Determining the global manifold structure of a continuous-time heterodimensional cycle

Figure 1.  Panel (a) shows the locus ${\mathit {PtoP}}$ (purple curve) of the heterodimensional cycle of system (1) in the $ (J, s) $-plane, together with the loci of Hopf bifurcation ${\mathit {H}}$ (red curve), of saddle-node bifurcation of limit cycles ${\mathit {SL}}$ (green curve) ending on ${\mathit {H}}$ at the point ${\mathit {DH}}$, and of period-doubling bifurcation ${\mathit {PD}}$ (blue curve); the curve ${\mathit {PD}}$ is tangent to ${\mathit {SL}}$ at the point ${\mathit {PS}}$, to the left of which ${\mathit {PD}}$ is dotted. The heterodimensional PtoP cycle for the indicated point $ (J^\ast, s^\ast) = (3.02661, 9.0) $ on ${\mathit {PtoP}}$ is shown in projections onto $ (c, v, c_t) $-space in panel (b) and onto $ (c, c_t, n) $-space in panel (c). It consists of a unique (and non-transverse) connecting orbit $ A $ (black curve) from $ \Gamma_1 $ to $ \Gamma_2 $ (green curves) and a two-dimensional topological cylinder $ B $ (purple surface) of trajectories from $ \Gamma_2 $ to $ \Gamma_1 $

Figure 2.  Sketch of a heterodimensional cycle in a three-dimensional discrete-time system. Here, two saddle fixed points $ \gamma_1 $ and $ \gamma_2 $ have two-dimensional manifolds $ W^s(\gamma_1) $ and $ W^u(\gamma_2) $ that intersect transversely in a curve $ B $ and one-dimensional manifolds $ W^u(\gamma_1) $ and $ W^s(\gamma_2) $ that intersect in a single orbit $ \left( a_k \right)_{k \in \mathbb{Z}} $. Reproduced from [63]

Figure 3.  The heterodimensional PtoP cycle, shown in panel (a) in projection onto $ (c, c_t, n) $-space with the section $ \Sigma $ (grey plane) defined by $ c = 0.15 $, while panel (b) shows its intersection sets in $ \Sigma $. The periodic orbits $ \Gamma_1 $ and $ \Gamma_2 $ (green curves) intersect $ \Sigma $ in the points $ \gamma_1^\pm $ and $ \gamma_2^\pm $, respectively; the connecting orbit $ A $ (black curve) intersects $ \Sigma $ in points $ a^\pm_k $ marked by $ \ast $ (these are extremely close to $ \gamma^\pm_1 $ for $ k \leq -1 $); and the cylinder $ B $ (purple surface) intersects $ \Sigma $ in two (purple) curves $ \widehat{B}^\pm $

Figure 4.  The two-dimensional stable manifold $ W^s(\Gamma_2) $ (blue surface) intersects $ \Sigma $ (grey plane) in the two primary intersection curves $ \widehat{W}_0^{s, \pm}(\Gamma_2) $ (blue curves). Panel (a) shows, in projection onto $ (c, c_t, n) $-space, the section $ \Sigma $ and the side of $ W^{s}(\Gamma_2) $ that comes very close to $ \Gamma_1 $ (green curve); panel (b) shows in $ \Sigma $ the intersection sets $ \widehat{W}_0^{s, \pm}(\Gamma_2) $, $ \gamma^\pm_1 $, $ \gamma^\pm_2 $ and $ a^\pm_k $

Figure 5.  The stable manifold $ W^s(\Gamma_2) $ (blue) returns to $ \Sigma $ (grey plane) in backward time creating additional intersection curves, of which $ \widehat{W}_{1}^{s, -}(\Gamma_2) $ is labeled. These backward-time returns accumulate very fast onto the intersection set $ \widehat{W}^{ss, \pm} $ (cyan curve) with $ \Sigma $ of the two-dimensional strong stable manifold $ W^{ss}(\Gamma_1) $ (cyan surface). Panel (a) shows a projection onto $ (c, c_t, n) $-space, and panel (b) shows the respective intersection sets in $ \Sigma $; compare with Fig. 4

Figure 6.  The two-dimensional unstable manifold $ W^{u}(\Gamma_1) $ (red surface) intersects $ \Sigma $ (grey plane) in the primary curve $ \widehat{W}^u_0(\Gamma_1) $ that contains the two points $ \gamma_1^\pm $ and crosses the tangency locus $ C $ in $ \Sigma $ twice. Panel (a) shows, in projection onto $ (c, v, n) $-space, the part of $ W^{u}(\Gamma_1) $ between $ \Gamma_1 $ (green curve) and the arc of $ \widehat{W}^u_0(\Gamma_1) $ (red curve) in $ \Sigma $ that contains the two points $ a^+_0 $ and $ a^-_0 $; panel (b) shows in $ \Sigma $ all of $ \widehat{W}^u_0(\Gamma_1) $ (red curve), $ \gamma^\pm_1 $, $ \gamma^\pm_2 $ and $ a^\pm_0 $

Figure 7.  A part of $ W^u(\Gamma_1) $ intersects $ \Sigma $ (grey plane) again in the closed curve $ \widehat{W}^u_1(\Gamma_1) $. Panel (a) shows, in projection onto $ (c, v, n) $-space, the periodic orbit $ \Gamma_1 $ (green curve) and the respective part of $ W^{u}(\Gamma_1) $ (red surface) up to $ \widehat{W}^u_1(\Gamma_1) $ (red curve) in $ \Sigma $; panel (b) shows in $ \Sigma $ the intersection sets $ \widehat{W}^{u}(\Gamma_0) $ and $ \widehat{W}^{u}(\Gamma_1) $ (red curves), $ \gamma^\pm_1 $, $ \gamma^\pm_2 $, $ a^\pm_0 $ and $ a^\pm_1 $

Figure 8.  A part of $ W^u(\Gamma_1) $ intersects $ \Sigma $ (grey plane) a second time in a spiraling curve $ \widehat{W}^u_2(\Gamma_2) $ that contains the points $ a^+_k $ for $ k \geq 2 $. Panel (a) shows, in projection onto $ (c, v, n) $-space, the periodic orbit $ \Gamma_1 $ (green curve) and the respective part of $ W^{u}(\Gamma_1) $ (red surface) up to $ \widehat{W}^u_2(\Gamma_2) $ (red curve) in $ \Sigma $; panel (b) shows in $ \Sigma $ the intersection sets $ \widehat{W}^u_0(\Gamma_0) $, $ \widehat{W}^u_1(\Gamma_1) $ and $ \widehat{W}^u_2(\Gamma_2) $ (red curves), $ \gamma^\pm_1 $, $ \gamma^\pm_2 $, and $ a^+_k $ for $ k \geq 0 $

Figure 9.  An enlargment of Fig. 8 showing how $ W^u(\Gamma_1) $ (red surface) spirals and accumulates onto the two-dimensional strong unstable manifold $ W^{uu}(\Gamma_2) $ (orange surface). Panel (a) shows a projection onto $ (c, v, n) $-space with $ \Sigma $ (grey plane); panel (b) shows in $ \Sigma $ the respective intersection sets $ \widehat{W}^u_2(\Gamma_1) $ (red curve), $ \widehat{W}_0^{uu}(\Gamma_2) $ (orange curve), $ \gamma^\pm_2 $, and $ a^\pm_k $ for $ k \geq 2 $

Figure 10.  An overall view in projection onto $ (c, c_t, n) $-space of how $ W^s(\Gamma_2) $ (blue surface) and $ W^u(\Gamma_1) $ (red surface) intersect in the (non-transverse) connecting orbit $ A $ (black curve), and how this generates the discrete intersection sets $ a^\pm_k $ in the section $ \Sigma $ (grey plane); compare with Fig. 4(a)

Figure 11.  Two views of $ \Sigma $ in panels (a) and (b) illustrate how $ \widehat{W}^{s, +}(\Gamma_2) $ and $ \widehat{W}^{s, -}(\Gamma_2) $ (blue curves) intersect $ \widehat{W}^u_0(\Gamma_0) $, $ \widehat{W}^u_1(\Gamma_1) $ and $ \widehat{W}^u_2(\Gamma_2) $ (red curves) in the points $ a^+_k $ and $ a^-_k $, respectively. Also shown are $ \widehat{W}_0^{ss, \pm}(\Gamma_1) $, $ \widehat{W}_0^{uu}(\Gamma_2) $ and the two intersection curves $ \widehat{B}^\pm $ of the cylinder $ B $; compare with Fig. 10

Figure 12.  A sketch of the invariant objects in the section $ \Sigma $ that give rise to the heterodimensional PtoP cycle; compare with Fig. 11