Self-portrait in the Blue Mountains

David C Groothuizen Dijkema

I am a doctoral candidate in mathematics at the University of Auckland, under the supervision of Professors Claire Postlethwaite and Vivien Kirk. I am interested in the mathematical theory of dynamical systems, and my thesis is about dynamics near structures known as heteroclinic networks.

My email: david.groothuizen.dijkema@auckland.ac.nz

My CV.

University of Auckland profile. GitHub. arXiv. MathSciNet. ORCiD.

Minisymposium on homoclinic and heteroclinic structures at SIAM DS 25

Together with Alexander Lohse at Universität Hamburg, I am organising a minisymposium at the 2025 SIAM Conference on Applications of Dynamical Systems (DS25) in Denver, CO, USA, May 11-15. The topic of our minisymposium is Dynamics near Homoclinic and Heteroclinic Structures:
Homoclinic and heteroclinic structures are commonly found in a variety of discrete and continuous dynamical systems, from homoclinic tangencies and butterflies to heterodimensional cycles and heteroclinic networks. These structures can consist of equilibria, periodic orbits, or even chaotic sets together with the non-empty intersections of their respective stable and unstable manifolds. Bifurcations involving them can lead to chaotic sets, and they frequently have intricate stability properties unlike other structures in dynamical systems. Such systems are of interest within the study of dynamical systems theory, but also have a range of applications. For example, heteroclinic cycles and networks are invoked as models of intransitive relations and intermittent phenomena, and are applied to population dynamics, mathematical neuroscience, as well as economic and game theory. Systems containing these structures are also studied with the addition of noise. Rich dynamics are often observed, typically requiring nuanced analysis and numerical computations. This minisymposium will bring together researchers to showcase recent results concerning homoclinic and heteroclinic structures, from both theoretical and applied perspectives, using a range of analytic techniques.

Publications and Preprints

  1. David C Groothuizen Dijkema, Vivien Kirk, Claire M Postlethwaite, and Alastair M Rucklidge. Continuity of projected maps near heteroclinic networks in ℝ4. In preparation.
  2. David C Groothuizen Dijkema, Vivien Kirk, and Claire M Postlethwaite. (2024). Analysis of dynamics near heteroclinic networks in ℝ4 with a projected map. Submitted to SIAM Journal on Applied Dynamical Systems. arXiv: 2410.21486 [math.DS]
  3. David C Groothuizen Dijkema and Claire M Postlethwaite. (2023). Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition. Nonlinearity, 36(12):6546—6588. DOI: 10.1088/1361-6544/ad0212. arXiv: 2208.05630 [math.DS]

Talks and Posters

Talks

  1. 3 February 2025: Analysing dynamics near heteroclinic networks with a projected map, Australia and New Zealand Industrial and Applied Mathematics Conference, Coffs Harbour, NSW, Australia. (Received an honourable mention for the ANZIAM T M Cherry Prize for Best Student Talk.)
  2. 10 December 2024: Analysing dynamics near heteroclinic networks with a projected map, Joint Meeting of the NZMS, AustMS, and AMS, University of Auckland, NZ. (Received the NZMS Aitken Prize for Best Student Talk.)
  3. 5 December 2024: Travelling waves in spatially-extended models of cyclic competition, Workshop on Pattern Formation, University of Auckland, NZ.
  4. 17 October 2024: Dynamics near heteroclinic networks, Applied Mathematics Unit Doctoral Talks, University of Auckland, NZ.
  5. 13 February 2024: Switching and cycling near heteroclinic networks as a piecewise-smooth dynamical system, Australia and New Zealand Industrial and Applied Mathematics Conference, Hahndorf, SA, Australia.
  6. 1 December 2023: Switching and cycling near heteroclinic networks as a piecewise-smooth dynamical system, New Zealand Mathematical Society (NZMS) Colloquium, Victoria University of Wellington, NZ. (Received an honourable mention for the NZMS Aitken Award for Best Student Talk.)
  7. 29 November 2023: Travelling waves and heteroclinic networks in spatially-extended models of cyclic competition, New Zealand Mathematics and Statistics Postgraduate Conference, Victoria University of Wellington, NZ.
  8. 27 June 2023: Dynamics near heteroclinic cycles and networks, Leeds Applied Nonlinear Dynamics Seminar, Department of Mathematics, University of Leeds, UK.
  9. 23 June 2023: Dynamics near heteroclinic cycles and networks, Centro de Matemática da Universidade do Porto Seminar, Universidade do Porto, Portugal.
  10. 12 May 2023: Switching near heteroclinic networks as a piecewise-smooth dynamical system, SIAM Conference on Applications of Dynamical Systems, Portland, OR, USA.
  11. 14 November 2022: Switching near heteroclinic networks as a piecewise-smooth system, Sydney Dynamics Group Workshop: Dynamical Systems in NZ - Castaways, Auckland, NZ.
  12. 6 October 2022: Switching near heteroclinic networks as a piecewise-smooth system, Applied Mathematics Unit Doctoral Talks, Department of Mathematics, University of Auckland, NZ.
  13. 7 June 2022: Stability and cycling near heteroclinic cycles and networks, Student Research Conference, Department of Mathematics, University of Auckland, NZ.
  14. 1 June 2022: Stability and cycling near heteroclinic cycles and networks, Postgraduate Student Seminar, Department of Mathematics, University of Auckland, NZ.
  15. 21 July 2021: Spatially extended models of competition between four species, Postgraduate Mid-year Talks, Department of Mathematics, University of Auckland, NZ. (Received one of four prizes for best talk.)
  16. 8 June 2021: A spatially-extended model of competition between four species, Student Research Conference, Department of Mathematics, University of Auckland, NZ.
  17. 4 February 2021: Heteroclinic networks in partial differential equations, Summer Research Students Conference, Department of Mathematics, University of Auckland, NZ.

Here is the animation that accompanied most of the above talks:

This animation is an example of a trajectory near the Guckenheimer-Holmes cycle, the prototypical example of a heteroclinic cycle. It was first considered by May and Leonard (though not using the terminology of heteroclinic cycles) in their study of nonlinear competition between three species. Each of the three coordinates represents the density of each of three species which compete with each other in a cyclic manner, just like the game of Rock-Paper-Scissors. In their words,

…[A] nonlinear study of this domain of parameter space reveals … a wider class of asymptotic solutions in which the system cycles from being composed almost wholly of population 1, to almost wholly 2, to almost wholly 3, back to 1, but with the time to complete the cycle becoming longer and longer (being proportional to the length of time the system has been running), and with the system coming in turn ever closer to the points with 1 alone, 2 alone and 3 alone, yet never actually converging on any one point.

May R and Leonard L 1975 SIAM J. Appl. Math. 29 243

Posters

  1. 4 December 2023: Travelling waves and heteroclinic networks in spatially-extended models of cyclic competition, New Zealand Mathematical Society (NZMS) Colloquium, Victoria University of Wellington, NZ. (Received the ANZIAM Award for Best Poster Presentation.)
  2. 13 October 2022: Switching near heteroclinic networks as a piecewise-smooth dynamical system, Antipodeal Dynamics Workshop, Centre for Systems, Dynamics and Control, Department of Mathematics, University of Exeter, UK and the Department of Mathematics, University of Auckland, NZ.

Cryptic Crossword

As part of the Joint Meeting of the NZMS, AustMS, and AMS I compiled a themed cryptic crossword with the help of two other members of the Department of Mathematics, Jonny Stephenson and Đorđe Mitrović. The crossword is here. The solution is here.

My surname

My surname is Dutch, and maximally unpronounceable. The way I say it in English is [ɡɹʉːθ'hɐʉ̯ˌzɪn dæɪ̯'kɪˌmɐ]. The correct way to say it in Dutch, however, is [ˈɣɾoːt.ɦœy̯.zə(n) ˈdɛi̯.kə.mɑ]. In the dialect where my family is from, it is said [ˈχɾɔu̯t.hœː.sə ˈdɛː.kə.mɑ].

If you don't know IPA, then the way to say my name in English is "grooth-HOW-zin day-KI-mə".

design shamelessly stolen from Alex Elzenaar.