Arnold’s Cat Map — A Study in Chaos and Dynamical Systems

Algebra & Coding Project
National Higher School of Mathematics
Marouf Haider
Bahidj Nafaa
📅 Academic Year: 2023–2024

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🔍 Project Overview

Arnold’s Cat Map is a discrete dynamical system known for exhibiting chaotic behavior. It transforms a unit square using a linear transformation followed by a modular reduction, resulting in image scrambling and periodicity.

This project explores:

• The mathematical foundations of the Cat Map
• Its behavior on the torus and images
• Periodicity and its connection to Fibonacci numbers
• A Python implementation for image encoding, decoding, and minimal period calculation

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📄 Contents

• report.pdf: Full academic write-up with theory, proofs, and visuals
• Implementation.ipynb: Jupyter notebook with:
  • Image encoding using Arnold's Cat Map
  • Calculation of minimal period
  • Image retrieval using reverse iterations
• assets: A folder with all test units used (including the one used in the report).
• outputs: Folder containing the results of applying the encoding to the test units.
• README.md: This file.

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🧠 Key Concepts

• Discrete Dynamical Systems
• Chaos Theory
• Matrix Maps on the Torus
• Image Scrambling and Periodicity
• Connections to Fibonacci and Pisano Periods

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💻 Technologies

• Python 3
• NumPy
• OpenCV (cv2)
• Jupyter Notebook

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How to Run the Code

1. Clone this repo or download the .ipynb notebook.
2. Install required dependencies:

   pip install numpy opencv-python

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🔬 Test Run

This repo includes an example input and output to demonstrate functionality.

• 📥 Input image (240*240): -📤 Output result (10 iterations):
• 📤 Retrieved image:

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📧 Contacts

• Haider Marouf-(haider.marouf@nhsm.edu.dz)/(ensmmarouf@gmail.com).
• Bahidj Nafaa-(N/A).

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Feel free to open issues for suggestions or comments. Contributions are welcome.