Period doubling system under fractal signal: Bifurcation in the renormalization group equation (Q1185432)

The period doubling system under the special fractal signal is investigated. As the period doubling system the logistic equation (1) is used \[ x_{n+1}=1-\lambda x^ 2_ n.\tag{1} \] The fractal signal is modelled by the \(y_ n\) given by \[ y_{2n}=b(1+y_ n),\;y_{2n+1}=a(1+y_ n).\tag{2} \] So the general model treated is \[ x_{n+1}=1-\lambda x^ 2_ n+cy_ n.\tag{3} \] By means of renormalization group (RG) technique it is shown that the bifurcation in the renormalization group equation is equivalent to the adequate transition from one critical situation to another. So the method of RG helps in studying critical dynamics of such systems.