The VRP^-2 dynamics, with the parameter alpha value approaching the value equal to the value of the dimensionless ﬁne-structure constant, features maximum variety with regard to the number of bifurcations with the minimum degree of chaosticity.

D. B. Volov Specific behavior of one chaotic dynamics near the fine-structure constant http://arxiv.org/abs/1205.6091

http://chaosandcorrelation.org/Chaos/DV_1_5_2012.pdf

Volov_D__VTP_5_2011.pdf

http://www.sciteclibrary.ru/rus/catalog/pages/11612.html

This files contains an articles describing the Verhulst-Ricker-Planck dynamic and its relation to the fine structure constant.

The MathCAD's text program for the bifurcation diagram "four rats" (D.B.Volov, Russia, Samara).

Source code in MATLAB to reproduce the bifurcation diagram "four rats" (A.P.Trounev, Toronto, Canada)

(smile must be replaced by a colon)

x(i+1)=-L(k)/(x(i)^2(exp(x(i))+alpha))

L=zeros(1,500);

y=zeros(1,128);
ly=zeros(1,128);
alpha=1/137.035999074;
dL=(3.7+exp(1.1989))/500;

for k=1:length(L)

for i=1:length(y)

      if (k==1)

         L(k)=-3.8;

         y(k,=1;
         ly(k,=log(y(k,);

      else

         if (i==1)

            L(k)=L(k-1)+dL;
            V=y(k-1,128)*y(k-1,128)*(exp(-y(k-1,128))+alpha);

            y(k,i)=exp(L(k))/V;
            ly(k,i)=log(y(k,i));

         else
            V1=y(k,i-1)*y(k,i-1)*(exp(-y(k,i-1))+alpha);
            y(k,i)=exp(L(k))/V1;
               ly(k,i)=log(y(k,i));

         end

      end

end
end

for i=1:128

plot(L,ly(:,i),'.k')

hold on

title('a=1/137');

xlabel('lnK');ylabel('lnx');

end