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朱利亚集合边界：复迭代稳定性可视化

朱利亚集合边界：复迭代稳定性可视化
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2016-10-20 22:11
函数复数滑块动画轨迹函数图像
① 本范例通过动态调整复数参数C，直观展示迭代z→z²+C的朱利亚集合边界，即复平面上点经迭代后是否趋于稳定的分界线。② 核心数学概念包括复平面、迭代、动力系统、稳定与混沌边界，以及朱利亚集合的经典分形结构。③ 作图要点：利用GeoGebra的复数对象（如A、C等）和大量点对象（A1至A38）追踪迭代轨迹，通过滑块控制C值，实时更新边界形态，展示不同C值下稳定区域的变化。④ 适合高中至大学阶段的数学或计算机图形学教学，用于理解复动力系统、分形几何以及参数变化对迭代行为的影响，也可作为探究性学习工具。

GeoGebra 命令及参数

  1. Point A(0.6020044513897775, 1.9253924034948553)
  2. A
  3. Point C(0.054085595542901974, 0.057278348579523214)
  4. (((Random[0, 1] * 2) - 1) * sqrt(A1 - C))
  5. (((Random[0, 1] * 2) - 1) * sqrt(A2 - C))
  6. (((Random[0, 1] * 2) - 1) * sqrt(A3 - C))
  7. (((Random[0, 1] * 2) - 1) * sqrt(A4 - C))
  8. (((Random[0, 1] * 2) - 1) * sqrt(A5 - C))
  9. (((Random[0, 1] * 2) - 1) * sqrt(A6 - C))
  10. (((Random[0, 1] * 2) - 1) * sqrt(A7 - C))
  11. (((Random[0, 1] * 2) - 1) * sqrt(A8 - C))
  12. (((Random[0, 1] * 2) - 1) * sqrt(A9 - C))
  13. (((Random[0, 1] * 2) - 1) * sqrt(A10 - C))
  14. (((Random[0, 1] * 2) - 1) * sqrt(A11 - C))
  15. (((Random[0, 1] * 2) - 1) * sqrt(A12 - C))
  16. (((Random[0, 1] * 2) - 1) * sqrt(A13 - C))
  17. (((Random[0, 1] * 2) - 1) * sqrt(A14 - C))
  18. (((Random[0, 1] * 2) - 1) * sqrt(A15 - C))
  19. (((Random[0, 1] * 2) - 1) * sqrt(A16 - C))
  20. (((Random[0, 1] * 2) - 1) * sqrt(A17 - C))
  21. (((Random[0, 1] * 2) - 1) * sqrt(A18 - C))
  22. (((Random[0, 1] * 2) - 1) * sqrt(A19 - C))
  23. (((Random[0, 1] * 2) - 1) * sqrt(A20 - C))
  24. (((Random[0, 1] * 2) - 1) * sqrt(A21 - C))
  25. (((Random[0, 1] * 2) - 1) * sqrt(A22 - C))
  26. (((Random[0, 1] * 2) - 1) * sqrt(A23 - C))
  27. (((Random[0, 1] * 2) - 1) * sqrt(A24 - C))
  28. (((Random[0, 1] * 2) - 1) * sqrt(A25 - C))
  29. (((Random[0, 1] * 2) - 1) * sqrt(A26 - C))
  30. (((Random[0, 1] * 2) - 1) * sqrt(A27 - C))
  31. (((Random[0, 1] * 2) - 1) * sqrt(A28 - C))
  32. (((Random[0, 1] * 2) - 1) * sqrt(A29 - C))
  33. (((Random[0, 1] * 2) - 1) * sqrt(A30 - C))
  34. (((Random[0, 1] * 2) - 1) * sqrt(A31 - C))
  35. (((Random[0, 1] * 2) - 1) * sqrt(A32 - C))
  36. (((Random[0, 1] * 2) - 1) * sqrt(A33 - C))
  37. (((Random[0, 1] * 2) - 1) * sqrt(A34 - C))
  38. (((Random[0, 1] * 2) - 1) * sqrt(A35 - C))
  39. (((Random[0, 1] * 2) - 1) * sqrt(A36 - C))
  40. (((Random[0, 1] * 2) - 1) * sqrt(A37 - C))
  41. (((Random[0, 1] * 2) - 1) * sqrt(A38 - C))
  42. (((Random[0, 1] * 2) - 1) * sqrt(A39 - C))
  43. (((Random[0, 1] * 2) - 1) * sqrt(A40 - C))
  44. (((Random[0, 1] * 2) - 1) * sqrt(A41 - C))
  45. (((Random[0, 1] * 2) - 1) * sqrt(A42 - C))
  46. (((Random[0, 1] * 2) - 1) * sqrt(A43 - C))
  47. (((Random[0, 1] * 2) - 1) * sqrt(A44 - C))
  48. (((Random[0, 1] * 2) - 1) * sqrt(A45 - C))
  49. (((Random[0, 1] * 2) - 1) * sqrt(A46 - C))
  50. (((Random[0, 1] * 2) - 1) * sqrt(A47 - C))
  51. (((Random[0, 1] * 2) - 1) * sqrt(A48 - C))
  52. (((Random[0, 1] * 2) - 1) * sqrt(A49 - C))
  53. (((Random[0, 1] * 2) - 1) * sqrt(A50 - C))
  54. (((Random[0, 1] * 2) - 1) * sqrt(A51 - C))
  55. (((Random[0, 1] * 2) - 1) * sqrt(A52 - C))
  56. (((Random[0, 1] * 2) - 1) * sqrt(A53 - C))
  57. (((Random[0, 1] * 2) - 1) * sqrt(A54 - C))
  58. (((Random[0, 1] * 2) - 1) * sqrt(A55 - C))
  59. (((Random[0, 1] * 2) - 1) * sqrt(A56 - C))
  60. (((Random[0, 1] * 2) - 1) * sqrt(A57 - C))