---
license: mit
language:
- en
library_name: chaossim
tags:
- chaos-theory
- mathematics
- simulation
- game-theory
- fibonacci
- bernoulli
- nash-equilibrium
- dynamical-systems
---
# 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

```mathematica
(* 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.