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The logistic function solves the logistic ODE which is the continuous version of the logistic map.

However, I was not able to find why any of these things are called "logistic".

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Doesn't the first link you supply answer the question? "A logistic function or logistic curve is a common sigmoid function, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth." –  EuYu Apr 11 '13 at 4:25
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The term is very old, and has been used, for example, to mean arithmetic. Verhulst introduced the term for the curve that is often used to model constrained growth. I believe it is not known why. –  André Nicolas Apr 11 '13 at 4:25
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@EuYu The fact that the name was "given by Verhulst" does not explain why it was given. Usually people pick names for a reason, but maybe Verhulst was an exception. –  Max Apr 11 '13 at 4:45
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From what I can tell, Verhulst wrote in French. I'm guessing that "logistic" here comes from logistique in French which in turn derives from "to lodge" or "to house". The curve originally arose from the study of population growths so this seems like a reasonable interpretation to me. I would probably ask a linguist to be sure. –  EuYu Apr 11 '13 at 4:56
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Ok, I looked at the French wikipedia, which states "Le nom de courbe logistique leur a été donné par Verhulst sans que l'on sache exactement pourquoi." - "The name "logistic curve" was given to it by Verhulst, but no one knows exactly why". The reference $href=rasch.org/rmt/rmt64k.htm$ gives the following commentary: –  Max Apr 11 '13 at 5:26

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Cite max

The fact that the name was "given by Verhulst" does not explain why it was given. Usually people pick names for a reason, but maybe Verhulst was an exception. Ok, I looked at the French wikipedia, which states "Le nom de courbe logistique leur a été donné par Verhulst sans que l'on sache exactement pourquoi." - "The name "logistic curve" was given to it by Verhulst, but no one knows exactly why". The reference rasch.org gives the following commentary:

"Verhulst writes "We will give the name logistic [logistique] to the curve" (1845 p.8). Though he does not explain this choice, there is a connection with the logarithmic basis of the function. Logarithm was coined by John Napier (1550-1617) from Greek logos (ratio, proportion, reckoning) and arithmos (number). Logistic comes from the Greek logistikos (computational). In the 1700's, logarithmic and logistic were synonymous. Since computation is needed to predict the supplies an army requires, logistics has come to be also used for the movement and supply of troops. So it appears the other meaning of "logistics" comes from the same logic as Verhulst terminology, but is independent (?). Verhulst paper is accessible; the definition is on page 8 (page 21 in the volume), and the picture is after the article (page 54 in the volume). "

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