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Lorenz System

Lorenz System
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2019-05-01 05:08
CalculusGraphing
This example demonstrates the Lorenz system, a classic example of a chaotic dynamical system. It involves ordinary differential equations in calculus. Using GeoGebra commands like NSolveODE, Length, Sequence, etc., the system's trajectory can be dynamically displayed. Suitable for high school and above students to learn about chaos theory and differential equations.

GeoGebra Commands & Parameters

  1. Textfield(A0) → InputBox5
  2. Textfield(B0) → InputBox6
  3. Textfield(β) → InputBox1
  4. Textfield(σ) → InputBox2
  5. Textfield(ρ) → InputBox3
  6. Textfield(tmax) → InputBox4
  7. NSolveODE({x', y', z'}, 0, {x(A0), y(A0), z(A0)}, tmax) → numericalIntegral1, numericalIntegral2, numericalIntegral3
  8. Length(numericalIntegral1) → len
  9. NSolveODE({x', y', z'}, 0, {x(B0), y(B0), z(B0)}, tmax) → numericalIntegral4, numericalIntegral5, numericalIntegral6
  10. Sequence((y(Point[numericalIntegral1, i]), y(Point[numericalIntegral2, i]), y(Point[numericalIntegral3, i])), i, 0, 1, 1 / len) → LA
  11. Element(LA, (n * len) + 1) → At
  12. Sequence((y(Point[numericalIntegral4, i]), y(Point[numericalIntegral5, i]), y(Point[numericalIntegral6, i])), i, 0, 1, 1 / len) → LB
  13. Element(LB, (n * len) + 1) → Bt
  14. Point({t, x(At)}) → xA
  15. Point({t, y(At)}) → yA
  16. Point({t, z(At)}) → zA
  17. Point({t, x(Bt)}) → xB
  18. Point({t, y(Bt)}) → yB
  19. Point({t, z(Bt)}) → zB
  20. Sequence(Segment[Element[LA, j], Element[LA, j + 1]], j, 1, len) → LAconnected
  21. Sequence(Segment[Element[LB, j], Element[LB, j + 1]], j, 1, len) → LBconnected

AI Drawing Prompt Reference

Create a 3D phase-space plot of the Lorenz system showing two chaotic trajectories starting from two distinct initial points. Each trajectory is a polyline formed by connected discrete points, winding through space. Additionally, project each coordinate (x, y, z) of both trajectories onto separate 2D plots over time, resulting in six projection curves. The image should highlight the sensitive dependence on initial conditions and the butterfly-like attractor structure.
The prompt above was compiled from the original drawing commands and is for reference only.