Abstract

Citation

Researchers should cite this work as follows:

• Goldwyn, E. (2022). 5-015-RunnersSynchronize-ModelingScenario. SIMIODE, QUBES Educational Resources. doi:10.25334/TF3Z-0556

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Article Context

Description

The idea is that instead of describing these oscillatory systems with several differential equations (one for each population or physical quantity), we can reduce the system to one differential equation that describes the phase of oscillation.

In this scenario we start with a toy model in which an oscillator moves around the unit circle at a constant speed.

Then we add a second oscillator and examine a system where the oscillators attract (or repel) each other (we use the ``Kuramoto Model").

Finally, we extend this concept to a more complicated predator-prey oscillator.

By only looking at the difference in phase between these two oscillators, we reduce what is usually modeled with four differential equations to one differential equation which can then be studied with traditional equilibrium techniques. The goal is for students to practice finding and classifying equilibria of a one-variable system in this modeling context while also introducing variables defined on a restricted domain (the circle).

This scenario also introduces bifurcations as the phase model described undergoes a saddle node bifurcation as the difference in frequencies between the two oscillators is increased. This scenario should be appropriate immediately after students have been introduced to finding and classifying equilibria of one-variable autonomous differential equations.