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1. Dynamic System examples

1. Fish in tank without predators and not breeding

Rate at which fish die is dependent on how many fish are in the tank and whether the tank’s carrying capacity has been exceeded. Basically if the rate of waste production is greater than the tank can sustain. For instance I got bad advice on my fishtank, put around 30 fish into the tank and half died overnight. After that the fish died off at a more regular rate and the last one has survived much longer that the others.

2. Tides

The interaction of the oceans on earth and the gravitational force exerted on those oceans causes the tides. The height of the tide is dependent on how close that particular part of the earth is to the moon. Throughout the day as the earth spins the height of the tides then changes all over as different parts of the earth move closer to and further away from the moon.

2. http://scihi.org/edward-lorenz-chaos-theory/

I just find the fact that such small changes in the initial conditions of a system can mean that it becomes impossible to predict events even past 3 days, even if you have a very accurate set of equations to describe the system. I also found it interesting that Lorenz’s discovery of this was made by accident, but ended up being such an important discovery.

4. What I find incredible is how accurate the Chinese astronomers were without the use of modern technology and long before many other civilizations were able to do anything near what they managed. For instance the fact that they determined the length of the year to within 30 seconds of what we know its length to be today.

Reply from Jonathan: I'm sorry to hear about your fish! Nice examples!

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During Strogatz's lecture he mentioned the problem of turbulence. I looked it up and it turns out that the equations governing fluid motion (Navier-Stokes equations) are non-linear and can also be chaotic which means that solutions to these equations have not been found for all possible initial conditions. This means that its actually really difficult to precisely answer questions like "At what velocity will a fluid flowing through a pipe transition from laminar to turbulent flow?" and "When will a wave transition from smoothly propagating on the surface of the ocean to breaking?" for all possible initial conditions. I think this makes fluid flow a cool example of a dynamical system. (Also interestingly there is a $1,000,000 prize for proving or disproving that there exists a unique, well-behaved solution to the Navier-Stokes equations for all initial conditions.) I'm pretty sure I butchered some of the explanations so here is an article and a YouTube video that I found when looking into this problem:

https://arstechnica.com/science/2018/10/turbulence-the-oldest-unsolved-problem-in-physics/

https://www.youtube.com/watch?v=ERBVFcutl3M

Reply from Jonathan: That Numberphile video is a very good one. Indeed there are other systems, like the double pendulum, which are chaotic, but we know that solutions exist. We will look at existence and uniqueness of solutions in a little while in the course. It's, as you say, the existence of solutions to the Navier-Stokes equations which we don't currently know how to prove.

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1. I found a really interesting video online that simulates logistic growth and competition in a population which I've linked below and highly recommend. It's very easy to keep up with and covers a lot of what we've done in class. The channel also has videos on simulations of epidemics as well as evolution which are all interesting dynamical systems.

3. What I found interesting about 3 body systems is the special cases where it is solvable such as when one body can be considered infinitely small. The Lagrangian case, where the three bodies form an equilateral triangle and the Eulerian case, where two of the bodies are motionless.

https://www.youtube.com/watch?v=uRTtlpD_U54

Reply from Jonathan: Nice, this looks like a great series, thank you!

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A system i assume would be dynamic would be the spread of COVID-19 over time. It is dependent on the "migration" of the population and its size.
What I find perplexing is how accurate/close to accurate their observations and calculations where. It brings it to dawn how simple and boxed we are in our times.

Reply from Jonathan: Yes, spot on. There are actually a lot of different ways of modelling the spread of disease. This paper is one particular example of it: https://www.frontiersin.org/articles/10.3389/fphy.2020.601459/full

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I found it really interesting that chaos is still quite a new science and that it had gone unnoticed for so long. I also found out that the name 'Laplace's demon' is given to an imaginary character that knows the position and momenta of every single particle in the universe. It was believed that this character would be able determine the future precisely, just through newtonian mechanics. I also read up on a Persian mathematician, al-khwarizmi, who had studied the motion of the sun and the planets (they knew about 5 at the time) in 830. He is also often described as the father of algebra as he introduced methods of 'reduction' and 'balancing', core principles of algebra. He demonstrated how to  complete the square, using geometric arguments. Chinese astronomy also dates back to 3000BC.

Reply from Jonathan: Nice work! In a similar vein, you might want to look up Maxwell's demon.

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i'm not really sure if this counts as one; but wouldn't how humans interact with the environment be an example? Over time how the human interacts with their environment has drastically changed and will continue to change till the end of time ( to our detriment ?). I find this particularly interesting, if it does count as an example.

Reply from Jonathan: Yes, absolutely it counts. With an example like this, one has to think very carefully about exactly what is changing in time, and what one can measure. Presumably if you were to model this, you wouldn't be able to model the individual actions of all humans, so you would have to average over populations of humans. Even there, there will be many things which would be hard to model: The influence of politics, of media, etc. This doesn't make it a bad example at all. It's a great example, but definitely a hard one to think about precisely how you would go about doing it.

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A not so unique example of a dynamical system is that of the predator-prey model. But a particular example, or application, of this model that I find interesting is that of sea otters and sea urchins. It might not seem particularly amazing, but otters are incredibly important in maintaining the population of sea urchins. The California sea otter was thought to be extinct a few years ago, but a small raft of them miraculously managed to survive. The otters were declared a protected species and later played a crucial role in balancing the population of sea urchins — which in the otters' absence was incredibly high.
Now the application of this story to dynamical systems are the differential equations that are used to model and monitor otter and urchin populations.

Chinese astronomy:

Something I find particularly interesting is how Chinese astronomy varies from Indo-European astronomy.

A circumpolar star is a star that, as viewed from a given latitude on Earth, never sets below the horizon due to its apparent proximity to one of the celestial poles (from occasionally untrustworthy Wikipedia). I find it fascinating that even though the same celestial bodies were observed and studied, the methods in which this was achieved differ quite a lot. The Indo-European method was to base their observations on the rising and setting of celestial bodies on the horizon, whereas the Chinese used circumpolar stars as their reference point.

Reply from Jonathan: Fascinating information, thank you!

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In reading up about the 3-body problem I found this simulator software which is really cool to play around with.
It's somewhat related to chaos in gravitational systems since one feature allows you to speed up time in a solar system and see how the orbits of planets change over time.

Reply from Jonathan: This looks stunning, thank you!

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SpaceEngine is a realistic virtual Universe you can explore on your computer. You can travel from star to star, from galaxy to galaxy, landing on any planet, moon, or asteroid with the ability to explore its alien landscape. You can alter the speed of time and observe any celestial phenomena you please. All transitions are completely seamless, and this virtual universe has a size of billions of light-years across and contains trillions upon trillions of planetary systems. The procedural generation is based on real scientific knowledge, so SpaceEngine depicts the universe the way it is thought to be by modern science. Real celestial objects are also present if you want to visit them, including the planets and moons of our Solar system, thousands of nearby stars with newly discovered exoplanets, and thousands of galaxies that are currently known.

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I think the popularity of a certain entity in relation to a population over time is a good example. For instance, the popularity of a certain word in language, the usage of a certain product, or just the general popularity of some entity in relation to a particular population and how it changes over time.

I'm not exactly sure how this would be modeled but a model similar to the logistic map which increases then decreases over time depending on certain variables might be a good way to model it.

Reply from Jonathan: Nice example. Indeed the issue is always what are the important factors and which can be ignored. Clearly we can't take into account every interaction with every human, so one might think about having some abstracted set of variables, where you average over the smaller interactions to give you the big-scale picture.

Simon Roper has a very interesting Youtube channel where he looks at the evolution of language, particularly within Europe: https://www.youtube.com/channel/UChnRk6mxWsSOGElm8phdSxw/videos

There is also a fascinating (though slightly outdated) book called Genes, Peoples and Languages by Sforza which looks at the relationships between languages over time.

Here is a paper looking at the mathematical modelling of memes: https://www.researchgate.net/publication/259922083_Mathematical_Models_of_Fads_Explain_the_Temporal_Dynamics_of_Internet_Memes