Homoclinic bifurcation analysis for logistic map

In this study, we have developed the method to obtain the homoclinic bifurcation parameter of an arbitrary targeted fixed point in the logistic map T-r. We have considered the geometrical structure of T-r around x = 0.5 and derived the core condition of the bifurcation occurrence. As the result of numerical experiment, we have calculated the exact bifurcation parameter of the fixed point of T-r(l) with l <= 256. We have also discussed the Feigenbaum constants found in the bifurcation parameter and the fixed point coordinate sequences. This fact implies the local stability of the fixed point and global structure around it are in association via the constants.