Modelling the Feigenbaum Diagram by the Trajectory of the

Modelling the Feigenbaum Diagram by the Trajectory of the Point ½ By Patrick Hurley Richard O’ Riordan Partially Supported by The Boole Centre for Research in Informatics (BCRI), UCC. Aims of the Experiment: 1. To quantify a statistical probability of describing the long-term trajectories of random initial values by the short-term trajectories of the point ½. To make assumptions about these statistics based on the data. 2. Background Information: Experiment: The whole experiment was started because of a project given to us by Dr. Tom Carroll in module MA2055, (Experimentation and Problem Solving). He asked us to investigate the behaviour of particular values of λ in the logistic map and discuss our findings. The aim of the course is to develop a research ethic among students and to foster a team based outlook to problem solving, and the freedom of the course allowed us to explore avenues not being explored in other courses. Our team compromised Patrick Hurley, Richard O’ Riordan and Eoin Mulholland. Our research led us to the Logistic Map and Feigenbaum Diagram. Logistic Map: Consider the Logistic Map: tn +1 = λ tn (1 − tn ) We fix t = t0 and derive polynomials in terms of λ. We thus plot f , n Î N in terms of λ. n t = By choosing 0 2 we obtain; t0 = 1 2 1 f 1 = t1 = l t0 1 - t0 = l 4 f 2 = t2 = l t1 1 - t1 = l2 4 - l3 16 f 3 = t3 = l t2 1 - t2 = l3 4 - l4 16 - f 4 = t4 = l t3 1 - t3 = l4 4 16 16 9 10 7l - 3l + 256 128 - l5 - H L H L H L H L l5 16 + l6 32 - l7 256 l6 - l7 32 l 11 8 + 7l + 512 + 256 7 l 12 2048 13 - 3l + l 14 2048 4096 - l 15 65536 etc. Using Mathematica's Nest function, we compute f each n∈Ν and plot these functions. All of the above functions take the value of 0 for λ=0 or λ=4, ∀ n≥2. Appendix 1 a. shows the first 5 curves overlayed n The Feigenbaum Diagram. The Feigenbaum Diagram is a very elegant way of examining the long term behaviour of the logistic map. It allows us to see the trajectories of a fixed point through discrete values of λ and see where periodic windows occur. It also exhibits properties not obvious in the simple cobweb diagram. For example, the self-similarity of the trajectories is very apparent in the Feigenbaum Diagram. (Appendix 1 b.) There is also another phenomenon visible in the Feigenbaum Diagram. This is the seemingly regular curves that sweep across the diagram. They are, intuitively, not supposed to be there, as each λ is supposed to exhibit independent behaviour and these curves would suggest an underlying structure to the randomness exhibited. This, however, is exactly what is going on. These curves act as attractors for the trajectories and we determined that these curves are described by the trajectory of the point ½. Appendix 1 (c.,d.) Using these curves we can determine that there is a point at which all but one curves intersect. This point is the fixed point for the trajectory of the point ½ and can also be shown to be the point before which no period of odd period can exist. There are many other interesting phenomena regarding these trajectories, (including their intersections on the complex plane and the points of origin of these curves as we iterate to -infinity), but these we leave for another discussion. Experimental Method: We will use numerical methods to determine the raw data needed for the analysis. Although Mathematica was used initially, it was determined that we may get more comprehensive results by using C++. Our experiment involves varying different parameters to attain a table of results. Our first act is to fix the initial value being used and ensure the trajectory is well defined. We do this by discarding the fist 10,000,000 iterates of the function. We can then assume that, at each value of λ for which the function has a fixed period n, at that value of λ the function is on an orbit of that period. We then obtain the next 100,000 points at each value of λ and compare these with the first 25 points obtained by using the trajectory of the point ½. We wish to find out how many of the 100,000 points are close to first 25 points of trajectory ½. We seek percentages for sequential closeness’ of 0.001, 0.002, 0.003, 0.004, and 0.005 of a unit. We then vary the different parameters above for different numbers of initial points discarded, numbers of points recorded, and number of curves. We then repeat the experiment for different initial values. Our data is compiled and sorted in Microsoft Excel Spreadsheets and graphs and tables formed therein. Appendix 2, Example 1 Appendix 1. a.) First 5 polynomial curves overlayed b.) Feigenbaum Diagram 0.99 + 8 ≤ λ ≤ 1.03 + 8 c.) Feigenbaum Diagram 1.02 + 8 ≤ λ ≤ 1.03 + 8 d.) First 25 Curves 1.02 + 8 ≤ λ ≤ 1.03 + 8 Lambda Distance=0.001 Distance=0.002 Distance=0.003 Distance=0.004 Distance=0.005 3.55 3.551 3.552 3.553 3.554 3.555 3.556 3.557 3.558 3.559 3.56 3.561 3.562 3.563 3.564 3.565 3.566 3.567 3.568 3.569 3.57 3.571 3.572 3.573 3.574 3.575 3.576 3.577 3.578 3.579 3.58 3.581 3.582 3.583 3.584 3.585 3.586 3.587 3.588 3.589 3.59 3.591 3.592 3.593 3.594 3.595 Init Value= 0.1 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 87.500% 50.000% 50.000% 93.750% 100.000% 100.000% 100.000% 81.250% 71.344% 60.509% 47.537% 52.067% 41.911% 37.782% 45.981% 35.891% 39.411% 38.484% 36.659% 33.582% 46.112% 100.000% 44.201% 35.781% 37.518% 34.307% 33.522% 29.513% 31.501% 28.170% 28.127% 31.662% 28.322% 27.679% Iter. Out=1e+007 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 87.500% 87.500% 100.000% 100.000% 100.000% 100.000% 87.500% 79.445% 77.368% 70.271% 68.361% 54.422% 50.537% 64.777% 53.092% 56.065% 55.156% 52.609% 45.506% 57.343% 100.000% 59.301% 52.697% 51.272% 49.566% 50.810% 42.327% 43.503% 42.629% 42.459% 43.335% 40.952% 44.544% Iter. In=100000 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 87.500% 100.000% 100.000% 100.000% 100.000% 96.875% 87.328% 87.023% 79.213% 79.780% 63.256% 58.503% 76.293% 64.318% 67.408% 65.048% 62.637% 54.068% 64.381% 100.000% 68.558% 63.753% 61.647% 61.441% 61.978% 50.753% 52.337% 52.951% 52.627% 52.154% 49.790% 56.795% No. Curve=25 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 96.875% 88.954% 90.805% 86.767% 85.186% 71.251% 65.438% 80.385% 75.204% 74.064% 71.310% 71.029% 61.612% 69.678% 100.000% 74.983% 69.473% 70.146% 70.358% 70.378% 57.647% 58.994% 59.636% 59.169% 58.377% 56.748% 64.060% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 96.875% 90.512% 93.856% 92.306% 88.641% 78.097% 71.659% 83.914% 81.496% 79.758% 77.359% 77.606% 66.766% 74.469% 100.000% 79.431% 73.990% 75.321% 75.292% 77.570% 62.550% 63.766% 66.080% 65.094% 62.555% 62.309% 70.607% Lambda 3.596 3.597 3.598 3.599 3.6 3.601 3.602 3.603 3.604 3.605 3.606 3.607 3.608 3.609 3.61 3.611 3.612 3.613 3.614 3.615 3.616 3.617 3.618 3.619 3.62 3.621 3.622 3.623 3.624 3.625 3.626 3.627 3.628 3.629 3.63 3.631 3.632 3.633 3.634 3.635 3.636 3.637 3.638 3.639 3.64 3.641 3.642 3.643 3.644 3.645 Distance=0.001 27.426% 24.755% 30.472% 25.845% 27.384% 27.711% 100.000% 27.404% 26.265% 21.756% 100.000% 42.676% 30.117% 25.380% 25.816% 25.670% 23.437% 26.454% 30.763% 22.860% 23.439% 23.901% 20.622% 21.885% 24.976% 22.604% 25.143% 22.280% 25.967% 26.931% 22.029% 100.000% 100.000% 100.000% 83.333% 100.000% 100.000% 79.169% 100.000% 36.969% 28.034% 26.191% 28.483% 21.739% 22.813% 20.872% 26.122% 21.685% 17.002% 24.127% Distance=0.002 44.596% 35.647% 43.709% 40.166% 43.551% 41.717% 100.000% 44.688% 39.395% 29.956% 100.000% 56.165% 43.879% 40.098% 39.652% 38.804% 36.378% 40.050% 42.601% 36.589% 35.821% 38.018% 32.829% 32.459% 38.268% 31.981% 39.147% 32.612% 38.171% 36.960% 27.860% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 88.889% 100.000% 50.007% 42.279% 38.212% 40.635% 33.291% 35.392% 33.482% 36.007% 31.248% 27.186% 35.815% Distance=0.003 Distance=0.004 57.687% 45.157% 53.570% 50.957% 54.320% 52.188% 100.000% 55.554% 48.133% 36.454% 100.000% 63.297% 53.853% 51.860% 50.304% 47.907% 45.647% 47.307% 51.737% 46.855% 45.913% 48.302% 42.857% 40.519% 46.864% 39.745% 49.186% 40.639% 46.639% 42.749% 31.915% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 95.833% 100.000% 58.410% 51.162% 45.339% 48.349% 42.729% 45.934% 42.332% 44.481% 37.305% 35.939% 44.428% 67.157% 53.183% 61.147% 59.576% 63.541% 59.776% 100.000% 62.543% 55.868% 42.493% 100.000% 68.802% 61.327% 62.180% 59.865% 55.541% 53.321% 53.434% 58.847% 55.928% 54.570% 57.136% 51.934% 47.578% 54.354% 45.941% 56.387% 47.157% 52.901% 46.791% 35.519% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 97.222% 100.000% 65.694% 58.814% 51.203% 54.427% 49.724% 54.277% 49.261% 51.434% 42.429% 43.233% 51.452% Distance=0.005 72.969% 58.414% 67.384% 67.639% 71.113% 66.689% 100.000% 68.531% 63.034% 47.570% 100.000% 73.961% 67.807% 69.336% 67.791% 61.574% 59.450% 58.519% 62.713% 61.584% 61.824% 64.647% 59.351% 53.634% 59.269% 50.572% 62.174% 52.636% 58.197% 49.710% 38.765% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 97.222% 100.000% 70.288% 65.783% 56.259% 60.291% 55.267% 61.009% 55.626% 55.899% 47.257% 47.843% 57.135% Lambda 3.646 3.647 3.648 3.649 3.65 3.651 3.652 3.653 3.654 3.655 3.656 3.657 3.658 3.659 3.66 3.661 3.662 3.663 3.664 3.665 3.666 3.667 3.668 3.669 3.67 3.671 3.672 3.673 3.674 3.675 3.676 3.677 3.678 3.679 3.68 3.681 3.682 3.683 3.684 3.685 3.686 3.687 3.688 3.689 3.69 3.691 3.692 3.693 3.694 3.695 Distance=0.001 21.951% 31.258% 26.077% 27.096% 20.541% 20.890% 16.826% 18.602% 18.551% 18.893% 100.000% 18.197% 19.136% 20.855% 18.926% 19.395% 15.505% 36.403% 21.281% 19.571% 17.704% 18.661% 17.586% 18.449% 17.436% 18.183% 16.438% 100.000% 16.480% 16.564% 15.577% 17.139% 17.269% 21.551% 18.018% 18.480% 20.460% 18.886% 16.947% 17.348% 13.686% 19.297% 19.953% 14.965% 14.678% 17.241% 15.559% 17.175% 16.301% 16.258% Distance=0.002 34.229% 38.441% 39.484% 39.865% 31.996% 30.813% 25.886% 28.985% 29.313% 28.899% 100.000% 26.326% 30.535% 28.623% 29.770% 28.898% 20.533% 46.827% 31.627% 27.696% 27.799% 26.925% 27.632% 29.214% 27.009% 28.021% 28.121% 100.000% 27.488% 25.632% 25.673% 26.922% 27.577% 28.532% 29.801% 30.440% 32.014% 29.109% 27.043% 27.019% 20.648% 27.945% 30.699% 25.267% 23.122% 28.167% 25.576% 27.528% 27.223% 26.357% Distance=0.003 Distance=0.004 43.946% 43.328% 48.857% 48.009% 41.487% 39.209% 33.529% 36.650% 37.130% 36.790% 100.000% 33.298% 40.662% 33.784% 37.592% 36.830% 24.528% 53.211% 40.451% 33.028% 35.902% 33.315% 37.095% 37.030% 33.925% 36.292% 36.482% 100.000% 36.989% 32.703% 34.421% 33.100% 35.749% 33.712% 36.934% 39.704% 40.570% 37.165% 34.469% 34.836% 25.353% 33.593% 38.404% 34.076% 30.689% 38.094% 33.728% 36.283% 35.651% 34.977% 51.704% 47.632% 55.407% 54.659% 49.638% 45.878% 39.837% 43.252% 43.298% 42.746% 100.000% 39.143% 48.486% 37.626% 44.190% 43.420% 27.938% 57.756% 47.202% 37.378% 43.125% 39.093% 45.382% 44.502% 39.297% 43.553% 43.114% 100.000% 44.330% 38.841% 41.936% 38.678% 42.671% 37.931% 43.340% 46.498% 47.125% 43.490% 40.815% 41.817% 29.683% 37.235% 43.385% 42.304% 37.772% 45.884% 40.507% 43.710% 43.425% 42.267% Distance=0.005 58.245% 51.290% 61.122% 60.280% 55.156% 51.650% 44.870% 48.337% 49.098% 47.969% 100.000% 44.334% 54.638% 41.253% 49.714% 49.654% 30.904% 61.858% 53.019% 41.507% 49.274% 44.053% 52.325% 51.028% 43.863% 49.728% 49.581% 100.000% 49.596% 44.140% 48.185% 43.403% 48.422% 41.501% 48.801% 51.373% 52.197% 48.652% 45.939% 48.617% 33.638% 40.073% 47.723% 49.119% 43.723% 51.365% 45.982% 49.997% 50.523% 48.095% Lambda 3.696 3.697 3.698 3.699 3.7 3.701 3.702 3.703 3.704 3.705 3.706 3.707 3.708 3.709 3.71 3.711 3.712 3.713 3.714 3.715 3.716 3.717 3.718 3.719 3.72 3.721 3.722 3.723 3.724 3.725 3.726 3.727 3.728 3.729 3.73 3.731 3.732 3.733 3.734 3.735 3.736 3.737 3.738 3.739 3.74 3.741 3.742 3.743 3.744 3.745 Distance=0.001 14.370% 16.824% 16.162% 15.761% 17.076% 17.416% 100.000% 29.869% 19.313% 20.568% 15.266% 13.933% 18.504% 14.830% 13.880% 19.070% 14.367% 15.229% 14.207% 14.025% 14.849% 14.231% 17.261% 14.976% 14.924% 16.939% 16.178% 13.919% 16.039% 16.023% 15.941% 14.479% 15.348% 14.253% 17.799% 16.135% 15.628% 15.317% 16.040% 16.847% 17.418% 19.879% 19.699% 100.000% 100.000% 80.000% 100.000% 88.750% 68.187% 29.672% Distance=0.002 23.918% 27.468% 26.965% 26.350% 26.971% 25.580% 100.000% 40.947% 29.351% 31.443% 25.342% 23.071% 30.219% 23.619% 22.883% 27.862% 22.031% 25.233% 24.129% 23.001% 21.489% 18.994% 26.603% 25.063% 23.563% 26.317% 25.211% 22.053% 25.898% 25.488% 24.991% 24.085% 24.995% 22.265% 28.342% 27.031% 24.390% 25.252% 25.011% 27.513% 26.688% 28.301% 25.002% 100.000% 100.000% 80.000% 100.000% 97.500% 81.354% 41.788% Distance=0.003 Distance=0.004 32.623% 36.205% 35.952% 34.958% 34.316% 31.476% 100.000% 48.412% 38.282% 40.133% 34.571% 31.586% 38.365% 30.413% 28.911% 34.836% 29.166% 34.092% 32.882% 31.027% 26.857% 22.682% 34.720% 33.845% 30.522% 34.663% 32.668% 29.452% 33.246% 33.126% 33.100% 32.695% 33.645% 29.155% 37.149% 35.138% 32.403% 33.178% 32.753% 36.273% 34.345% 34.131% 28.545% 100.000% 100.000% 100.000% 100.000% 98.750% 88.306% 49.129% 39.788% 44.275% 43.275% 43.163% 40.701% 36.887% 100.000% 54.058% 45.806% 47.020% 43.088% 38.969% 43.412% 36.362% 34.288% 40.742% 35.732% 41.543% 41.309% 38.554% 31.792% 25.830% 40.705% 41.408% 36.343% 41.016% 39.472% 35.978% 39.743% 39.544% 39.455% 40.241% 41.438% 34.833% 43.561% 41.412% 39.455% 40.066% 40.033% 42.794% 39.827% 39.168% 31.576% 100.000% 100.000% 100.000% 100.000% 100.000% 93.023% 55.534% Distance=0.005 45.625% 49.432% 50.320% 49.941% 45.825% 41.384% 100.000% 58.868% 51.730% 53.252% 50.087% 45.664% 47.741% 42.147% 39.488% 45.812% 41.847% 47.228% 48.368% 45.072% 36.337% 28.783% 45.762% 47.214% 41.614% 46.524% 45.457% 41.853% 44.883% 45.293% 45.085% 46.726% 48.195% 40.098% 49.289% 46.799% 46.337% 46.413% 46.378% 47.942% 44.598% 42.907% 33.974% 100.000% 100.000% 100.000% 100.000% 100.000% 94.705% 61.164% Lambda 3.746 3.747 3.748 3.749 3.75 3.751 3.752 3.753 3.754 3.755 3.756 3.757 3.758 3.759 3.76 3.761 3.762 3.763 3.764 3.765 3.766 3.767 3.768 3.769 3.77 3.771 3.772 3.773 3.774 3.775 3.776 3.777 3.778 3.779 3.78 3.781 3.782 3.783 3.784 3.785 3.786 3.787 3.788 3.789 3.79 3.791 3.792 3.793 3.794 3.795 Distance=0.001 20.516% 20.412% 17.803% 18.179% 19.018% 19.123% 15.039% 15.453% 15.323% 15.315% 17.869% 13.685% 15.482% 13.473% 15.708% 15.882% 17.146% 14.595% 15.011% 14.773% 15.305% 13.800% 14.404% 14.257% 13.391% 12.513% 14.216% 14.242% 14.900% 30.946% 18.651% 14.060% 16.813% 14.514% 12.325% 14.139% 15.167% 12.931% 13.202% 13.778% 18.549% 15.717% 12.548% 12.737% 13.380% 13.642% 13.438% 12.593% 13.480% 14.737% Distance=0.002 32.513% 30.871% 28.615% 28.598% 29.111% 29.361% 22.448% 24.653% 25.325% 25.078% 26.338% 21.582% 21.792% 22.226% 22.573% 24.093% 26.304% 23.142% 24.779% 21.755% 25.531% 23.303% 23.852% 22.994% 20.495% 18.469% 21.557% 23.248% 18.571% 41.884% 28.704% 22.642% 25.745% 24.163% 19.407% 21.270% 24.036% 22.011% 22.275% 23.603% 25.793% 25.173% 20.352% 20.741% 20.984% 22.500% 20.952% 21.396% 20.837% 22.384% Distance=0.003 Distance=0.004 39.499% 37.949% 37.024% 36.792% 36.793% 38.377% 28.635% 31.482% 34.436% 34.147% 32.844% 27.585% 27.342% 29.863% 27.929% 30.574% 34.339% 30.371% 31.883% 27.293% 34.243% 31.508% 31.824% 30.262% 26.914% 23.305% 27.417% 31.102% 21.607% 47.797% 37.301% 29.652% 32.452% 31.947% 25.496% 27.452% 31.549% 30.418% 30.728% 31.254% 32.072% 32.446% 27.283% 28.264% 27.470% 29.693% 27.151% 28.889% 26.914% 28.534% 45.787% 42.886% 42.846% 43.039% 42.530% 46.251% 34.176% 37.528% 42.602% 42.772% 38.715% 32.758% 32.298% 36.826% 32.950% 35.765% 42.047% 36.813% 38.283% 32.317% 41.298% 38.579% 38.627% 36.153% 32.336% 27.398% 32.742% 38.301% 24.369% 52.320% 44.547% 34.548% 37.779% 37.270% 31.003% 32.464% 37.676% 38.426% 37.999% 37.970% 37.016% 38.540% 33.643% 35.419% 32.541% 35.405% 32.515% 35.317% 31.763% 34.095% Distance=0.005 51.638% 47.155% 47.827% 47.533% 47.346% 51.177% 39.220% 43.187% 47.984% 49.556% 43.913% 37.629% 36.592% 42.384% 37.469% 40.025% 48.662% 42.880% 44.183% 36.392% 47.367% 44.637% 45.281% 41.278% 37.162% 30.692% 37.924% 43.841% 26.826% 55.462% 49.528% 38.791% 41.795% 41.660% 35.875% 36.504% 43.314% 45.321% 43.893% 43.380% 41.404% 44.327% 39.429% 41.838% 37.135% 40.378% 37.312% 40.534% 36.585% 38.749% Lambda 3.796 3.797 3.798 3.799 3.8 3.801 3.802 3.803 3.804 3.805 3.806 3.807 3.808 3.809 3.81 3.811 3.812 3.813 3.814 3.815 3.816 3.817 3.818 3.819 3.82 3.821 3.822 3.823 3.824 3.825 3.826 3.827 3.828 3.829 3.83 3.831 3.832 3.833 3.834 3.835 3.836 3.837 3.838 3.839 3.84 3.841 3.842 3.843 3.844 3.845 Distance=0.001 12.118% 12.696% 12.527% 12.423% 14.789% 92.327% 15.965% 12.809% 14.108% 12.743% 14.453% 14.576% 14.331% 14.205% 13.232% 12.889% 13.571% 14.168% 14.036% 13.026% 17.937% 13.891% 14.027% 13.276% 14.688% 17.420% 16.449% 16.301% 17.955% 19.253% 22.595% 26.323% 24.803% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 66.666% 33.334% 83.334% 100.000% 100.000% 100.000% Distance=0.002 20.356% 21.066% 19.371% 21.175% 23.288% 97.504% 25.924% 20.978% 23.830% 21.693% 22.988% 22.400% 22.518% 23.438% 22.639% 21.247% 22.537% 23.075% 23.309% 20.936% 25.729% 22.559% 21.953% 20.912% 23.858% 25.809% 25.078% 24.612% 26.825% 27.677% 30.496% 33.398% 29.501% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 66.666% 66.667% 100.000% 100.000% 100.000% 100.000% Distance=0.003 Distance=0.004 27.686% 28.642% 24.270% 29.353% 30.218% 98.956% 32.565% 28.430% 31.580% 28.683% 29.665% 29.256% 29.497% 30.043% 30.585% 28.511% 29.751% 30.290% 30.736% 26.868% 31.430% 29.553% 27.988% 26.854% 31.434% 32.175% 32.081% 30.899% 33.388% 33.049% 35.552% 37.016% 33.060% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 66.667% 100.000% 100.000% 100.000% 100.000% 34.722% 34.825% 28.725% 36.840% 36.664% 100.000% 38.408% 35.134% 37.910% 34.996% 35.552% 35.083% 34.842% 36.155% 37.459% 34.297% 35.880% 36.010% 36.575% 31.549% 35.145% 35.661% 32.930% 30.941% 37.497% 37.864% 38.088% 35.820% 39.157% 37.095% 39.344% 39.941% 35.670% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% Distance=0.005 40.576% 40.287% 32.715% 42.657% 42.144% 100.000% 44.204% 41.527% 42.347% 40.967% 39.754% 39.684% 38.815% 41.741% 42.594% 39.717% 41.507% 40.740% 42.237% 35.584% 38.567% 40.819% 36.358% 34.196% 42.312% 42.622% 42.381% 40.238% 44.683% 40.622% 42.774% 41.644% 37.685% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% Lambda 3.846 3.847 3.848 3.849 3.85 3.851 3.852 3.853 3.854 3.855 3.856 3.857 3.858 3.859 3.86 3.861 3.862 3.863 3.864 3.865 3.866 3.867 3.868 3.869 3.87 3.871 3.872 3.873 3.874 3.875 3.876 3.877 3.878 3.879 3.88 3.881 3.882 3.883 3.884 3.885 3.886 3.887 3.888 3.889 3.89 3.891 3.892 3.893 3.894 3.895 Distance=0.001 100.000% 83.334% 100.000% 91.667% 88.746% 57.819% 58.166% 59.280% 96.920% 100.000% 100.000% 28.038% 32.978% 23.694% 15.727% 25.351% 20.227% 18.739% 17.307% 20.142% 20.696% 15.297% 13.200% 19.082% 18.589% 18.806% 20.877% 15.283% 14.740% 16.176% 14.953% 13.310% 13.748% 12.334% 13.543% 13.013% 14.629% 12.377% 13.500% 13.503% 11.955% 14.312% 14.685% 11.166% 11.845% 13.051% 14.291% 12.418% 12.490% 13.412% Distance=0.002 100.000% 100.000% 100.000% 100.000% 89.999% 74.364% 73.238% 74.191% 98.894% 100.000% 100.000% 41.973% 47.605% 37.492% 23.810% 35.618% 30.285% 29.338% 26.332% 29.508% 30.771% 22.481% 19.569% 28.830% 27.434% 29.196% 32.575% 22.871% 25.083% 23.381% 22.851% 20.862% 21.069% 19.202% 19.374% 21.474% 22.807% 19.227% 19.768% 22.272% 15.541% 22.535% 20.809% 17.377% 19.758% 18.612% 22.117% 20.035% 18.338% 21.769% Distance=0.003 Distance=0.004 100.000% 100.000% 100.000% 100.000% 96.666% 80.311% 81.929% 81.783% 100.000% 100.000% 100.000% 53.924% 55.711% 44.614% 30.565% 42.523% 38.556% 37.036% 33.324% 36.955% 38.580% 28.566% 24.016% 35.821% 35.249% 37.744% 39.364% 29.733% 32.610% 28.762% 29.102% 27.221% 26.532% 24.678% 24.124% 28.876% 29.203% 25.198% 25.594% 29.148% 18.467% 28.990% 25.138% 22.579% 25.857% 22.838% 29.050% 26.988% 23.671% 29.437% 100.000% 100.000% 100.000% 100.000% 96.666% 85.006% 86.751% 86.109% 100.000% 100.000% 100.000% 58.618% 60.849% 49.531% 36.772% 49.053% 44.451% 43.236% 39.041% 42.674% 45.012% 34.059% 28.209% 41.079% 40.835% 44.254% 44.221% 35.472% 38.266% 33.434% 34.769% 33.304% 31.810% 28.192% 28.609% 34.663% 34.659% 30.704% 29.861% 35.482% 20.964% 35.199% 29.016% 27.474% 31.038% 26.516% 35.287% 33.435% 28.562% 36.118% Distance=0.005 100.000% 100.000% 100.000% 100.000% 96.666% 89.263% 91.097% 89.517% 100.000% 100.000% 100.000% 62.677% 65.581% 54.221% 41.941% 54.205% 49.786% 48.402% 43.959% 47.722% 49.483% 38.143% 32.249% 44.831% 46.101% 48.528% 48.852% 40.940% 43.223% 37.999% 38.910% 38.399% 36.703% 31.568% 31.909% 39.949% 39.322% 35.695% 33.886% 41.422% 23.212% 40.987% 32.586% 31.802% 35.587% 30.212% 39.838% 38.402% 33.204% 41.095% Lambda 3.896 3.897 3.898 3.899 3.9 3.901 3.902 3.903 3.904 3.905 3.906 3.907 3.908 3.909 3.91 3.911 3.912 3.913 3.914 3.915 3.916 3.917 3.918 3.919 3.92 3.921 3.922 3.923 3.924 3.925 3.926 3.927 3.928 3.929 3.93 3.931 3.932 3.933 3.934 3.935 3.936 3.937 3.938 3.939 3.94 3.941 3.942 3.943 3.944 3.945 Distance=0.001 14.153% 13.470% 13.929% 12.764% 14.747% 15.382% 12.598% 14.167% 14.299% 16.127% 100.000% 27.309% 21.143% 13.651% 17.143% 17.338% 15.196% 17.159% 12.316% 11.470% 12.054% 11.903% 12.434% 11.855% 12.499% 11.422% 12.184% 12.774% 11.292% 10.276% 12.029% 11.245% 11.640% 10.170% 10.569% 12.168% 10.019% 10.743% 10.985% 12.039% 10.302% 11.679% 13.649% 11.408% 12.367% 9.353% 10.462% 11.302% 11.675% 10.516% Distance=0.002 22.572% 22.153% 21.397% 19.979% 22.898% 25.060% 21.056% 22.072% 21.168% 24.449% 100.000% 36.138% 29.211% 21.725% 25.814% 25.599% 22.215% 25.656% 18.060% 16.388% 17.361% 18.963% 19.071% 20.368% 19.718% 18.905% 19.427% 21.504% 18.941% 16.958% 19.318% 17.186% 18.649% 16.964% 18.031% 19.364% 16.536% 17.256% 17.658% 19.096% 16.339% 18.135% 20.576% 18.592% 19.144% 15.020% 16.946% 18.869% 18.622% 17.344% Distance=0.003 Distance=0.004 28.944% 28.813% 27.498% 26.318% 29.580% 32.922% 28.621% 28.748% 26.207% 30.546% 100.000% 42.028% 35.192% 27.310% 30.066% 30.549% 27.799% 31.621% 22.435% 20.348% 22.301% 24.881% 23.688% 27.405% 26.038% 25.157% 25.697% 29.053% 25.667% 22.959% 25.887% 22.058% 23.962% 23.203% 24.590% 26.058% 22.270% 22.959% 23.883% 25.782% 20.914% 23.457% 26.681% 24.625% 25.057% 19.918% 22.356% 24.209% 24.934% 22.186% 34.172% 34.759% 33.041% 32.428% 35.865% 39.386% 34.439% 35.343% 30.891% 35.004% 100.000% 46.763% 40.421% 31.841% 34.131% 34.394% 31.725% 36.947% 26.440% 23.897% 26.391% 30.173% 28.016% 32.710% 31.950% 30.921% 30.913% 34.623% 31.997% 28.267% 31.604% 26.144% 28.519% 29.014% 31.090% 32.279% 27.789% 28.335% 29.102% 31.543% 24.617% 27.167% 32.482% 29.237% 30.593% 24.375% 26.974% 29.150% 30.715% 26.558% Distance=0.005 39.248% 39.735% 37.850% 37.878% 40.904% 44.977% 39.675% 41.310% 35.385% 39.285% 100.000% 50.326% 44.675% 36.063% 38.221% 37.511% 35.172% 40.531% 30.323% 26.791% 29.793% 34.071% 31.764% 37.723% 36.788% 36.506% 35.188% 39.813% 37.619% 33.533% 36.566% 30.011% 32.447% 33.958% 35.857% 37.064% 32.945% 32.907% 33.696% 36.331% 27.692% 30.451% 37.598% 33.257% 35.367% 28.296% 30.915% 33.907% 35.725% 30.874% Lambda 3.946 3.947 3.948 3.949 3.95 3.951 3.952 3.953 3.954 3.955 3.956 3.957 3.958 3.959 3.96 3.961 3.962 3.963 3.964 3.965 3.966 3.967 3.968 3.969 3.97 3.971 3.972 3.973 3.974 3.975 3.976 3.977 3.978 3.979 3.98 3.981 3.982 3.983 3.984 3.985 3.986 3.987 3.988 3.989 3.99 3.991 3.992 3.993 3.994 3.995 Distance=0.001 11.660% 10.886% 10.387% 10.060% 10.093% 12.020% 10.961% 11.139% 11.662% 11.402% 10.884% 11.915% 12.421% 13.357% 15.301% 100.000% 21.701% 15.412% 12.740% 13.838% 12.952% 13.293% 11.789% 18.621% 13.402% 11.032% 11.635% 9.554% 10.257% 10.273% 11.051% 10.605% 12.640% 11.406% 9.573% 10.704% 10.012% 9.368% 11.181% 10.786% 9.975% 10.622% 10.737% 10.076% 11.096% 13.146% 10.351% 10.662% 8.764% 10.314% Distance=0.002 17.984% 18.989% 18.043% 16.775% 16.664% 16.106% 18.592% 18.172% 18.078% 18.449% 17.100% 19.095% 19.534% 19.646% 17.957% 100.000% 28.389% 21.861% 20.544% 21.677% 20.737% 20.261% 19.399% 22.196% 21.221% 18.838% 18.408% 15.910% 17.418% 16.473% 16.776% 18.446% 18.441% 18.868% 16.219% 17.373% 16.015% 15.311% 18.018% 18.365% 16.063% 16.728% 16.644% 17.315% 17.532% 19.986% 16.700% 18.123% 14.792% 16.976% Distance=0.003 Distance=0.004 23.409% 24.226% 24.382% 21.844% 22.407% 19.441% 24.804% 24.199% 23.472% 23.713% 22.530% 24.753% 24.335% 24.753% 19.826% 100.000% 32.755% 26.642% 26.981% 27.923% 26.796% 26.338% 25.559% 24.705% 26.995% 25.642% 24.798% 21.689% 24.189% 21.691% 21.345% 25.573% 23.757% 25.370% 22.224% 22.967% 21.104% 20.941% 23.114% 23.915% 21.749% 22.054% 21.608% 22.869% 23.400% 24.918% 21.708% 23.444% 20.181% 21.763% 28.567% 28.491% 29.696% 26.268% 27.042% 22.476% 30.094% 29.678% 28.263% 28.243% 25.901% 29.805% 28.539% 29.165% 21.318% 100.000% 35.776% 30.806% 31.597% 32.764% 31.301% 31.613% 30.969% 27.024% 31.924% 31.389% 29.906% 27.105% 30.598% 26.638% 24.875% 30.175% 28.821% 30.077% 27.885% 28.421% 25.568% 26.114% 28.145% 28.830% 26.829% 27.317% 26.441% 27.082% 28.696% 29.540% 26.466% 28.024% 25.198% 26.443% Distance=0.005 33.141% 32.647% 34.774% 30.536% 30.844% 24.913% 34.147% 34.949% 31.844% 32.350% 29.080% 34.571% 32.305% 32.998% 22.609% 100.000% 38.467% 34.554% 35.424% 37.474% 35.134% 35.300% 35.675% 29.071% 36.108% 36.836% 33.963% 31.858% 35.986% 31.529% 28.351% 34.085% 33.201% 33.621% 32.519% 33.379% 29.999% 31.106% 33.064% 32.959% 31.592% 32.187% 31.037% 31.006% 32.060% 33.863% 31.061% 32.195% 29.885% 30.834% Lambda 3.996 3.997 3.998 3.999 4 Distance=0.001 8.328% 8.787% 10.138% 9.458% 4.041% Distance=0.002 13.162% 15.014% 16.566% 14.903% 5.766% Distance=0.003 Distance=0.004 17.261% 20.932% 22.420% 20.114% 7.033% 21.078% 26.299% 26.540% 24.105% 8.125% Distance=0.005 24.933% 31.509% 30.233% 27.988% 9.095% Percentage of Points within certain Distance of Curves 100.000% 120.000% 20.000% 40.000% 60.000% 80.000% 0.000% 3.55 3.56 3.58 3.59 3.60 3.62 3.63 3.64 3.65 3.67 3.68 3.69 3.71 3.72 3.73 3.75 3.76 3.77 3.78 3.80 3.81 3.82 3.84 3.85 3.86 3.88 3.89 3.90 3.91 3.93 3.94 3.95 3.97 3.98 3.99 Lambda Percentage Closeness vs. Lambda 0.001 0.002 0.003 0.004 0.005