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I have just begun to read about Chaos theory and have come across the statement that "Period three implies chaos."

My question: Does any odd period imply chaos or only 3? If so, how can populations of cicadas that cycle every 17 years exist?

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Can you give a bit more background? What period 3? –  Chris Feb 12 '14 at 18:20
@chris Period 3 means a cycle of 3 years. –  biogirl Feb 12 '14 at 18:23
Ahh, now. Lifecycles based in the number 3 overlap with several others - 6, 9, 12, yearly, and so on. If the lifecycle is only 17 years, then a parasite needs to have exactly the same lifecycle (its a primer number so there are no other fractions) to meet its host. –  Chris Feb 12 '14 at 18:25
@Chris Ummm..I have read that before and I am not trying to ask the reason why 17 year cycle are beneficial, i am asking how can they EXIST in the first place ? –  biogirl Feb 12 '14 at 18:27
@Chris It will become clearer to you what exactly am asking after reading this –  biogirl Feb 12 '14 at 18:28

1 Answer 1

up vote 4 down vote accepted

The chaotic behaviour you are referring to (at least the one described in your link in the comments) is a property of the discrete version of the logistic equation, where you get chaotic dynamics at growth rates above ~3.55 (see the logistic map). The behaviour of this equation has been described in a classic paper by Robert May (1976). As you increase growth rate (a in your link) from one you go from a stable attractor (approached directly or by dampened occilations) to cycling behaviour (between 2 to 4 to 8 states as growth rate increase) to chaotic dynamics, which is shown in this bifurcation diagram.

enter image description here

And to be clear, nothing of this has anything to do with period/cycle length per se (in the sense of species interactions or lagging feedback loops), but is a property of the model and the population growth rate. The statement about period 3 implying chaos probably refers to a brief window around r ~3.83 ($1+\sqrt{8})$ where you get cycling between three values, and at higher values then this you only find chaotic behaviour (which you get at lower values as well though).

Addition: I now realize that the statement "Period three implies chaos" comes from the theoretical paper with the same name (Li & Yorke, 1975). This paper proves that all one-dimensional models (not only the equation above) that has a period 3 cycle will also show chaotic behaviour. This proof is a special case of the Sharkovsky's theorem, which is older but was unknown to Li & Yorke at the time.

There is also a question at MathSE that deals with the same problem, and some of the answers there link to useful resourses:

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