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296 chapter 121b=1.0b=4.0b=5.0X0–1–100Figure 12.3 A bifurcation plot for the Gaussian map. (Courtesy of W. Hager.)4. Print your x ∗ values to no more than three or four decimal places. You willnot be able to resolve more places than this on your plot, and this restrictionwill keep your output files smaller by permitting you to remove duplicates.It is hard to control the number of decimal places in the output with Java’sstandard print commands (although printf and DecimalFormat do permitcontrol). A simple approach is to multiply the x i values by 1000 and thenthrow away the part to the right of the decimal point. Because 0 ≤ x n ≤ 1, thismeans that 0 ≤ 100 ∗ x n ≤ 1000, and you can throw away the decimal part bycasting the resulting numbers as integers:Ix[i]= (int)(1000*x[i]) // Convert to 0 ≤ ints ≤ 1000You may then divide by 1000 if you want floating-point numbers.5. You also need to remove duplicate values of (x, µ) from your file (they justtake up space and plot on top of each other). You can do that in Unix/Linuxwith the sort -u command.6. Plot your file of x ∗ versus µ. Use small symbols for the points and do notconnect them.7. Enlarge sections of your plot and notice that a similar bifurcation diagramtends to be contained within each magnified portion (this is calledself-similarity).8. Look over the series of bifurcations occurring atµ k ≃ 3, 3.449, 3.544, 3.5644, 3.5688, 3.569692, 3.56989, ... . (12.13)The end of this series is a region of chaotic behavior.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 296