ChaosSim

A sophisticated chaos simulation software utilizing the Wolfram Programming Language to model randomized chaotic systems through mathematical principles.

Overview

ChaosSim combines Bernoulli numbers, Fibonacci sequences, and game-sum theory (Nash equilibrium) to simulate and visualize complex chaotic patterns and behaviors in mathematical systems.

Features

• Bernoulli Number Integration: Leverage Bernoulli numbers for probabilistic chaos modeling
• Fibonacci-Based Patterns: Generate chaotic sequences based on Fibonacci number properties
• Nash Equilibrium Analysis: Apply game theory principles to simulate equilibrium states in chaotic systems
• Advanced Visualizations: Create stunning visual representations of chaotic patterns
• Customizable Parameters: Adjust simulation parameters for different chaos scenarios

Requirements

• Wolfram Mathematica (version 12.0 or higher recommended)
• Wolfram Engine or Wolfram Desktop

Project Structure

ChaosSim/
├── README.md                  # Project documentation
├── ChaosSim.nb               # Main simulation notebook
├── MathUtils.wl              # Mathematical utility functions
├── Visualizations.nb         # Visualization examples
└── Examples.nb               # Sample simulations

Getting Started

1. Open ChaosSim.nb in Wolfram Mathematica
2. Evaluate all cells to initialize the simulation environment
3. Explore different chaos scenarios by adjusting parameters
4. Check Examples.nb for pre-built simulation demonstrations

Usage

Basic Chaos Simulation

(* Generate Bernoulli-based chaos *)
bernoullliChaos = SimulateBernoulliChaos[iterations, complexity]

(* Create Fibonacci pattern *)
fibonacciPattern = GenerateFibonacciChaos[depth, variance]

(* Analyze Nash equilibrium *)
nashState = AnalyzeNashEquilibrium[payoffMatrix, players]

Mathematical Foundation

Bernoulli Numbers

Used for generating probabilistic distributions in chaos modeling, providing smooth transitions between chaotic states.

Fibonacci Sequences

Creates self-similar patterns and golden ratio-based chaos structures, fundamental to natural chaotic systems.

Nash Equilibrium

Models strategic interactions in multi-agent chaotic systems, determining stable states in game-theoretic scenarios.

Examples

See Examples.nb for complete demonstrations including:

• Multi-dimensional chaos attractors
• Bernoulli-weighted random walks
• Fibonacci spiral chaos patterns
• Game-theoretic equilibrium in chaotic markets

License

MIT License - Feel free to use and modify for your research and projects.

Contributing

Contributions are welcome! Please feel free to submit pull requests or open issues for bugs and feature requests.

Author

Created for advanced chaos theory research and mathematical simulation.