Defining logistic map equation

[ChuckNoise]

Hi folks

I am reading a book about deterministic chaos and following a statement in the book I have tried to solve an example which I just can't wrap my head around. So I hoped someone in here could shed a bit of light on the problem for me

So we got this periodical kicked rotator for which a two-dimensional return map is defined. I am told in the book that one of these equations (x_n+1) can be turned into a logistic map I just can't figure out how!

Here's a little screenshot from the pdf of the book:



So my problem is that I can't figure out how we form eq. 2.19 from eq. 2.18b. Is there anyone who can guide me through this?

Thanks

 

[lex]

It comes from 2.18b if you do what they say, let [MATH]f(x_n)=(r-1)x_n-r x_n^2[/MATH], let [MATH]\frac{K}{\Gamma}[/MATH] be 1, let [MATH]\Gamma \to \infty[/MATH](Multiply out the bracket in 2.18b before you start applying these).