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

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  • A heterodimensional cycle consists of two saddle periodic orbits with unstable manifolds of different dimensions and a pair of connecting orbits between them. Recent theoretical work on chaotic dynamics beyond the uniformly hyperbolic setting has shown that heterodimensional cycles may occur robustly in diffeomorphisms of dimension at least three. We consider the first explicit example of a heterodimensional cycle in the continuous-time setting, which has been identified by Zhang, Krauskopf and Kirk [Discr. Contin. Dynam. Syst. A 32(8) 2825-2851 (2012)] in a four-dimensional vector-field model of intracellular calcium dynamics.

    We show here how a boundary-value problem set-up can be employed to determine the organization of the dynamics in a neighborhood in phase space of this heterodimensional cycle, which consists of a single connecting orbit of codimension one and an entire cylinder of structurally stable connecting orbits between two saddle periodic orbits. More specifically, we compute the relevant stable and unstable manifolds, which we visualize in different projections of phase space and as intersection sets with a suitable three-dimensional Poincaré section. In this way, we show that, locally near the intersection set of the heterodimensional cycle, the manifolds interact as described by the theory for three-dimensional diffeomorphisms. On the other hand, their global structure is more intricate, which is due to the fact that it is not possible to find a Poincaré section that is transverse to the flow everywhere. More generally, our results show that advanced numerical continuation techniques enable one to investigate how abstract concepts â€" such as that of a heterodimensional cycle of a diffeomorphism â€" arise and manifest themselves in explicit continuous-time systems from applications.

    Mathematics Subject Classification: Primary: 34C37, 37D10, 37M20, 37G20; Secondary: 92-10.


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  • 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

    Table 1.  Parameter values used for the intracellular calcium model (1)

    $ D $ $ \alpha $ $ k_f $ $ \phi_1 $ $ \gamma $ $ k_s $ $ \varepsilon $ $ k_p $ $ \phi_2 $
    $ 25.0 $ $ 0.05 $ $ 20.0 $ $ 2.0 $ $ 5.0 $ $ 20.0 $ $ 0.2 $ $ 20.0 $ $ 1.0 $
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