Phase control maps

 The work of a curious fellow

The many gain picture...
 In the previous section we examined a function called the logistic function and observed that its graph depended on the parameter we called gain. As the gain increased, the height of the graph (its amplitude) increased. You may think of adjusting the gain parameter as though you were turning the volume control on your stereo. In fact parameters like gain are sometimes called control parameters. Review this function and the effect of gain by playing around with the Attractor Demonstration display. In the function graphs displayed in the last section, we picked a value of gain and then varied the independent variable x, calculating and plotting the dependent variable y at each x location. What we propose now is to pick a value for the independent variable x and hold it fixed. Then vary the control parameter gain and plot the dependent variable at each g location. This in effect interchanges the role of parameter and independent variable. This exchange of variables will allow us to see the effect on the function of iterating from at a particular point with a whole set of gain values. If we carry out the scheme described above for the logistic function, what would the graph look like? The logistic function is y=g*x*(1-x) We have restricted our exploration to a domain of x to be greater than 0 and less than 1. Let us take any x in that interval, say 0.5, and hold it as a constant in the function so y=g*0.5(1-0.5) Multiplying this out we get y=g*0.25 This is a pretty simple result. It says if x is 0.5, y is 1/4 of the gain. If we picked some other x, y would still be a fixed fraction of the gain. The graph would be a straight line.
 The effect of varying the initial value of x, x0, may be seen in the next display. You will see the slope of the straight line map is controlled by x0. You will also notice that increasing x0 from 0.5 or decreasing it from 0.5 has the same effect. This follows from the symmetry about the line x=0.5 of the logistic function. For low iteration numbers the map clearly depends on the value of x0. Run the Phase-Control Maps display, varying x0 between 0 and 1 with the iteration control set to iteration zero to illustrate what we were saying. We will begin referring to iterations by number at this point. The zeroth iteration, symbolized by i0, is just the function itself. i1 is the first iteration, i2 the second and so on. This plot of y vs. gain is given the name phase-control map. This hyphenated name comes from the two dimensions of the map. The horizontal axis now represents the control parameter rather than the independent variable. The vertical axis still represents the dependent variable but the term phase is applied to that dimension of the map. This terminology arises from the study of dynamical systems that we will get into later. In effect we have created a new space measured in dimensions of gain and the unitless number y, a phase-control space. We take the domain of the phase-control map to be 0
 The logistic function itself, y0=g*x0*(1-x0) is zeroth iteration, i0. For the first iteration, i1, y1=g*x1*(1-x1) remembering that iterating means x1=y0. Substituting y0 for x1 gives us: y1=g * (g*x0*(1-x0)) * (1-(g*x0*1-x0)) So y1 depends only on g and x0. By multiplying this out and rearranging you can see: y1=g^2*(x0-x0^2)-g^3*(x0-x0^2)^2 To get our map we fixed the value of x0 at some number between 0 and 1, so x0-x0^2 is just some positive constant less than 1, call it "u". Now y1 may be written: y1=u*g^2-u^2*g^3 In i0, y depended only on the first power of g. in i1, y depends on the second and third power of g. The map of i0 was just a straight line. The map of i1 bends upward for small values of g then downward as the negative term takes over. Remember for g<1 small powers of g are larger than high powers of g.
 Run the Phase-Control Maps display again and explore the effect of initial x value on iterates higher than zero. For i2 the dependence on g is given by the following mess: y2=u*g^3-u^2*g^4-u^2*g^5+2*u^3*g^6-u^4*g^7 I suspect that in general the pattern we detect in the i2 expression holds for higher iterates, but it is too hard for me to verify that the first term of the nth iteration (in) will involve g^(n+1). The last term will involve g^(2^n+n+1). There will be 2^n+1 terms with coefficients decreasing as power of g increases and a mixture of plus and minus signs. In any event it is evident that the complication of the expression goes up exponentially.
 The implication of this explosion of complexity for maps of higher iterations is troubling. Remember that our assumption in graphing was that the function was well behaved in the region we were trying to plot. You will see that at the higher values of g the map becomes more and more jagged as the iteration number increases. Imagine what the map of the 500th iteration or the 5000th might look like. These functions of g would be so badly behaved that a graph even with a very high resolution, that is with the domain divided into very many steps, could not accurately reflect the function. There would be points on the function completely missed by the graph. Run the Phase Control Maps display one more time.
 Now we want to extend our phase-control map to look at several iterations at one time. This is done by picking a value of gain corresponding to each pixel all the way across the display and carrying out the iteration as often as desired using that gain. For each iteration between the first and last to be plotted the y value is plotted above the chosen gain. Then we move to the next gain value and repeat the whole process. Each calculated point is marked by a lighted pixel of color based on the iteration. For the higher iteration numbers, depending on your computer, you might begin to notice that the plot takes longer to develop. In fact since Java is an interpreted language, requiring a lot of the work to be done at run time, we are going to push the system pretty hard with some of the later displays. If you are curious about this stuff and want to see a full powered (fast) version, try our Order program. Run the Quadratic Iterations i0-i11 display next. These first 12 iterations begin to display some of the interesting characteristics of the logistic phase-control map. Think about the features of this map and try to reason out why you see what you do. We will be examining the details of this map as we get into the next section. Some folks are surprised by the complexity inherent in iterating the simple logistic function. You should not be surprised at this point because you have seen that iteration produces complex equations. In the next lesson we will explore the phase control map of the logistic function in more detail. Are there any questions?
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