# Chaos theory and population cycles of odd periodicity

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?

• 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. 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 ? Feb 12 '14 at 18:27
• @Chris It will become clearer to you what exactly am asking after reading this Feb 12 '14 at 18:28

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).