Images from the r-b-plane
In this page, we are still considering the function f(x) = b * sin2(x + r). But now, we want to completly forget all those possibilities to switch between A and B values for r Mario Markus has introduced. In this paper, if an iteration starts with a certain r, it is stuck to that r (see first and third chapter of the Mathematical Introduction Page).
Since there is no A-B-plane any more, we can’t anymore draw any images from that plane. But there remains some fundamental questions. Like this: given a certain value b and starting with a certain initial value x (like 0, for example), for which values of r do we have a positive or a negative Lyapunov exponent? Let us consider a simple case first: be r = 0. f has a fixpoint wherever f(x) = x holds. Depending on b, this can be the case only for x = 0 or many times:
f(x) = b * sin2x with b = 1, 2 resp. 5, plotted from 0 to 2pi
Some fixpoints are repelling, some are attractive, depending on their first derivation (being absolute smaller or greater 1). 0 is always an attractive fixpoint. That means, if we start with an initial x = 0 or small and r = 0, than for any b, the iteration will allways be stable and the Lyapunov exponent will be negative infinity.
With positive r, the whole function is shifted, attractive fixpoints may become repelling or vice versa, or get lost at all.
f(x) = b * sin2(x + 1.5) with b = 1, 2 resp. 5, plotted from 0 to 2pi
Instead of being attracted by a fixpoint, an x can also be attracted be a 2-cycle. A point is part of a 2-cycle, if f2(x) = x. More general, a point belongs to a n-cycle if fn(x) = x holds. A n-cycle can be attractive or repelling. If an initial x becomes attracted by a n-cycle with small absolute first derivations, the Lyapunov Exponent becomes negative. The following image shows f2 instead of f:
f2(x) for f(x) = b * sin2x with b = 1, 2 resp. 5, plotted from 0 to 2pi
Now we can draw a map of all possible cases for r and b. After r = pi, the map starts repeating, of course. We draw the Lyapunov exponent of the r-b-plane, with yellow indicating negative exponents (the brighter, the more negative), blue indicating postive exponents (the darker, the greater). Here it is:
(r * b)-map, r from 0 to pi, b from 0 to 2pi
(An even larger image of the r-b-map can found here. Just for aesthetical reasons, b grows from top to bottom instead of from bottom to top, as you might expect, while r grows from left to right.)
This is computed for 0 as initial x. For r = 0, 0 is always an attractive fixpoint, this explains the prominent vertical branch. For small b, every r leads fast to an attractive fixpoint, that is why the top of the image is completly white. For every b, there is a r so that the first maxima of f is a fixpoint (a maxima or minima has a first derivation of zero and is therefor not only attractive, but super-attractive, if it is a fixpoint). The more b grows, the more r must shrink (modulo pi) to adjust the maxima as a fixpoint. This explains the prominent diagonal branch.
This image depends on the chosen x, of course, but not crucially: if you chose a different x, x might be attracted by a different fixpoint or cycle, and therefor, dependend on x, different branches overlap each other. This is illustrated by the following image, with an initial x = 2.25:
(r * b)-map, r from 0 to pi, b from 0 to 2pi, with x = 2.25
(Once again, b grows from top to bottom instead of from bottom to top.) As you can see, at the first crossing of the largest vertical and the largest diagonal branch, now the diagonal branch overlaps the vertical branch, but the main features of both images are the same.
To investigate the meaning of the initial x a bit further, the next image shows a r-x-plane. r grows from left to right, x grows from bottom to top, both from 0 to pi:
(r * x)-map, r from 0 to pi, x from 0 to pi, with b = 2.75
This is computed for b = 2.75. Other values of b show even less global influence of x (the local influence of x of course can be immense).