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arxiv: 1906.11438 · v1 · pith:HKYUM4KWnew · submitted 2019-06-27 · 🧮 math.DS

Global manifold structure of a continuous-time heterodimensional cycle

Andy Hammerlindl , Bernd Krauskopf , Gemma Mason , Hinke M. Osinga This is my paper

Pith reviewed 2026-05-25 14:52 UTC · model grok-4.3

classification 🧮 math.DS
keywords heterodimensional cyclestable and unstable manifoldscontinuous-time dynamical systemsPoincaré sectioncalcium dynamicsheteroclinic connectionsfour-dimensional vector fieldboundary-value problem
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The pith

In a four-dimensional calcium model, the manifolds of a heterodimensional cycle follow three-dimensional diffeomorphism theory locally but display more intricate global structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the stable and unstable manifolds around a heterodimensional cycle formed by two saddle periodic orbits in a continuous-time four-dimensional vector field. It establishes that near the intersection set the manifolds interact exactly as predicted by existing theory for three-dimensional maps. Away from that set the global arrangement becomes more complex because no single Poincaré section can remain transverse to the flow at every point. This matters for applied models because it shows how an abstract dynamical object appears concretely and why continuous-time geometry differs from its discrete-time counterpart.

Core claim

A heterodimensional cycle in the calcium model consists of one codimension-one connecting orbit and a cylinder of structurally stable connecting orbits. Locally near the intersection set the associated stable and unstable manifolds interact as described by the theory for three-dimensional diffeomorphisms. Their global structure is more intricate because it is impossible to find a Poincaré section transverse to the flow everywhere.

What carries the argument

The heterodimensional cycle (pair of heteroclinic connections between saddle periodic orbits with unstable manifolds of different dimensions) together with its stable and unstable manifolds computed via boundary-value problems.

If this is right

  • Local manifold geometry near the cycle intersection matches the predictions of three-dimensional discrete-time theory.
  • Global manifold structure in continuous time must be analyzed without assuming a globally transverse section.
  • Heterodimensional cycles can be located and visualized numerically in concrete application models such as calcium dynamics.
  • The abstract notion of a heterodimensional cycle is realized and studyable in four-dimensional flows from applications.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Continuous-time systems may require additional geometric tools beyond those sufficient for discrete maps when studying global manifold arrangements.
  • Numerical continuation methods for manifolds in flows should incorporate checks for regions where transversality to a section breaks down.
  • Similar global intricacies could appear in other high-dimensional application models once heterodimensional cycles are located.

Load-bearing premise

The heterodimensional cycle and its connecting orbits exist in the model exactly as located earlier, and the chosen boundary-value problem plus Poincaré section capture the true manifold geometry without significant numerical artifacts.

What would settle it

A computation or observation showing that the manifolds near the intersection set fail to interact as predicted by three-dimensional diffeomorphism theory, or that a single Poincaré section transverse to the flow can be constructed everywhere.

Figures

Figures reproduced from arXiv: 1906.11438 by Andy Hammerlindl, Bernd Krauskopf, Gemma Mason, Hinke M. Osinga.

Figure 1
Figure 1. Figure 1: Sketch of a heterodimensional cycle in a three-dim [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Panel (a) shows the locus PtoP (purple curve) of the heterodimensional cycle of system (1) in the (J, s)-plane, relative to the loci of Hopf bifurcation H (red curve), of saddle-node bifurcation of limit cycles SL (green curve) ending on H at the point DH, and of period-doubling bifurcation PD (blue curve); the curve PD is tangent to SL at the point PS, to the left of which PD is dotted. The heterodimensio… view at source ↗
Figure 3
Figure 3. Figure 3: The heterodimensional PtoP cycle, shown in panel ( [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The two-dimensional stable manifold Ws (Γ2) (blue surface) intersects Σ (grey plane) in the two primary intersection curves Wcs,± 0 (Γ2) (blue curves). Panel (a) shows, in projection onto (c, ct , n)-space, the section Σ and the side of Ws (Γ2) that comes very close to Γ1 (green curve); panel (b) shows in Σ the intersection sets Wcs,± 0 (Γ2), γ ± 1 , γ ± 1 and a ± k . 12 [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 5
Figure 5. Figure 5: The stable manifold Ws (Γ2) (blue) returns to Σ (grey plane) in backward time creating additional intersection curves, of which the first, Wcs,± −1 , is shown. These backward-time returns accumulate very fast onto the intersection set Wcss,± (cyan curve) with Σ of the two-dimensional strong stable manifold Wss(Γ1) (cyan surface). Panel (a) shows a projection onto (c, ct , n)-space, and panel (b) shows the … view at source ↗
Figure 7
Figure 7. Figure 7: figure 7. Panel (a) shows, in projection onto ( [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: The two-dimensional unstable manifold Wu (Γ1) (red surface) intersects Σ (grey plane) in the primary curve Wcu 0 (Γ1) that contains the two points γ ± 1 and crosses the tangency locus C in Σ twice. Panel (a) shows, in projection onto (c, v, n)-space, the part of Wu (Γ1) between Γ1 (green curve) and the arc of Wcu 0 (Γ1) (red curve) in Σ that contains the two points a + 0 and a − 0 ; panel (b) shows in Σ al… view at source ↗
Figure 7
Figure 7. Figure 7: A part of Wu (Γ1) intersects Σ (grey plane) again in the closed curve Wcu 1 (Γ1). Panel (a) shows, in projection onto (c, v, n)-space, the periodic orbit Γ1 (green curve), the respective part of Wu (Γ1) (red surface) up to Wcu 1 (Γ1) (red curve) in Σ; panel (b) shows in Σ the intersection sets Wcu (Γ0) and Wcu (Γ1) (red curves), γ ± 1 , γ ± 2 , a ± 0 and a ± 1 . other hand, the fact that all points a ± k f… view at source ↗
Figure 8
Figure 8. Figure 8: A part of Wu (Γ1) intersects Σ (grey plane) a second time in a spiralling curve Wcu 2 (Γ2) that contains the points a + k for k ≥ 2. Panel (a) shows, in projection onto (c, v, n)-space, the periodic orbit Γ1 (green curve) and the respective part of Wu (Γ1) (red surface) up to Wcu 2 (Γ2) (red curve) in Σ; panel (b) shows in Σ the intersection sets Wcu 0 (Γ0), Wcu 1 (Γ1) and Wcu 2 (Γ2) (red curves), γ ± 1 , … view at source ↗
Figure 9
Figure 9. Figure 9: An enlargment of figure 8 showing how Wu (Γ1) (red surface) spirals and accumu￾lates onto the two-dimensional strong unstable manifold Wuu(Γ2) (orange surface). Panel (a) shows a projection onto (c, v, n)-space with Σ (grey plane); panel (b) shows in Σ the respective intersection sets Wcu 2 (Γ0) (red curve), Wcuu 0 (Γ2) (orange curve), γ ± 2 , and a ± k for k ≥ 2. 3.5 Interaction between Wcs,±(Γ2) and Wcu … view at source ↗
Figure 10
Figure 10. Figure 10: An overall view in projection onto (c, ct , n)-space of how Ws (Γ2) (blue surface) and Wu (Γ1) (red surface) intersect in the (non-transverse) connecting orbit A (black curve), and how this generates the discrete intersection sets a ± k in the section Σ (grey plane); compare with figure 4(a). from the intersection Ws (Γ2)∩ Wu (Γ1) [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Two views of Σ in panels (a) and (b) illustrate how [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: A sketch of the invariant objects in the section Σ t [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
read the original abstract

A heterodimensional cycle consists of a pair of heteroclinic connections between two saddle periodic orbits with unstable manifolds of different dimensions. 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 study a concrete example of a heterodimensional cycle in the continuous-time setting, specifically in a four-dimensional vector field model of intracellular calcium dynamics. By employing advanced numerical techniques, Zhang, Krauskopf and Kirk [Discr. Contin. Dynam. Syst. A 32(8) 2825--2851 (2012)] found that a heterodimensional cycle exists in this model. We investigate the geometric structure of the associated stable and unstable manifolds in the neighbourhood of this heterodimensional cycle, consisting of a single connecting orbit of codimension one and an entire cylinder of structurally stable connecting orbits between two saddle periodic orbits. We employ a boundary-value problem set-up to compute their stable and unstable manifolds, which we visualize in different projections of phase space and as intersection sets with a suitable three-dimensional Poincar\'e section. 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 Poincar\'e section that is transverse to the flow everywhere. Our results show that the abstract concept of a heterodimensional cycle arises and can be studied in continuous-time models from applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper numerically studies the stable and unstable manifolds of a heterodimensional cycle in a four-dimensional continuous-time model of intracellular calcium dynamics. Relying on the cycle previously located by Zhang, Krauskopf and Kirk (2012), the authors employ boundary-value problem methods to compute the manifolds associated with a codimension-one connecting orbit and a cylinder of structurally stable connections. These are visualized in phase-space projections and as intersections with a chosen three-dimensional Poincaré section. The central claim is that, locally near the cycle, the manifolds interact as predicted by theory for three-dimensional diffeomorphisms, while the global structure is more intricate because no Poincaré section can be transverse to the flow everywhere.

Significance. If the computations are accurate, the work supplies a concrete continuous-time example from an applied model that illustrates both the local validity of discrete-time heterodimensional cycle theory and the additional geometric complexities induced by the flow. The explicit use of BVP techniques for global manifold computation and the direct comparison of local versus global behavior constitute a useful case study bridging abstract theory and applications.

major comments (1)
  1. [Numerical methods and results sections (around the BVP setup and Poincaré section description)] The BVP formulation and Poincaré section are central to all reported manifold geometry, yet the manuscript provides no convergence checks, tolerance settings, mesh adaptation details, or validation by recovering the known heteroclinic connections from Zhang et al. (2012). This absence is load-bearing for the claim of an 'intricate' global structure, as discretization artifacts could alter the reported intersection sets.
minor comments (1)
  1. [Abstract and §1] The abstract and introduction could more explicitly state the dimension of the vector field and the precise codimensions of the connecting orbits to aid readers unfamiliar with the 2012 reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the constructive comment on the numerical methods. We address the point below.

read point-by-point responses

  1. Referee: [Numerical methods and results sections (around the BVP setup and Poincaré section description)] The BVP formulation and Poincaré section are central to all reported manifold geometry, yet the manuscript provides no convergence checks, tolerance settings, mesh adaptation details, or validation by recovering the known heteroclinic connections from Zhang et al. (2012). This absence is load-bearing for the claim of an 'intricate' global structure, as discretization artifacts could alter the reported intersection sets.

    Authors: We agree that the manuscript would benefit from explicit documentation of the numerical parameters and validation steps. In the revised version we will add a short subsection (or appendix) that specifies the collocation tolerances (absolute and relative), the mesh-adaptation strategy employed by the BVP solver, residual-norm convergence checks performed under successive mesh refinements, and a direct numerical recovery of the codimension-one and cylinder connections reported by Zhang et al. (2012). These additions will confirm that the reported intersection sets are not discretization artifacts and thereby strengthen the distinction between local and global manifold geometry. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a numerical case study that computes stable/unstable manifolds of a pre-located heterodimensional cycle via boundary-value problems and Poincaré sections. It cites Zhang et al. (2012) only for cycle existence and location; the geometric claims about local interaction (matching 3D diffeomorphism theory) and global intricacy (due to non-transverse sections) are obtained directly from the independent computations without any fitted parameters renamed as predictions, self-definitional reductions, or load-bearing self-citations that collapse the result to its inputs. The derivation chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on an existing model and standard results from dynamical systems without introducing new fitted parameters or postulated entities; the main external dependency is the prior numerical location of the cycle itself.

axioms (2)
  • standard math Stable and unstable manifolds of hyperbolic periodic orbits exist and can be computed via boundary-value problems.
    Invoked throughout the numerical construction of the manifolds near the cycle.
  • domain assumption The heterodimensional cycle consisting of one codimension-one connection and a cylinder of connections exists in the calcium model as located by Zhang et al. (2012).
    The entire study is built on this prior computational finding; the current paper does not re-verify existence.

pith-pipeline@v0.9.0 · 5823 in / 1614 out tokens · 35348 ms · 2026-05-25T14:52:58.094953+00:00 · methodology

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Lean theorems connected to this paper

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    Relation between the paper passage and the cited Recognition theorem.

    We investigate the geometric structure of the associated stable and unstable manifolds in the neighbourhood of this heterodimensional cycle, consisting of a single connecting orbit of codimension one and an entire cylinder of structurally stable connecting orbits between two saddle periodic orbits.

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Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    Anosov D V 1967 Geodesic flows on closed Riemannian manifo lds with negative curvature Proc. Inst. Steklov 90 1–235

    work page 1967

  2. [2]

    Atri A, Amundsen J, Clapham D and Sneyd J 1993 A single-poo l model for intracellular calcium oscillations and waves in the Xenopus laevis oocyte Biophysical Journal 65 1727– 1739

    work page 1993

  3. [3]

    Bam´ on R, Kiwi J and Rivera-Letelier J 2006 Wild Lorenz-l ike attractors Preprint, arXiv:0508045

    work page 2006

  4. [4]

    Beyn W-J 1990 The numerical computation of connecting or bits in dynamical systems IMA J. Numer. Anal. 10(3) 379–405

    work page 1990

  5. [5]

    Beyn W-J 1994 On well-posed problems for connecting orbi ts in dynamical systems Chaotic Numerics Contemporary mathematics 172 ed P E Kloeden (Providence, RI: American Mathematical Society) pp 131–168

    work page 1994

  6. [6]

    Birkhoff G D 1917 Dynamical systems with two degrees of fre edom Trans. Amer. Math. Soc. 18 199-300

    work page 1917

  7. [7]

    Bonatti C and Crovisier S 2004 R´ ecurrence et g´ en´ ericit´ eInvent. Math. 158(1) 33–104

    work page 2004

  8. [8]

    a blender? Notices Amer

    Bonatti C, Crovisier S D ´ ıaz L J and Wilkinson A 2016 What i s. . . a blender? Notices Amer. Math. Soc. 63(10) 1175–1178 23

    work page 2016

  9. [9]

    Bonatti C and D ´ ıaz L J 1996 Persistent nonhyperbolic tra nsitive diffeomorphisms Ann. of Math. 2nd Series 143(2) 357–396

    work page 1996

  10. [10]

    Bonatti C and D ´ ıaz L J 2008 Robust heterodimensional cycles and C 1-generic consequences J. Inst. Math. Jussieu 7 469-525

    work page 2008

  11. [11]

    A Global Geometric and Probabilistic Perspective Encyclopaedia Math

    Bonatti C, D ´ ıaz L J and Viana M 2005 Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective Encyclopaedia Math. Sci. 102 (Berlin Heidelberg: Springer-Verlag)

    work page 2005

  12. [12]

    Champneys A R, Kuznetsov Yu A and Sandstede B 1996 A numer ical toolbox for homoclinic bifurcation analysis Int. J. Bifurcation and Chaos 6(5) 867–887

    work page 1996

  13. [13]

    Christov I C, Lueptow R M, Ottino J M and Sturman R 2014 A st udy in three-dimensional chaotic dynamics: granular flow and transport in a bi-axial s pherical tumbler SIAM J. Appl. Dynam. Syst. 13(2) 901–943

    work page 2014

  14. [14]

    Crovisier S and Pujals E R 2015 Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms. Invent. Math. 201(2) 385–517

    work page 2015

  15. [15]

    D ´ ıaz L J 1995 Robust nonhyperbolic dynamics and heterodimensional cycles. Ergod. Theory Dyn. Syst. 15 291-315

    work page 1995

  16. [16]

    D ´ ıaz L J, Kiriki S and Shinohara K 2014 Blenders in centr e unstable H´ enon-like families: with an application to heterodimensional bifurcations Nonlinearity 27(3) 353–378

    work page 2014

  17. [17]

    Dieci L and Rebaza J 2004 Point-to-periodic and periodi c-to-periodic connections BIT Numerical Mathematics 44(1) 4162; with erratum in 44(3) 617–618

    work page 2004

  18. [18]

    Doedel E J 1981 Auto, a program for the automatic bifurcation analysis of autono mous systems Congr. Numer. 30 265–384

    work page 1981

  19. [19]

    Doedel E J 2007 Auto-07P: Continuation and bifurcation software for ordinary differe ntial equations, with major contributions from Champneys A R, Fai rgrieve T F, Kuznetsov Yu A, Oldeman B E, Paffenroth R C, Sandstede B, Wang X J and Zhang C; a vailable at http://cmvl.cs.concordia.ca/auto/

    work page 2007

  20. [20]

    Doedel E J, Kooi B W, van Voorn G A K and Kuznetsov Yu A 2008 C ontinuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connecti ons Int. J. Bifurcation and Chaos 18(7) 1889–1903

    work page 2008

  21. [21]

    Doedel E J, Kooi B W, van Voorn G A K and Kuznetsov Yu A 2009 C ontinuation of connecting orbits in 3D-ODEs: (II) Cycle-to-cycle connect ions Int. J. Bifurcation and Chaos 19(1) 159–169

    work page 2009

  22. [22]

    Doedel E J, Krauskopf, B and Osinga, H M 2015 Global organ ization of phase space in the transition to chaos in the Lorenz system Nonlinearity 28(11) R113-R139

    work page 2015

  23. [23]

    Dullin H R and Wittek A 1995 Complete Poincar´ e sections and tangent sets J. Phys. A 28 7157-7180

    work page 1995

  24. [24]

    Edelstein-Keshet L 1988 Mathematical Models in Biology Birk¨ auser Mathematics Series (Boston: McGraw-Hill)

    work page 1988

  25. [25]

    Ermentrout G B and Terman D H 2010 Mathematical Foundations of Neuroscience Inter- disciplinary Applied Mathematics 35 (New York: Springer-Verlag) 24

    work page 2010

  26. [26]

    England J P, Krauskopf B and Osinga H M 2005 Bifurcations of stable sets in noninvertible planar maps Int. J. Bifurcation and Chaos 15(3) 891–904

    work page 2005

  27. [27]

    England J P, Krauskopf B and Osinga H M 2005 Computing one -dimensional global man- ifolds of Poincar´ e maps by continuation SIAM J. Appl. Dynam. Syst. 4(4) 1008–1041

    work page 2005

  28. [28]

    Galias Z 1997 Positive topological entropy of Chua’s ci rcuit: a computer assisted proof Int. J. Bifurcation and Chaos 7(2) 331–349

    work page 1997

  29. [29]

    Gonchenko S V, Meiss J D and Ovsyannikov I I 1006 Chaotic d ynamics of three-dimensional H´ enon maps that originate from a homoclinic bifurcation Regular and Chaotic Dynamics 11(2) (2006) 191–212

    work page 2006

  30. [30]

    Gonchenko S V, Ovsyannikov I I, Sim´ o C and Turaev D 2005 T hree-dimensional H´ enon-like map and wild Lorenz-like attractors Int. J. Bifurcation and Chaos 15(11) 3493–3508

    work page 2005

  31. [31]

    Gonchenko S V, Shilnikov L P and Turaev D 2009 On global bi furcations in three- dimensional diffeomorphisms leading to wild Lorenz-like att ractors Regular and Chaotic Dynamics 14(1) 137–147

    work page 2009

  32. [32]

    Guckenheimer J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems and Bi- furcations of Vector Fields (New York: Springer-Verlag)

    work page 1983

  33. [33]

    Hittmeyer S, Krauskopf B and Osinga H M 2013 Interacting global invariant sets in a planar map model of wild chaos SIAM J. Appl. Dynam. Syst. 12(3) 1280–1329

    work page 2013

  34. [34]

    Hittmeyer S, Krauskopf B, Osinga H M and Shinohara K 2018 Existence of blenders in a Hnon-like family: geometric insights from invariant mani fold computations Nonlinearity 31(10) R239-R267

    work page 2018

  35. [35]

    Hobbs C A and Osinga H M 2008 Bifurcations of the global st able set of a planar endo- morphism near a cusp singularity Int. J. Bifurcation and Chaos 18(8) 2207–2222

    work page 2008

  36. [36]

    Homburg A J and Sandstede B 2010 Homoclinic and heterocl inic bifurcations in vector fields Handbook of Dynamical Systems III ed B Hasselblatt, H W Broer and F Takens (North Holland: Elsevier) pp 379–524

    work page 2010

  37. [37]

    Jackson T and Radunskaya A 2015 Applications of Dynamical Systems in Biology and Medicine The IMA Volumes in Mathematics and its Applications 158 (New York: Springer- Verlag)

    work page 2015

  38. [38]

    Just W and Kantz H 2000, Some considerations on Poincar´ e maps for chaotic flows J. Phys. A 33 163-170

    work page 2000

  39. [39]

    Kostelich E J, Kan I, Grebogi C, Ott E and Yorke J A 1997 Uns table dimension variability: a source of nonhyperbolicity in chaotic systems Physica D 109(1–2) 81–90

    work page 1997

  40. [40]

    Krauskopf B and Osinga H M 2007 Computing invariant mani folds via the continuation of orbit segments; Numerical Continuation Methods for Dynamical Systems ed B Krauskopf, H M Osinga and J Gal´ an-Vioque (Springer Netherlands) pp 117 –154

    work page 2007

  41. [41]

    Krauskopf B, Osinga H M, Doedel E J, Henderson M E, Gucken heimer J, Vladimirsky A, Dellnitz M and Junge O 2005 A survey of methods for computing ( un)stable manifolds of vector fields Int. J. Bifurcation and Chaos 15(3) 763–791 25

    work page 2005

  42. [42]

    Krauskopf B, Osinga H M and Peckham P 2007 Unfolding the c usp-cusp bifurcation of planar endomorphisms SIAM J. Appl. Dynam. Syst. 6(2) 403–440

    work page 2007

  43. [43]

    Krauskopf B and Rieß T 2008 A Lin’s method approach to find ing and continuing hetero- clinic connections involving periodic orbits Nonlinearity 21(8) 1655–1690

    work page 2008

  44. [44]

    Kuznetsov Yu A 2004 Elements of Applied Bifurcation Theory 3rd edition (New York: Springer-Verlag)

    work page 2004

  45. [45]

    Lee C M, Collins P J, Krauskopf B and Osinga H M 2008 Tangen cy bifurcations of global Poincar´ e mapsSIAM J. Appl. Dynam. Syst. 7(3) 712–754

    work page 2008

  46. [46]

    Lin X-B 1990 Using Melnikovs method to solve Shilnikovs problems Proc. R. Soc. Edinb. A 116(3–4) 295–325

    work page 1990

  47. [47]

    Oldeman B E, Champneys A R and Krauskopf B 2003 Homoclini c branch switching: a numerical implementation of Lin’s method. Int. J. Bifurcation and Chaos 13(10) 2977–2999

    work page 2003

  48. [48]

    Palis J 1991 Homoclinic bifurcations, sensitive-chao tic dynamics and strange attractors; Dynamical systems and related topics (Nagoya, 1990) Advanced Series in Dynamical Sys- tems 9 ed K Shiraiwa (Singapore: World Scientific Publishing) pp 46 6–473

    work page 1991

  49. [49]

    Palis J and de Melo W 1982 Geometric Theory of Dynamical Systems (New York: Springer- Verlag)

    work page 1982

  50. [50]

    Palis J and Takens F 1993 Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Cambridge: Cambridge University Press)

    work page 1993

  51. [51]

    Rademacher J D M 2005 Homoclinic orbits near heteroclin ic cycles with one equilibrium and one periodic orbit J. Diff. Eqns. 218(2) 390–443

    work page 2005

  52. [52]

    Rademacher J D M 2010 Lyapunov-Schmidt reduction for un folding heteroclinic networks of equilibria and periodic orbits with tangencies J. Diff. Eqns. 249(2) 305–348

    work page 2010

  53. [53]

    Plasmas 10(9) 3796–3799

    Roeder R K W, Rapoport B I and Evans T E 2003 Explicit calcu lations of homoclinic tangles in tokamaks Phys. Plasmas 10(9) 3796–3799

    work page 2003

  54. [54]

    Smale S 1967 Differentiable dynamical systems Bull. Amer. Math. Soc. 73 747–817

    work page 1967

  55. [55]

    Turaev D and Shilnikov L P 1998 An example of a wild strang e attractor Sbornik: Mathe- matics 189(2) 291–314

    work page 1998

  56. [56]

    Viana R L, Da Silva E C, Kroetz T, Caldas I L, Roberto M and S anju´ an M A F 2011 Fractal structures in nonlinear plasma physics Philos. Trans. Royal Soc. A 369(1935) 371–395

    work page 2011

  57. [57]

    Zhang W B 2006 Discrete Dynamical Systems, Bifurcations and Chaos in Econo mics Math- ematics in Science and Engineering 204 (North Holland: Elsevier)

    work page 2006

  58. [58]

    Zhang W, Krauskopf B and Kirk V 2012 How to find a codimensi on-one heteroclinic cycle between two periodic orbits Discrete Contin. Dyn. Syst. Ser. A 32(8) 2825–2851 26

    work page 2012