🌀 Lorenz Attractor Visualizer

An interactive Lorenz attractor simulator in Unity 🎮, designed to explore chaos, the butterfly effect, and velocity dynamics in a visually intuitive way.

butt.mp4

Recording.2025-10-28.191557.mp4

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🔬 Features

• Numerical Integration: Uses RK4 (Runge-Kutta 4th order) for accurate time evolution of the Lorenz system.
• Dynamic Mesh Rendering: The attractor trail is drawn with a Mesh rather than a LineRenderer for performance (even with 25k+ points).
• Per-Point Velocity Coloring:
  Visualize instantaneous velocity magnitude along the trail using a customizable Gradient 🎨.
• Butterfly Effect Mode 🦋:
  Slight rounding of initial conditions demonstrates sensitive dependence on initial conditions — chaos in action!
• Compare Two Attractors:
  Spawn multiple attractors (normal vs. butterfly-effect) to observe divergence over time.
• Orbit Camera:
  Smooth, mouse-controlled orbit camera that automatically centers on the active attractor points.

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⚙️ Usage

1. Setup Attractor:

   • Create a ScriptableObject (LorenzAttractorSettings) for each attractor.
   • Configure:
     • timeStep
     • initialPosition
     • maxPoints
     • velocityGradient
     • enableButterflyEffect ✅ to apply rounding to initial conditions.

2. Add Attractor to Scene:

   • Attach LorenzAttractor.cs to a GameObject.
   • Assign the corresponding LorenzAttractorSettings.

3. Camera Setup:

   • Attach OrbitCamera.cs to your main camera.
   • Set target to one of the Lorenz attractor GameObjects.
   • Enable Use Dynamic Center to follow the attractor trail.

4. Visual Exploration:

   • Drag the mouse to rotate the camera.
   • Scroll to zoom in/out.
   • Watch attractors diverge in real time, colored by velocity magnitude.

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📊 Scientific Notes

• The Lorenz system is defined by:

[ \begin{cases} \frac{dx}{dt} = \sigma (y - x) \ \frac{dy}{dt} = x (\rho - z) - y \ \frac{dz}{dt} = xy - \beta z \end{cases} ]

with typical parameters:
σ = 10, ρ = 28, β = 8/3.

• Butterfly Effect: Small differences in initial conditions (here via rounding) grow exponentially, illustrating chaotic dynamics.
• Velocity Visualization:
  Each vertex is colored according to the magnitude of its instantaneous derivative, showing where the system moves faster or slower in 3D space.

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🧩 Scripts

• LorenzAttractor.cs — Generates and updates the attractor mesh in real-time.
• LorenzAttractorEquationsSolver.cs — Implements RK4 integration and derivative calculation.
• LorenzAttractorSettings.cs — ScriptableObject storing parameters, including butterfly effect option.
• OrbitCamera.cs — Smooth orbit camera centered on the attractor’s active points.

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🚀 Demo Ideas

• Spawn two attractors:
  • One normal
  • One with butterfly effect enabled
• Observe divergence over time 🦋.
• Experiment with gradient coloring to highlight high-velocity regions.
• Increase maxPoints for high-resolution trails.

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💻 Requirements

• Unity 2023+ (or compatible version)
• Standard 3D pipeline
• Optional: HDRP/URP for improved visuals