the $ (\alpha, \hat{\beta}) $-plane, the stable manifold of $ W^s(\mathbf{0}) $ approaches, but does not connect to $ \mathbf{q}_\infty $, because it lies outside $ S_{\rm c} $ (a). To the right of $ \mathbf{Het^\infty} $, it lies inside $ S_{\rm c} $ and so connects to $ \mathbf{q}_\infty $. The illustration in compactified Poincaré coordinates is for $ \alpha = 5.3 $ with $ \beta = 2.9 $ in panel (a) and $ \beta = 2.8 $ in panel (b)

Figure 11.  Set-up with Lin's method to compute a connecting orbit from $ \mathbf{q}_\infty $ to a saddle periodic orbit $ \Gamma_o $ (green curve) with two orbit segments that meet in the common Lin section $ \Sigma $ (green plane), illustrated in compactified Poincaré coordinates for $ \alpha = 6.2 $ and $ \beta = 1.6 $. Panel (a) shows the initially chosen orbit segments $ \mathbf{u} $ (cyan) to $ \Gamma_o $ and $ \mathbf{u}_{\rm B} $ (magenta) from $ \mathbf{q}_\infty $ that define the Lin space $ Z $ (which appears curved in this representation); note that the Lin gap $ \eta $ is initially nonzero. Panel (b) shows the situation where $ \eta = 0 $ and $ \mathbf{u} $ and $ \mathbf{u}_{\rm B} $ connect in $ \Sigma $ to form the heteroclinic connection