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I have a question concerning chaos. As you may well know, the logistic mapping $$x_{n+1} = rx_n (1-x_n) $$ exhibits chaos, depending on the value of r. This logistic mapping is a reparametrized version of the difference equation

$$x_{n+1} = x_n + k x_n (1 - \frac{x_n}{M}) $$

describing exponential growth with a maximum. The continuous limit of this difference equation, where x=x(t), is the differential equation

$$\frac{dx}{dt} = kx \Bigl(1 - \frac{x}{M}\Bigr) $$

In the continuous limit this differential equation somehow does not exhibit chaotic behaviour anymore. Intuitively I can understand this, because the solution of this differential equation, when x_0 < M, describes an S-like curve between x(t)=x_0 and x(t)=M, and hence one does not have periodic solutions anymore. My question is: why exactly does the chaotic behaviour disappear in the continuous limit? Are there some theorems about this (e.g. theorems forbidding chaotic behaviour for scalar differential equations, as opposed to sets of differential equations like the one of Lorenz)? I tried to look for some resources and found a lot of notes describing the logistic mapping, but hardly any of them is describing the continuous limit (i.e., chaotic behaviour if one goes from the difference equation to the differential equation). Any help is appreciated!

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# A Chaos: difference vs differential equation

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