Part of June 22, 2010 Handout
MORE ON SIMPLY CHAOTIC *
Below are alternate ways to express the behavior of the equation: x[n+1] = r*x[n]*(1-x[n])
Chaotic Behavior (Values stay close until the 4th step, then they diverge) |
Non-Chaotic Oscillatory Behavior (By the 15th step the values become identical) |
Results for r=4.000, x=0.600 (m) at 0 x = 0.60100, 0.60000, 0.59900 at 1 x = 0.95920, 0.96000, 0.96080 at 2 x = 0.15656, 0.15360, 0.15067 at 3 x = 0.52819, 0.52003, 0.51187 at 4 x = 0.99682, 0.99840, 0.99944 at 5 x = 0.01267, 0.00641, 0.00225 at 6 x = 0.05004, 0.02547, 0.00899 at 7 x = 0.19014, 0.09927, 0.03564 at 8 x = 0.61594, 0.35767, 0.13748 at 9 x = 0.94623, 0.91897, 0.47431 at 10 x = 0.20352, 0.29786, 0.99736 at 11 x = 0.64840, 0.83656, 0.01053 at 12 x = 0.91191, 0.54692, 0.04169 at 13 x = 0.32131, 0.99120, 0.15979 at 14 x = 0.87227, 0.03491, 0.53704 at 15 x = 0.44565, 0.13476, 0.99451 at 16 x = 0.98818, 0.46640, 0.02183 |
Results for r=3.500, x=0.600 (m) at 0 x = 0.60100, 0.60000, 0.59900 at 1 x = 0.83930, 0.84000, 0.84070 at 2 x = 0.47207, 0.47040, 0.46874 at 3 x = 0.87227, 0.87193, 0.87158 at 4 x = 0.38995, 0.39083, 0.39175 at 5 x = 0.83261, 0.83329, 0.83399 at 6 x = 0.48779, 0.48622, 0.48459 at 7 x = 0.87448, 0.87434, 0.87417 at 8 x = 0.38418, 0.38456, 0.38499 at 9 x = 0.82805, 0.82835, 0.82871 at 10 x = 0.49834, 0.49764, 0.49683 at 11 x = 0.87499, 0.87498, 0.87496 at 12 x = 0.38284, 0.38286, 0.38290 at 13 x = 0.82696, 0.82698, 0.82701 at 14 x = 0.50085, 0.50080, 0.50072 at 15 x = 0.87500, 0.87500, 0.87500 at 16 x = 0.38282, 0.38282, 0.38282 |
Another Way of Looking at the System
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On the left diagram values of x move away from the point of equilibrium (marked by a circle) while on the right diagram the values of x move towards equilibrium.
* Copyright ©2010 by Theo Pavlidis