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I'll soon be teaching a (basic) course on mathematical control theory. The first part of the course will focus on mathematical modelling of dynamical systems. More precisely, I will present examples of mathematical models of dynamical systems encountered in engineering, physics, biology, etc., and discuss some of their basic properties (e.g., existence of equilibria, stability, etc.)

Now, examples that are commonly used include: damper-spring-mass system, RLC circuits, pendulum, etc. However, I would like to come up with something different. Thus, I was thinking about models used to describe unconventional/unusual dynamical phenomena.

Examples of "unconventional" modelling that came to my mind are, for instance, dynamical models of love or hate displayed by individuals in a romantic relationship (see, e.g., this paper). However, I'm curious to hear about more unconventional examples. Thanks in advance for your input!

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  • $\begingroup$ Would you consider the Ising model (describing phase transitions) as an unconventional example? $\endgroup$
    – Konstantinos Kanakoglou
    Commented Aug 16, 2019 at 0:52
  • $\begingroup$ @KonstantinosKanakoglou: Yes, it could fit. But I’d be more interested in dynamical models that are easy to explain, as I have to present them to students that do not have a solid physics/math background. $\endgroup$
    – Ludwig
    Commented Aug 16, 2019 at 1:51
  • $\begingroup$ When you say biology are you also thinking about population dynamics? $\endgroup$
    – user347489
    Commented Aug 16, 2019 at 8:58
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    $\begingroup$ Chemical oscillators are very interesting. $\endgroup$
    – Rodrigo de Azevedo
    Commented Aug 16, 2019 at 10:29
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    $\begingroup$ @KonstantinosKanakoglou On the contrary, I would say that the Ising model is -- together with percolation -- the most conventional model in statistical mechanics $\endgroup$
    – Jules Lamers
    Commented Aug 17, 2019 at 0:50

15 Answers 15

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Economics offers many examples of simple models for various systems not based on physical principles. One nice example is Hotelling's line model, which gives rise to Hotelling's "law". It considers two firms deciding where to locate their business to capture the most customers along a stretch of road.

Considering such models over a period of sequential decision making yields a dynamics and it is most common to study equilibria solutions.From Wikipedia:

Another example of the law in action is that of two takeaway food pushcarts, one at each end of a beach. If there is an equal distribution of rational consumers along the beach, each pushcart will get half the customers, divided by an invisible line equidistant from the carts. But, each pushcart owner will be tempted to push his cart slightly towards the other, moving the invisible line so that the owner is on the side with more than half the beach. Eventually, the pushcart operators will end up next to each other in the center of the beach.

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    $\begingroup$ This also comes up in the Median Voter Theorem, based on the same model by Hotelling: en.wikipedia.org/wiki/Median_voter_theorem#History $\endgroup$
    – user44143
    Commented Aug 15, 2019 at 7:24
  • $\begingroup$ If the Wikipedia article is accurate, Hotelling explicitly disclaimed considering any variables but location. This makes the model nearly useless. Further, it suggests that each will move slightly closer to the center. But if indeed moving cost is irrelevant, wouldn’t the first to move go a lot further than “slightly”? $\endgroup$
    – WGroleau
    Commented Aug 15, 2019 at 17:26
  • $\begingroup$ @WGroleau perhaps an interesting idea would be to include a certain decay of customer attraction with distance. The equilibrium in that case would not be for the two vendors to be next to each other. $\endgroup$
    – Kai
    Commented Aug 16, 2019 at 14:54
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The book

Enns, Richard H., It’s a nonlinear world., Springer Undergraduate Texts in Mathematics and Technology. New York, NY: Springer (ISBN 978-0-387-75338-6/hbk; 978-0-387-75340-9/ebook). xii, 383 p. (2011). ZBL1214.00007.

contains many nice examples, also for models of sports (Ch. 6) and war (Ch. 11).

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    $\begingroup$ Hi, can you make more detailed references? which models would fit the quetion best? $\endgroup$
    – András Bátkai
    Commented Aug 15, 2019 at 11:20
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    $\begingroup$ Ah yes, Chapters 6 and 11 contain many unconventional examples from the areas of sports and war. $\endgroup$
    – Bastian Seifert
    Commented Aug 15, 2019 at 11:30
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    $\begingroup$ Can you please give a sentence or two about one or two such examples you find the most telling? (for those of us who don't have quick access to the book) $\endgroup$
    – Mitch
    Commented Aug 17, 2019 at 15:39
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My favorite example is the use of hidden Markov models to analyze defensive strategies in basketball, as explained in Characterizing the spatial structure of defensive skill in professional basketball, by Alexander Franks, Andrew Miller, Luke Bornn, and Kirk Goldsberry, Ann. Appl. Stat. Volume 9, Number 1 (2015), 94–121. Making various modeling assumptions (e.g., that the defenders are employing a man-to-man defensive strategy), they use the EM algorithm on some optical tracking data from actual basketball games to infer who is guarding whom.

Note: I learned about this research from an article in one of the volumes of What's Happening in the Mathematical Sciences.

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  • $\begingroup$ The optical tracking data itself is particularly impressive...tracking ball and player locations over time at pretty high resolution makes this analysis possible. $\endgroup$
    – R Hahn
    Commented Aug 17, 2019 at 20:13
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It appears that there are applications of dynamical systems to music. See, e.g., Rick Bidlack, Music from chaos: nonlinear dynamical systems as generators of musical materials or Boon and Decroly, Dynamical systems theory for music dynamics or David Burrows, A Dynamical Systems Perspective on Music or just type

dynamical systems in music

into Google, as I did, and see what comes back at you.

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    $\begingroup$ Or type $$\rm dynamical\ systems\ in\ baseball$$ or $$\rm dynamical\ systems\ in\ winemaking$$ or whatever. $\endgroup$
    – Gerry Myerson
    Commented Aug 15, 2019 at 6:58
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    $\begingroup$ When I Google dynamical systems in winemaking, I get one result which is on-topic but too difficult...and then one nonsense hit ("leaders of standard download dynamical systems released a 'education opinion ' in which the energy-related club..."), multiple hits on dynamic systems which are not dynamical systems in the mathematical sense, one wikipedia article which lacks the word "winemaking", and one mathematical festschrift in which the reference to wine seems to be an inside joke. It may all be amusing, but as a pedagogical suggestion, it fails. $\endgroup$
    – user44143
    Commented Aug 15, 2019 at 22:24
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    $\begingroup$ @Matt, I get Two modelling approaches of winemaking: first principle and metabolic engineering, tandfonline.com/doi/abs/10.1080/13873954.2010.514701 and "We deal with an application of partial differential equations to the correct definition of a wine cellar," tandfonline.com/doi/abs/10.1080/0020739x.2017.1396626 and Dynamical modeling of alcoholic fermentation and its link with nitrogen consumption, sciencedirect.com/science/article/pii/S1474667016304049 and... $\endgroup$
    – Gerry Myerson
    Commented Aug 15, 2019 at 23:30
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    $\begingroup$ ...Complexity and Dynamics of the Winemaking Bacterial Communities, ncbi.nlm.nih.gov/pmc/articles/PMC4907434 $\endgroup$
    – Gerry Myerson
    Commented Aug 15, 2019 at 23:31
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    $\begingroup$ In that last example, “Complexity and Dynamics of the Winemaking Bacterial Communities”, do you see any differential equations? Or anything that one could call “mathematical control theory”, as in the question? I do not. $\endgroup$
    – user44143
    Commented Aug 16, 2019 at 2:00
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There are a lot of applications of ideas in dynamical systems to social media for things like event detection and forecasting. Some of the literature has to be taken with a grain of salt because of the difficulties in gathering data, but this paper looks nice:

https://arxiv.org/pdf/1603.00074.pdf

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Although it's presented as a game, the fox-and-duck problem may be cast as a control problem from the perspective of the fox. The aim of the fox is to catch the duck, while the duck conversely tries to escape the fox. The duck can swim in any direction on a circular pond with speed $v$. The fox, who cannot swim, can run around the pond with speed $\lambda v$ (or less). The duck cannot take off from water but can immediately fly, and escape the fox, as soon as it lands at a point at the edge of the pond that the fox cannot reach in time. Initially the duck is at the centre of the pond.

For $\lambda=4$, the duck has an escape strategy. However, for a slightly larger value of $\lambda$, the fox can “control” the duck to stay within the bounds of the pond.

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    $\begingroup$ There is a cool numberphile video and simulation about this game. $\endgroup$
    – Tadashi
    Commented Aug 20, 2019 at 0:43
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You mention the motion of a pendulum as conventional. Perhaps extending to a chaotic double pendulum would serve as a useful contrast:

          Pendulum-2
          Images from minutelabs.
          ChaoticPend

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There are papers studying a zombie outbreak using infectious disease modelling (an SIR continuous-time Markov model): see e.g.

P. Munz et al, When zombies attack!: Mathematical modelling of an outbreak of zombie infection, Infectious Disease Modelling Research Progress, 133-150 (2009)

and the review at https://jgeekstudies.org/2015/05/18/zombie-model/

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Martin Gardner's Lady in the lake problem should serve as an illustrative example of a differential game of evasion.

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I suggest models in the social sciences like the Krause-Hegselmann model of consensus dynamics. Also the Kuramoto model for synchronizing oscillators is nice and can be used to model other things e.g. the synchronization of fireflies.

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Fellow teacher on mathematical control theory here. If you need some interesting examples of discrete dynamical systems, I'd recommend this article from IEEE Control System Magazine.

Discrete-time models arise from the time sampling of measurements made on such systems, usually via an analog-to-digital converter, for the purpose of digital computation. Granted, even with the plethora of example applications that textbooks and instructors can offer, students are sometimes unconvinced that applicability in the real world is worth the significant effort required to master the requisite concepts and methods. Perhaps not as commonly discussed are example applications that represent abstractions of the man-made, nonnatural world. In such a setting, the time index might not even represent “physical” time. Examples of these abstraction models come from topics such as multiagent systems, manufacturing, the Internet, computer systems, logistics, social networks, supply chains, project management, and finance.

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Depending on the type of course you are teaching you might be interested in machine learning models (in terms of the meta variable heuristic optimizations) or more generally, you might want to consider well known industrial cases like chemical kinetics in still-pots for perfumery. Then again you can make things arbitrarily complex and talk about multi-scale models for atomic simulations up to finite element simulations based on the Navier Stokes, i.e. going from Lennard Jones fluids right up to the Transport Equations (Heat, mass and fluid transfer).

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    $\begingroup$ Can you give good refeences for these topics? $\endgroup$
    – András Bátkai
    Commented Aug 16, 2019 at 7:43
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A control systems lecturer at my university has done some interesting work with mathematical modelling on when to develop treatment strategies to reduce the disease burden in a population. Very interesting work.

Jeffrey, A.M., Xia, X., and Craig, I.K., When to Initiate HIV Therapy: A Control Theoretic Approach, IEEE Transactions on Biomedical Engineering, Vol.50, No. 11, 2003, pp. 1213- 1220.

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John Sterman has many mathematical models as dynamical system of business processes such as supply chain and productivity. These are just two. I recommend perusing his publications for others.

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An interesting class of unconventional, unusual, non standard dynamical phenomena are the Canards (from the French canard = duck) that occur in slow-fast, stiff, singularly perturbated dynamical systems. They are properly modelled using non standard analysis.

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