Trajectory-Scaling Functions for Generalized Logistic Maps with z-Unimodality
This Demonstration shows Feigenbaum's trajectory scaling function (TSF) [1–6],[more]
of a unimodal map as a step-like function. Here
where is the iteration number, is the iterate of starting from the initial condition (i.e. ), is the control parameter, is the point in the domain of the function where , is the period-doubling bifurcation order starting from ( and are not defined for because becomes ), is the superstable parameter value for each bifurcation order (e.g. for period 2, for period 4, for period 8, etc.), and is the integer that relates all superstable orbits between and periods (i.e. the domain of the scaling function is ). By introducing a new variable , the TSF can also be defined as
1. The test map used in this Demonstration, , generalizes the well-known logistic map [7–14]. Here ; is the subcontrol parameter (which determines the unimodality, the degree of the local maximum of ).
2. The program for this Demonstration presents a large collection of super-stable parameter values for period-doubling periodic attractors within and , which were obtained using a high-precision Newton algorithm with fixed precision 50. These values are accurate up to 45 decimal places, so you can use them for your own research or study.
3. For , the lower bound of the domain of approaches (i.e. ) and approaches the universal scaling function (i.e. ) satisfying the following properties:
universality: , , , ,
symmetry: for , for and ,
where is the second Feigenbaum constant as a function of [1–6,15].
4. For the practical purpose of numerical simulation, is good enough to observe the universality [1, 3]. In 1988, Andrew L. Belmonte et al. used (i.e. cycles obtained from the logistic map) for comparison with their experimental data .
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