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I am looking for any publicly available dataset that has a "cyclical" structure to it, in the sense that if I plot the data in a certain way, a loop becomes visible. A good example of this would be the Lotka-Volterra predator-prey model, which has a very pronounced cycle, shown below. Are there any other good examples that demonstrate this?
The search term you are looking for is limit cycle. You can find some examples (legged-locomotion, digital control systems, etc.) here: https://www.sciencedirect.com/topics/engineering/limit-cycle.
