DR. Strogatz, thank you for an excellent analysis and demonstration of Feigenbaum's renormalization analysis of period doubling. From watching and analyzing this topic, I found it very difficult to follow.
A linear form like g(x)=x doesn't possess a quadratic maxima and hence can't be a solution of the functional equation needed to describe the onset of chaos!
Feigenbaum delta, Fd=4.669201609, can determine the "edge of chaos", the Feigenbaum point, Fp =3.5699, by subtracting unity, inverting, then adding unity, to the natural log e power: e^(((1/(Fd-1))+1)=3.5699045
DR. Strogatz, thank you for an excellent analysis and demonstration of Feigenbaum's renormalization analysis of period doubling. From watching and analyzing this topic, I found it very difficult to follow.
constants are the jewels in nature that make it work.
1:06:11 This functional equation, g(x)=alpha g^{2}(x/alpha), looks to have a simple solution: g(x)=x. Could there be any other closed form for g(x)?
A linear form like g(x)=x doesn't possess a quadratic maxima and hence can't be a solution of the functional equation needed to describe the onset of chaos!
Thankyou
thank you
Feigenbaum delta, Fd=4.669201609, can determine the "edge of chaos", the Feigenbaum point, Fp =3.5699, by subtracting unity, inverting, then adding unity, to the natural log e power: e^(((1/(Fd-1))+1)=3.5699045
good
R = F(ANGEL)