July  2022, 9(3): 393-419. doi: 10.3934/jcd.2022008

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

1. 

School of Mathematical Sciences, Monash University, Melbourne VIC 3800, Australia

2. 

Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

3. 

National Institute of Water and Atmospheric Research Ltd (NIWA), Private Bag 99940, Newmarket, Auckland 1149, New Zealand

Received  October 2021 Revised  February 2022 Published  July 2022 Early access  April 2022

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.

Citation: Andy Hammerlindl, Bernd Krauskopf, Gemma Mason, Hinke M. Osinga. Determining the global manifold structure of a continuous-time heterodimensional cycle. Journal of Computational Dynamics, 2022, 9 (3) : 393-419. doi: 10.3934/jcd.2022008
References:
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C. BonattiS. CrovisierL. J. Díaz and A. Wilkinson, What is $\ldots$ a blender?, Notices Amer. Math. Soc., 63 (2016), 1175-1178.  doi: 10.1090/noti1438.

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show all references

References:
[1]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Inst. Steklov, 90 (1967), 209 pp.

[2]

A. AtriJ. AmundsenD. Clapham and J. Sneyd, A single-pool model for intracellular calcium oscillations and waves in the Xenopus laevis oocyte, Biophysical J., 65 (1993), 1727-1739.  doi: 10.1016/S0006-3495(93)81191-3.

[3]

R. Bamón, J. Kiwi and J. Rivera-Letelier, Wild Lorenz-like attractors, Preprint, 2006, arXiv: math/0508045.

[4]

W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405.  doi: 10.1093/imanum/10.3.379.

[5]

W.-J. Beyn, On well-posed problems for connecting orbits in dynamical systems, In Chaotic Numerics, (eds. P. E. Kloeden and K. J. Palmer), Contemporary Mathematics, American Mathematical Society, Rhode Island, 172 (1994), 131-168. doi: 10.1090/conm/172/01802.

[6]

G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc., 18 (1917), 199-300.  doi: 10.1090/S0002-9947-1917-1501070-3.

[7]

C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104.  doi: 10.1007/s00222-004-0368-1.

[8]

C. BonattiS. CrovisierL. J. Díaz and A. Wilkinson, What is $\ldots$ a blender?, Notices Amer. Math. Soc., 63 (2016), 1175-1178.  doi: 10.1090/noti1438.

[9]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143 (1996), 357-396.  doi: 10.2307/2118647.

[10]

C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and $C^1$-generic consequences, J. Inst. Math. Jussieu, 7 (2008), 469-525.  doi: 10.1017/S1474748008000030.

[11]

C. Bonatti and L. J. Díaz, Abundance of $C^1$-robust homoclinic tangencies, Trans. Amer. Math. Soc., 364 (2012), 5111-5148.  doi: 10.1090/S0002-9947-2012-05445-6.

[12]

C. BonattiL. J. Díaz and S. Kiriki, Stabilization of heterodimensional cycles, Nonlinearity, 25 (2012), 931-960.  doi: 10.1088/0951-7715/25/4/931.

[13]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Springer-Verlag, Berlin, 2005.

[14]

A. R. ChampneysY. A. Kuznetsov and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 867-887.  doi: 10.1142/S0218127496000485.

[15]

I. C. ChristovR. M. LueptowJ. M. Ottino and R. Sturman, A study in three-dimensional chaotic dynamics: Granular flow and transport in a bi-axial spherical tumbler, SIAM J. Appl. Dynam. Syst., 13 (2014), 901-943.  doi: 10.1137/130934076.

[16]

S. Crovisier and E. R. Pujals, Essential hyperbolicity and homoclinic bifurcations: A dichotomy phenomenon/mechanism for diffeomorphisms, Invent. Math., 201 (2015), 385-517.  doi: 10.1007/s00222-014-0553-9.

[17]

L. J. Díaz, Robust nonhyperbolic dynamics and heterodimensional cycles, Ergod. Theory Dyn. Syst., 15 (1995), 291-315.  doi: 10.1017/S0143385700008385.

[18]

L. J. DíazS. Kiriki and K. Shinohara, Blenders in centre unstable Hénon-like families: With an application to heterodimensional bifurcations, Nonlinearity, 27 (2014), 353-378.  doi: 10.1088/0951-7715/27/3/353.

[19]

L. J. Díaz and S. A. Pérez, Hénon-like families and blender-horseshoes at nontransverse heterodimensional cycles, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1930006, 22 pp. doi: 10.1142/S0218127419300064.

[20]

L. J. Díaz and S. A. Pérez, Blender-horseshoes in center-unstable Hénon-like families, In New Trends in One-Dimensional Dynamics, Springer Proc. Math., Springer, Cham, 285 (2019), 137-163. doi: 10.1007/978-3-030-16833-9_8.

[21]

L. J. Díaz and S. A. Pérez, Nontransverse heterodimensional cycles: Stabilisation and robust tangencies, Preprint, arXiv: 2011.08926, 2020.

[22]

L. Dieci and J. Rebaza, Point-to-periodic and periodic-to-periodic connections, BIT Numerical Mathematics, 44 (2004), 41-62; with erratum in 44 (2004), 617-618. doi: 10.1023/B:BITN.0000046846.33609.da.

[23]

E. J. Doedel, Auto, a program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284. 

[24]

E. J. Doedel, Auto-07P: Continuation and bifurcation software for ordinary differential equations, With major contributions from A.R. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. E. Oldeman, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. Zhang, Concordia University, 2007; available at http://cmvl.cs.concordia.ca/auto/.

[25]

E. J. DoedelB. W. KooiG. A. K. van Voorn and Y. A. Kuznetsov, Continuation of connecting orbits in 3D-ODEs: (Ⅰ) Point-to-cycle connections, Int. J. Bifur. Chaos, 18 (2008), 1889-1903.  doi: 10.1142/S0218127408021439.

[26]

E. J. DoedelB. W. KooiG. A. K. van Voorn and Y. A. Kuznetsov, Continuation of connecting orbits in 3D-ODEs: (Ⅱ) Cycle-to-cycle connections, Int. J. Bifur. Chaos, 19 (2009), 159-169.  doi: 10.1142/S0218127409022804.

[27]

H. R. Dullin and A. Wittek, Complete Poincaré sections and tangent sets, J. Phys. A, 28 (1995), 7157-7180.  doi: 10.1088/0305-4470/28/24/015.

[28]

L. Edelstein-Keshet, Mathematical Models in Biology, Birkäuser Mathematics Series, Random House, Inc., New York, 1988.

[29]

G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, , Interdisciplinary Applied Mathematics 35, Springer, New York, 2010. doi: 10.1007/978-0-387-87708-2.

[30]

J. P. EnglandB. Krauskopf and H. M. Osinga, Computing one-dimensional global manifolds of Poincaré maps by continuation, SIAM J. Appl. Dynam. Syst., 4 (2005), 1008-1041.  doi: 10.1137/05062408X.

[31]

Z. Galias, Positive topological entropy of Chua's circuit: A computer assisted proof, Internat. J. Bifur. Chaos, 7 (1997), 331-349.  doi: 10.1142/S0218127497000224.

[32]

S. V. GonchenkoJ. D. Meiss and I. I. Ovsyannikov, Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation, Regul. Chaotic Dyn., 11 (2006), 191-212.  doi: 10.1070/RD2006v011n02ABEH000345.

[33]

S. V. GonchenkoI. I. OvsyannikovC. Simó and D. Turaev, Three-dimensional Hénon-like map and wild Lorenz-like attractors, Int. J. Bifur. Chaos, 15 (2005), 3493-3508.  doi: 10.1142/S0218127405014180.

[34]

S. V. GonchenkoL. P. Shilnikov and D. Turaev, On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors, Regul. Chaotic Dyn., 14 (2009), 137-147.  doi: 10.1134/S1560354709010092.

[35]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[36]

S. HittmeyerB. Krauskopf and H. M. Osinga, Interacting global invariant sets in a planar map model of wild chaos, SIAM J. Appl. Dynam. Syst., 12 (2013), 1280-1329.  doi: 10.1137/120902860.

[37]

S. Hittmeyer, B. Krauskopf, H. M. Osinga and K. Shinohara, Existence of blenders in a Hénon-like family: Geometric insights from invariant manifold computations, Nonlinearity, 31 (2018), R239-R267. doi: 10.1088/1361-6544/aacd66.

[38]

S. HittmeyerB. KrauskopfH. M. Osinga and K. Shinohara, How to identify a hyperbolic set as a blender, Discrete Contin. Dyn. Syst., 40 (2020), 6815-6836.  doi: 10.3934/dcds.2020295.

[39]

A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, In Handbook of Dynamical Systems, (eds. B. Hasselblatt, H. W. Broer and F. Takens), Elsevier North Holland, 3 (2010), 379-524. doi: 10.1016/S1874-575X(10)00316-4.

[40]

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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 $
$ 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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