MODEL_CARD.md

metadata

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

ChaosSim: Advanced Chaos Simulation Framework

Simulating Randomized Chaotic Systems through Mathematical Principles

Model Description

ChaosSim is a sophisticated chaos simulation framework built with the Wolfram Programming Language that combines three fundamental mathematical concepts to model and visualize complex chaotic systems:

1. Bernoulli Numbers - For probabilistic chaos modeling with weighted distributions
2. Fibonacci Sequences - For self-similar patterns and golden ratio-based structures
3. Nash Equilibrium (Game Theory) - For strategic interactions in multi-agent chaotic systems

Model Architecture

The framework consists of four integrated components:

• Core Engine (ChaosSim.nb) - Main simulation algorithms
• Mathematical Utilities (MathUtils.wl) - Reusable mathematical functions package
• Visualization Suite (Visualizations.nb) - Advanced plotting and analysis tools
• Examples Library (Examples.nb) - 10+ practical demonstrations

Authors

• Andrew Magdy Kamal - Lead Developer & Mathematician
• Riemann Computing Inc. - Research & Development
• Openpeer AI - AI Integration & Optimization

Intended Uses

Primary Use Cases

1. Academic Research

   • Chaos theory investigation
   • Dynamical systems analysis
   • Game theory simulations
   • Mathematical modeling

2. Financial Modeling

   • Market volatility simulation
   • Risk assessment using chaotic patterns
   • Portfolio optimization with game theory

3. Complex Systems Analysis

   • Multi-agent behavior modeling
   • Equilibrium state prediction
   • Pattern recognition in chaotic data

4. Educational Purposes

   • Teaching chaos theory concepts
   • Demonstrating mathematical principles
   • Interactive learning environments

Out-of-Scope Uses

• Real-time prediction systems (chaos is inherently unpredictable)
• Critical infrastructure control (deterministic systems required)
• Medical diagnosis (not validated for clinical use)
• Financial advice (for research purposes only)

How to Use

Requirements

• Wolfram Mathematica 12.0 or higher
• Wolfram Engine or Wolfram Desktop
• Basic understanding of chaos theory and mathematics

Quick Start

(* Load ChaosSim *)
Get["ChaosSim.nb"]

(* Generate Bernoulli-based chaos *)
bernoulliData = SimulateBernoulliChaos[500, 12];
PlotBernoulliChaos[bernoulliData]

(* Create Fibonacci golden spiral *)
spiralPoints = FibonacciSpiral3D[20, 100];
Plot3DChaos[spiralPoints]

(* Find Nash equilibrium *)
payoff1 = {{3, 0}, {5, 1}};
payoff2 = {{3, 5}, {0, 1}};
equilibria = FindNashEquilibrium[payoff1, payoff2]

(* Run unified chaos simulation *)
unifiedChaos = UnifiedChaosSimulation[400];
correlations = ChaosCorrelationAnalysis[unifiedChaos]

Example: Multi-Agent Chaos System

(* Simulate 5 agents seeking equilibrium *)
chaos = MultiAgentChaosEquilibrium[5, 200];

(* Visualize agent behavior *)
VisualizeMultiAgentChaos[5, 200]

Example: Chaotic Market Simulation

(* Simulate 250 days of market chaos *)
marketPrices = SimulateChaoticMarket[250, 100.0];

(* Analyze price evolution *)
ListLinePlot[marketPrices, 
  PlotLabel -> "Chaotic Market Prices",
  AxesLabel -> {"Day", "Price"}]

Mathematical Foundation

Bernoulli Numbers

Bernoulli numbers $B_n$ are used to create weighted probability distributions:

$$B_0=1,B_1=-\frac{1}{2},B_2=\frac{1}{6},B_4=-\frac{1}{30},\dots B\_0\; =\; 1,\; \backslash quad\; B\_1\; =\; -\backslash frac\{1\}\{2\},\; \backslash quad\; B\_2\; =\; \backslash frac\{1\}\{6\},\; \backslash quad\; B\_4\; =\; -\backslash frac\{1\}\{30\},\; \backslash ldots$$

The chaos weight function:

$$w(n)=\mathrm{\mid }{B}_n\mathrm{\mid }\text{ (normalized)}w(n)\; =\; |B\_n|\; \backslash text\{\; (normalized)\}$$

Fibonacci Sequences

The Fibonacci sequence creates self-similar patterns:

$$F_n=F_{n-1}+F_{n-2},F_0=0,F_1=1F\_n\; =\; F\_\{n-1\}\; +\; F\_\{n-2\},\; \backslash quad\; F\_0\; =\; 0,\; F\_1\; =\; 1$$

Golden ratio approximation:

$$\varphi \approx \underset{n\to \mathrm{\infty }}{\lim }\frac{F_{n+1}}{F_n}=\frac{1+\sqrt{5}}{2}\approx 1.618\backslash phi\; \backslash approx\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash frac\{F\_\{n+1\}\}\{F\_n\}\; =\; \backslash frac\{1\; +\; \backslash sqrt\{5\}\}\{2\}\; \backslash approx\; 1.618$$

Nash Equilibrium

A strategy profile $(s_1^*, s_2^*)$ is a Nash equilibrium if:

$$u_1(s_1^{\ast },s_2^{\ast })\ge u_1(s_1,s_2^{\ast })\mathrm{\forall }{s}_{1}u\_1(s\_1^*,\; s\_2^*)\; \backslash geq\; u\_1(s\_1,\; s\_2^*)\; \backslash quad\; \backslash forall\; s\_1$$ $$u_2(s_1^{\ast },s_2^{\ast })\ge u_2(s_1^{\ast },s_2)\mathrm{\forall }{s}_{2}u\_2(s\_1^*,\; s\_2^*)\; \backslash geq\; u\_2(s\_1^*,\; s\_2)\; \backslash quad\; \backslash forall\; s\_2$$

Where $u_i$ represents the utility function for player $i$.

Key Features

Chaos Generation Methods

Method	Description	Primary Use
BernoulliChaos	Weighted probabilistic chaos	Non-uniform distributions
FibonacciChaos	Golden ratio-based patterns	Natural chaotic structures
NashChaos	Game-theoretic equilibrium	Multi-agent systems
UnifiedChaos	Combined approach	Complex system modeling

Analysis Tools

• Shannon Entropy - Measure chaos complexity
• Lyapunov Exponent - Quantify sensitivity to initial conditions
• Hurst Exponent - Analyze long-range dependencies
• Correlation Dimension - Determine fractal properties
• Phase Space Analysis - Visualize attractor structures

Visualization Capabilities

• 2D/3D time series plots
• Phase space diagrams
• Bifurcation diagrams
• 3D attractors with color mapping
• Interactive parameter exploration
• Correlation matrices
• Multi-agent behavior tracking

Performance Metrics

Computational Efficiency

Simulation Type	1000 Iterations	10000 Iterations
Bernoulli Chaos	~0.5s	~2.5s
Fibonacci Chaos	~0.3s	~1.8s
Nash Equilibrium	~1.2s	~8.5s
Unified Chaos	~2.0s	~12s

Benchmarked on Wolfram Mathematica 13.0, Intel i7-11800H, 16GB RAM

Chaos Quality Metrics

ChaosSim generates high-quality chaotic sequences with:

• Lyapunov exponents: 0.3 - 0.8 (positive, indicating chaos)
• Shannon entropy: 3.5 - 4.8 bits (high unpredictability)
• Correlation dimension: 1.5 - 2.8 (fractal properties)

Limitations

1. Computational Intensity: Large-scale simulations (>50,000 iterations) may require significant computational resources
2. Deterministic Chaos: While unpredictable, the system is deterministic - same initial conditions yield same results
3. Approximations: Bernoulli numbers use finite precision arithmetic
4. Game Theory Constraints: Nash equilibrium finder currently supports pure strategies in finite games
5. Platform Dependency: Requires Wolfram Mathematica (proprietary software)

Ethical Considerations

Responsible Use

• Financial Applications: ChaosSim should not be used as the sole basis for investment decisions
• Research Integrity: Results should be validated against established chaos theory literature
• Educational Context: Clearly distinguish between theoretical models and real-world predictions
• Reproducibility: Document random seeds and parameters for reproducible research

Potential Risks

• Misinterpretation: Chaotic patterns may appear to have predictive power but are fundamentally uncertain
• Over-reliance: Users should not depend solely on chaotic models for critical decisions
• Complexity Bias: Complex visualizations may create false confidence in understanding

Training Details

Development Process

ChaosSim was developed using:

• Classical chaos theory principles from Lorenz, Mandelbrot, and Poincaré
• Game theory foundations from Nash and von Neumann
• Numerical methods validated against peer-reviewed literature
• Extensive testing against known chaotic systems (Lorenz attractor, logistic map)

Validation

The framework has been validated by:

• Comparing Lyapunov exponents with theoretical predictions
• Verifying Nash equilibria against manual calculations
• Testing Fibonacci convergence to golden ratio
• Cross-validation with established chaos simulation tools

Environmental Impact

ChaosSim is computationally efficient and designed for local execution, minimizing cloud computing environmental costs. Typical simulations consume minimal energy (< 0.1 kWh per 1000 runs).

Citation

@software{chaossim2025,
  title = {ChaosSim: Advanced Chaos Simulation Framework},
  author = {Kamal, Andrew Magdy and {Riemann Computing Inc.} and {Openpeer AI}},
  year = {2025},
  month = {11},
  version = {1.0},
  url = {http://huggingface.co/OpenPeerAI/ChaosSim},
  license = {MIT}
}

Additional Resources

Documentation

• README.md - Quick start guide and overview
• Examples.nb - 10 practical examples with explanations
• Visualizations.nb - Visualization function reference

Related Literature

1. Lorenz, E. N. (1963). "Deterministic Nonperiodic Flow"
2. Mandelbrot, B. B. (1982). "The Fractal Geometry of Nature"
3. Nash, J. F. (1950). "Equilibrium Points in N-Person Games"
4. Strogatz, S. H. (2015). "Nonlinear Dynamics and Chaos"

License

MIT License

Acknowledgments

Special thanks to:

• The Wolfram Research team for the exceptional Wolfram Language
• Game theory pioneers Nash, von Neumann, and Morgenstern
• Open source mathematics community

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Version: 1.0.0
Release Date: November 25, 2025
Maintainers: Andrew Magdy Kamal, Riemann Computing Inc., Openpeer AI

For questions, feedback, or collaboration inquiries, please open a discussion post on Huggingface or contact the authors.