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Lorenz-84 Attractor

Lorenz-84 Attractor
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2017-08-30 19:11
VectorFunctionCalculusParametric EquationGeometric Construction
In collections UQ-Applied Mathematical Analysis
This example demonstrates the three-dimensional dynamic trajectory of the Lorenz-84 attractor, showcasing the complex motion patterns of chaotic systems. Core mathematical concepts include differential equations, parametric equations, and 3D spatial visualization. Construction highlights from the GeoGebra command summary involve objects like buttons (button1, button2, etc.), sliders (strength, n, etc.), variables (t, A, k1, j1, etc.), and function definitions (f, g, F, G, H). Suitable for high school students and above in mathematics and physics, as well as enthusiasts interested in chaos theory, for educational demonstration and exploration of how parameters influence system behavior.

GeoGebra Commands & Parameters

  1. Numeric strength(2)
  2. Numeric n(12)
  3. Numeric h(0.005)
  4. Numeric speed(1)
  5. Numeric t(1.1)
  6. Numeric b(4)
  7. x(CornerThreeD[-1, 3])
  8. x(CornerThreeD[-1, 1])
  9. y(CornerThreeD[-1, 3])
  10. y(CornerThreeD[-1, 1])
  11. z(CornerThreeD[-1, 5])
  12. z(CornerThreeD[-1, 3])
  13. {(1.698887608931714, 2.0504010462068276, 0.5760998599060181), (-0.32313175376458236, -2.9202990154971715, -0.18578885707181514), (-0.703865778294956, -1.9430964880606907, -0.6836063941861303), (-1.380
  14. {0.17970358504534012, 1.0858554568201608, 1.8656741461820978, -2.105328338669458, -0.09429440885513785, 1.1553698756657758, -3.8389210160410334, -0.6083397544876137, 0.7633749698157235, 0.740578123608
  15. {-15.374318016002096, 13.888842084295998, 14.593751230568177, -3.956097146486112, -12.745440615259469, 4.042605164879367, -6.132644574233076, -6.935372589792351, -0.8253948753445224, -3.25692887787045
  16. {7.046504078530084, -9.860146301731675, 5.395055257497853, 20.34802139700003, 11.332781222476173, 14.719495976770112, 6.647815896271328, 16.05799165648909, -0.13194602490549057, -0.304139995327789, 26
  17. {0.14263420190131537, 1.073177675798263, 1.8403453563275223, -2.171186219318251, -0.13727512694310473, 1.1145728801495793, -3.8385632775683156, -0.6590283959830945, 0.7706154521125699, 0.7497481302166
  18. {-15.726236659530038, 14.353924922114915, 14.448768875013045, -4.616472714708148, -13.21199693493403, 3.5084403417098216, -6.13440647299943, -7.485936756429744, -0.8324320543325774, -3.263152289459468
  19. {6.482256590243258, -9.381851393274065, 6.2411372386916195, 20.212863071054336, 10.911175078624257, 14.961235724562984, 6.4222146369011845, 15.86450484742283, -0.1182903059856168, -0.3302432585945161,
  20. {0.1502311220547381, 1.0795865572520538, 1.845259545686293, -2.1603127872217405, -0.12966825270830995, 1.1198349047290317, -3.835477678314005, -0.6512566394969888, 0.7705397797892708, 0.75006368309443
  21. {-15.70507786014791, 14.337310623421345, 14.401281785974415, -4.613770116461824, -13.197670537907337, 3.4966889075101313, -6.131417587875186, -7.480774067436194, -0.8325618877186312, -3.26307527780630
  22. {6.465423640740355, -9.361284829889847, 6.240397848061697, 20.188910928780572, 10.891556635928817, 14.94125097106652, 6.422583295873837, 15.844231563426336, -0.11821747880329625, -0.33015616596802744,
  23. {0.1138063390998365, 1.0661571733780622, 1.8190163531705332, -2.2258139322216843, -0.17212378235391235, 1.0786380161723486, -3.834105788804237, -0.7016486354797227, 0.7776879484214403, 0.7592771377217
  24. {-16.034813147968713, 14.785603337838339, 14.207710131870245, -5.265389160877023, -13.648449476125858, 2.9506561786623635, -6.127283968118942, -8.023809956148822, -0.8397332219654894, -3.2692749213130
  25. {5.885793998503074, -8.862409810988387, 7.083563768640467, 20.030457222642156, 10.451626583236497, 15.162952864578756, 6.199377096016536, 15.631318733389202, -0.10456024123868085, -0.35641582464420085
  26. Numeric a(0.25)
  27. Numeric f(8)
  28. Numeric g(1)
  29. F(x, y, z) = (strength * (((-a) * x) - (y * y) - (z * z) + (a * f)))
  30. G(x, y, z) = (strength * ((-y) + (x * y) - ((b * x) * z) + g))
  31. H(x, y, z) = (strength * ((-z) + ((b * x) * y) + (x * z)))

AI Drawing Prompt Reference

Draw a three-dimensional dynamic trajectory showing the complex motion patterns of the Lorenz-84 attractor, defined by parametric equations involving variables x, y, z and parameters like strength, n, h, etc., demonstrating chaotic system behavior in 3D space with a coordinate system background.
The prompt above was compiled from the original drawing commands and is for reference only.