This Demonstration shows a table of superstable parameter values of a period-doubling periodic attractor. The test map is defined as
, which generalizes the well-known logistic map
is an iteration number,
is the unimodality (or the degree) of the local maximum of
. The superstable parameter values are used for the renormalization group analysis of many low-dimensional dynamical systems with chaotic behavior. See the references [1–4].
The algorithm used is a high-precision Newton algorithm with fixed precision arithmetic. For
, superstable parameter values for
are exactly the same as those for
. In this Demonstration, the required number types and their interval ranges are as follows:
is a rational number between 1 and 4.
is an integer, with 0 for the fastest calculation and the poorest visualization; 50 for moderate speed and moderate visualization; and 500 for the slowest speed with good visualization.
is an integer,
. You can select higher values manually, but at your own risk. For example, with
, the author's computer calculated the super-stable parameter values for
for one week! The minimum precision length used for this calculation was 40.
is an integer used in the fixed precision arithmetic. The Newton algorithm for finding superstable parameter values with a low precision may result in wrong answers. High precision is good but it requires a long calculation time. This Demonstration is designed for