Bifurcation in a coupled logistic map. Some analytic and numerical results

We study the bifurcation pattern, two- and four-cycle generation, and supertrack functions in the case of the coupled logistic system given byX_n+1=x_n(1–2y_n) +y_n,Y_n+1=y_n(1-y_n), which is of immense importance in various biophysical processes. We deduce analytic formulas for the two -and four-cycle fixed points and cross-check them numerically. The agreement is quite good. Next the bifurcation pattern is explained with the help of analytically derived supertrack functions. To discuss the stability of the system in the various zones defined by the parameter values (, ), the Lyapunov exponents are evaluated, showing a nice transition from the stable to the unstable region. An interesting phenomena occurs at=4, where the logistic itself is chaotic. We then show that near the fixed point an analytic solution can be obtained for the renormalization group equation. In the special case=1,=4 a neat analytic formula can be deduced for then-times iterated values of (x_i,y_i).