After the second stretching the central part of the third period
doubling bifurcation coincides with the first pictures.
For n → ∞ the two scaling constants converge to
α = 2.5029 in the horizontal x direction (dynamical space)
and δ =4.669 in the vertical c direction (parameter space).
Moreover self-similarity and these constants are universal (don't depend on detailes of map with quadratic minimum). To test the universality look at bifurcation cascade which arises for the quadratic-like map f^{ o3} in the biggest period-3 window. |
You see below two successive zooms of the first diagram. The three pictures begin from the superstable period-1,2,4 orbits correspondingly and end at preperiodic point m_{0,1,2}). The biggest windows have periodicity 3, 6, 12. It is evident that all diagrams (but not only ordinary period doubling cascade) are self-similar. |
In the diagrams shown before the biggest windows of regular dynamics have period 3, 6, 12. Superstable periodic orbits corresponding to these windows undergo reverse period doubling cascade too. You see that ordinary and reverse bifurcations lead to orbits of different kind. |