## Dynamical Systems

In Jr. High School I was in a musical chorus group called “Dynamics.” We were the auditioned choir and we were supposed to be good at vocal dynamics- altering our vocal volume over time. Now, I can’t say that we were especially good at fulfilling this portion of our singing obligation, but we did have a very dynamic group in another sense. We were constantly changing and evolving over time. Our vocal coach became pregnant, the boys’ voices changed, and we were all wild teenagers.

Mathematicians like to study the dynamics of a system. So in retrospect, I’m inspired to consider this punny case of a “dynamic” system. For this system I’m going to study the energy level of the students over time. One of the first question mathematics like to ask about dynamics is: are there cycles to the energy level of the group? For this system, I would expect a periodic flow in energy. I would guess that every year looks something like the graph below.

The next logical question is; why does the graph look like this? Immediately I consider when school breaks happen. Every December we had winter break, so it’s logical that we were excited before the break. Similarly in June there is a rise in energy before summer break. This is an example of how External Forcing can effect a system. Can you think of other external forcings in a school year? Would it help if I told you my choir was in charge of distributing singing Valentine’s?

The last basic principle of dynamic systems is about the stability of a system. How will the overall energy levels change if a minor external effect altered the group. For example, how much of an effect did my choral director’s pregnancy have? She told she was pregnant in October and the baby was due in June.

The news had an immediate effect on the system and a delayed effect on the system during the year of the perturbation. However, do you think you would see a graph that looks more like the first or the second during the year after the baby was born? I would bet that the graph for the year after the pregnancy looks more like the original graph. This would mean that the external forcings of school breaks are much stronger than the perturbation of pregnancy and that the system is pretty stable.

But then again, “Energy” isn’t a very quantitative measure. However, even through the rose colored glasses of adulthood, I think nothing is quite as exciting as vacation and the vacation caused dynamics were certainly more exciting that our vocal dynamics.

## About Samantha from SocialMath

Applied Mathematician and writer of socialmathematics.net.
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