Computer Simulation on the Gumowski-Mira Transformation

摘要: We perform a computer simulation on the Gumowski-Mira transformation (hereafter abbreviated as GM tranformation) and present variety of 2-dimensional patterns obtained from transformation. Among these patterns, there are images which resemble very much “living marine creatures”. 1. Definition Transformation One encounters various nonlinear phenomena in nature. Computer provides powerful tool understanding such (KINZEL REENTS, 1998). In this study, we (GUMOWSKI MIRA, 1980) images, (cross sections of) The is discrete dynamic system, expressed by following recurrent formula: x y f n + = − 1 2 0 05 ( . ) ), μ 1.1. with 0.008 following, set 0.008. Changing value μ, one obtains diversified attractors x-y plane above changing – 1.0 until 0.9 122 K. OTSUBO et al. step ∆μ 0.1, shown order Fig. more detailed dependence can be seen 2, 0.01 range 0.2 ≤ 0.29. Still 3, 0.0001 0.29 0.2909. Figures 4 5 represent 0.00001 Fig.1. 0.008, =– ~ 0.9, (step size) 0.1 x1 y1 0.1. 3. pattern 0.2909 0.0001. 2. 0.01. Simulation 123 0.29009, 0.0000001 0.2900009, respectively. From pictures presented above, see that even small change causes rather drastic changes patterns. Resulting depend sensitively μ. Figure 6 represents other values (– 0.8 0.71). 6. 0.71 5. 0.2900009 0.0000001. 4. 0.29009 0.00001. 124 Patterns Next 0. case examine how look like 7–13 several 8–12 slice tomato, wings butterfly, black cut end an orange, plankton, A Few Remarks 2.1. Possible realization nature It interesting to notice remind us some kind creatures” jellyfish, starfish, or plankton. widely recognized fractal geometry plays crucial role forming self-similar objects fern leaf, ramification tree, coastline so forth. Analogously tempted assume shapes appear considerations, argue mathematics complexity systems chaos underlies living world. 7. 0.7, 0.5. 8. 0, 0.15, 9. 0.2, 0.5 10. 0.22 11. 0.31, 12. 0.55, 13. 0.23 125 2.2. Bifurcation chart 14 bifurcation chart, vicinity 1.0004. This picture similar logistic map 2.3. 3-Dimensional image (x, y, μ) space Next, show 3-dimensional superimposing drawn size 0.04 (Fig. 15). y) fixed themselves cross-sections image, possess symmetries not easy imagine image. Concluding Remark As easily contrive transformations, transformation, study types transformations might realize world should worthy. emphasize help get deeper insight into hidden (STEWART, 15. 14. 1.0004, 126