This simulation analyzes the logistic sequence in detail. There are two corresponding windows I and II, in which the calculation can be performed with different parameters: the initial value of the iterations x0, the number of iterations not shown in the charts suppressed ≥ 0, the number of iterations above the suppressed ones shown visible ≥ 1. This way one can keep conditions constant in one window, while changing parameters in the other one.
The intial value is defined by a slider in the range 0.1 < xo < 0.99 in ticks of 0.01. The numbers of iterations are inserted as integers into two number fields.
Any change in the parameters clears all traces. Play starts the calculation for the parameters defined in both windows. It stops when the the r range has been covered. The left chart in each window displays the whole range 0 < r < 1, the right chart the bifurcation range 0.75 < r < 1 only.
When the simulation is opened, default values are x = 0.5, with 50 iterations suppressed, 1 iteration visible (this one will be the 51 st iteration). With this setting, there is only 1 point for each r. While the bifurcation areas are indicated by kinks in the curves, there is no bifurcation. There is still a single, well defined "limit curve" up to the third bifurcation area, where points suddenly scatter.
When the initial value is changed in one of the windows, the curves before the first bifurcation area will be indistinguishable for the eye - 50 iterations already smooth out the influence of the initial value. Yet the behavior beyond may be drastically different, showing other single branches of the limit curve for certain ranges of xo.
When the single iterations shown are differing by one in each window (r.g. suppressed = 50 in I, = 51 in II) and the initial value of both is equal, each window will show a different branch of the first bifurcation. This may not be the case if the initial values differ, and can be true even after a large number of iterations (e.g. suppressed = 1000). The logistic map has only two limit branches after the first and before the second bifurcation, but which one will result for a specific r and a specific iteration depends on the initial conditions.
When the number of visible iterations is 2, both branches of the first bifurcation are seen, independent of the initial value. Yet it depends on the initial condition which value of r is represented for a specific iteration in which branch.
When the number of visible iterations is 4 the four branches of the second bifurcation appear, and so on. Now a certain r may lead to any of the four branches in a specific iteration, but to one only.
It is interesting to study the behavior with no or only a few iterations suppressed (you should close and reopen the simulation if numerical artifacts disturb the chart with these changes). Keep visible = 1 constant, start with suppressed = 0 and increase by steps of 1. Now you will see the single iterations in consecution of their index . You will recognize polynomials of increasing order, which above order 1 (initial value) are quite close together up to r = 0.7, and which start to fluctuate increasingly wild when r = 0.9 is approached. Between both is the range of clear bifurcation, above the range of no discernable regularity .
Choosing suppressed = 0, one can compare in one chart the N successive first iterations by setting visible = N (1, 2, 3, 4...). Using a different initial value in the second window, one can see its influence at low iterations. It looks more pronounced in the area of the "limit curves" than for high iterations, but remember that that is not true above the first bifurcation! There the value may jump great distances from one branch to the other, and in the "chaotic" area any value is possible for distinct r and iteration, depending on the initial value.
To get the common impression of the logistic curve, set suppressed = 1000, visible = 500. It gives the impression that in the first bifurcation range there are two solutions for each r, independent of the initial value and the iteration. You know better now: there are two alternatives!