Logistic Map — Bifurcation Diagram (C++ Simulation)

This project is a small exploration of the logistic map and its bifurcation behaviour.
I used C++ to simulate the system, iterate over a range of r values, and generate the bifurcation diagram.

The logistic map is defined by the iterative equation:

x_{n+1} = r * x_n * (1 - x_n)

This simple recurrence moves from stable behaviour → oscillations → period-doubling → chaos.
Inside the chaotic region, you can also see periodic windows, band-merging points, and regions where the system becomes stable again.

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Output Files

All generated files (PNG, PPM, CSV) have been included as a compressed archive:

• outputs.zip

Inside the ZIP:

• bifurcation.png — final image
• bifurcation.ppm — raw PPM output
• bifurcation.csv — (r, x) data from the simulation

The PNG is the main visualization.
The raw files are included for completeness.
Note: A copy of the bifurcation PNG is kept in the out/ folder for easy access.

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Build & Run

Compile (requires g++)

g++ -O2 logistic_bifurcation.cpp -o logistic_bifurcation

Run

./logistic_bifurcation

This will generate fresh versions of:

• bifurcation.ppm
• bifurcation.csv

You can convert the PPM to PNG using ImageMagick: magick bifurcation.ppm -resize 4000x2500 -gamma 0.9 -sharpen 0x1 bifurcation.png

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What to Look For in the Diagram

• A long stable region before chaos begins
• Period-doubling leading into chaos
• Periodic windows (e.g., the period-3 window)
• Chaotic bands merging at specific r-values (band-merging points)
• Stable regions appearing inside chaos

These patterns also appear in real systems like population models, stock markets, control loops, neural networks, and weather models.

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Notes

• This project was done while exploring dynamical systems out of personal curiosity.
• The code is kept simple on purpose for clarity.
• The ZIP file contains all the outputs so the simulation does not need to be rerun.

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