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Additional resources for Strange Attractors. Creating Patterns in Chaos
Sample text
With m not equal to n, the horizontal and vertical dimensions are scaled differently, but since the computer rescales the attractor to fit the screen, the visual result is the same. These degeneracies show that there are many ways to code a particular attractor. Although this is true, there are so many different possible combinations of coefficients that it is very unlikely that two degenerate cases will be found spontaneously. Thus the examples displayed in this chapter represent but a tiny fraction of the possibilities, and you will be generating many other cases, almost none of which have been seen before.
Changes required in PROG03 to search for strange attractors in one-dimensional maps of order up to OMAX% 1000 REM ONE-D MAP SEARCH (Polynomials up to 5th Order) 1060 OMAX% = 5 'Maximum order of polynomial 1720 XNEW = A(O% + 1) 1730 FOR I% = O% TO 1 STEP -1 1830 XNEW = A(I%) + XNEW * X 1930 NEXT I% 2650 O% = 2 + INT((OMAX% - 1) * RND) 2660 CODE$ = CHR$(63 + O%) 2680 M% = O% + 1 41 2910 DF = 0 2930 FOR I% = O% TO 1 STEP -1 2940 DF = I% * A(I% + 1) + DF * X 2970 NEXT I% 3000 DF = ABS(DF) PROG04 produces an interesting array of shapes, samples of which are shown in Figures 2-5 through 2-10.
Furthermore, the magnitude of the negative exponent has to be greater than the positive one so that initial conditions scattered throughout the basin of attraction contract onto an attractor that occupies a negligible portion of the plane. The area of the ellipse continually decreases even as it stretches to an infinite length. There is a proper way to calculate both of the Lyapunov exponents. For the mathematically inclined, the procedure involves summing the logarithms of the eigenvalues of the Jacobian matrix of the linearized transformation with occasional Gram-Schmidt reorthonormalization.