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Lorenz Attractor — Chaotic Dynamics Visualized (C++ Simulations)

This project is an exploration of the Lorenz system, one of the most iconic chaotic dynamical systems.
Using C++, I simulate the attractor in different ways:

• clean 2D projection

• velocity-colored rendering

• parameter sweeps over σ, ρ, β

Every program outputs frames or images in PPM format, which can be turned into PNGs or GIF animations.

Folder Structure

lorenz_attr/
│
├── lorenz_attr_clean/
│   ├── main.cpp
│   └── lorenz.ppm
│
├── lorenz_attr_vel/
│   ├── main.cpp
│   └── lorenz_speed.ppm
│
└── lorenz_attr_sweep/
    ├── beta_sweep/
    │   ├── main.cpp
    │   └── frame_000.ppm ...
    ├── rho_sweep/
    │   ├── main.cpp
    │   └── frame_000.ppm ...
    └── sigma_sweep/
        ├── main.cpp
        └── frame_000.ppm ...

Each folder contains one fully standalone simulation. .ppm files replaced by .png or .gif for easy viewing, they can still be generated by running the code.

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What the Lorenz System Represents

The Lorenz system is defined by:

dx/dt = σ (y − x)
dy/dt = x (ρ − z) − y
dz/dt = x y − β z

This simple set of equations produces:

• butterfly-shaped attractors

• sensitivity to initial conditions

• chaotic switching between “wings”

• fractal interior structures

• emergent order inside chaos

Every visualization in this repo is a different lens on the same underlying physics.

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Included Programs

1. lorenz_attr_clean

Simple 2D projection (x vs z).
Shows the pure shape of the attractor with white pixels on black background.

2. lorenz_attr_vel

Colors each point by instantaneous speed ‖dx/dt‖.
Reveals fast and slow regions of the dynamics.

3. lorenz_attr_sweep

Three sub-projects that animate the attractor as:

• σ varies

• ρ varies

• β varies

This produces a morphing butterfly — a visual story of how chaos emerges as parameters change.

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🔧 Build & Run

All programs follow the same pattern:

Compile

g++ -O2 main.cpp -o lorenz

Run

./lorenz

The program will output .ppm image(s).

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🎞 Converting Frames → GIF

If you generate animation frames (from parameter sweeps),
use ImageMagick:

magick -delay 6 -loop 0 frame_*.ppm lorenz.gif

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Notes

• All simulations use Euler integration unless otherwise noted.

• No external libraries required — everything is pixel-by-pixel rendered.

• The PPM format is used intentionally for simplicity and speed.

• These programs are part of a broader exploration of chaotic systems, numerical ODEs, and scientific visualization.

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