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Julia Set Boundaries: Visualizing Stability in Complex Iteration

Julia Set Boundaries: Visualizing Stability in Complex Iteration
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2016-10-20 22:11
FunctionComplex NumberSlider AnimationLocusGraphing
① This example dynamically demonstrates the boundary of stability for the iteration z → z² + C by moving the complex parameter C, revealing the Julia set's fractal edge. ② Core mathematical concepts include complex numbers, iteration, dynamical systems, stability vs. chaos, and fractal geometry. ③ Construction highlights: GeoGebra objects such as A, C, and a series of points A1–A38 are used to trace iteration paths; a slider controls C, updating the boundary in real time to show how stable regions change with the parameter. ④ Suitable for high school to university-level teaching in mathematics or computer graphics, helping students understand complex dynamics, fractal structures, and parameter sensitivity, and serving as an exploratory tool for self-study.

GeoGebra Commands & Parameters

  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))