Analytical renormalization results for the cross-over behavior of periodic doubling, from conservative to dissipative systems (Q1114205)

It has been shown that there is a universal scaling function describing the cross-over of the effective Feigenbaum convergence rate \(\delta\) from its conservative value \((\delta =8.721097...)\) to its dissipative value \((\delta =4.669201...)\), 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 Hénon map), at all values of the Jacobian.