Analytical Renormalization Results for Th E Cross-over Behavior of Period Doubling, from Conservative to Dissipative Systems

It has been shown that there is a universal scaling function describing the cross-over of the effective Feigenbaum convergence rate 8 from its conservative value (8 = 8.721097 . . . . ) to its dissipative value (8 = 4.669201 . . . . ), as a function of the "effective dissipation". Using renormalization theory I obtain an explicit analytical expression for this cross-over function and show that it's not monotonic but has a minimum, just before it reaches its asymptotic dissipative value. I also derive an analytical expression for the (period-doubling) bifurcation values in a particular map (the Htnon map), at all values of the Jacobian.