From Fractals to Financial Chaos: Mathematical Patterns Behind Market Complexity

Abstract

Financial markets are often treated as random systems, yet real-world data shows repeating patterns, long-term dependencies, and scale-invariant behavior. This study aims to understand this complexity using mathematical ideas borrowed from physics. The paper uses fractal geometry and chaos theory to examine market behavior. Fractals explain self-similarity in price movements through measures like fractal dimension and the Hurst exponent. At the same time, nonlinear models such as the logistic map show how simple deterministic systems can produce chaotic outcomes. Concepts like power laws and volatility clustering further highlight how markets deviate from traditional random models. The findings suggest that financial markets behave like complex adaptive systems, where interactions between participants create structured but nonlinear dynamics. This supports the idea that markets are not purely random but influenced by underlying mathematical patterns. Overall, this work offers a unified framework for understanding market complexity using fractals and chaos theory. Future work can focus on testing these ideas with real data and developing computational models for better financial analysis.