Unconventional examples of mathematical modelling 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!
 A: 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. 
A: 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.
A: 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
A: 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.
A: You mention the motion of a pendulum as conventional. Perhaps extending to
a chaotic double pendulum would serve as a useful contrast:

          


          

Images from minutelabs.


         


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

A: 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.

A: 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).
A: 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).
A: 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. 
A: 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.
A: 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.
