Abstract

This study goes into the mathematical principles of chaos theory, focusing on its application to complex systems characterized by unpredictability and sensitivity to initial conditions. The literature review helped to raise key sub-research questions around nonlinearity, feedback loops, strange attractors, fractals, Lyapunov exponents, and the broader implications of chaos mathematics. The research applies a quantitative methodology, with statistical analysis, simulations, and mathematical modeling to study the role of chaos theory in real-world systems across fields like physics, biology, engineering, and economics. The results confirm several hypotheses: nonlinearity, feedback loops, and strange attractors are significant, but the practical utility of fractals and Lyapunov exponents in system analysis is also underlined. The findings confirm that chaos mathematics provides transformational insights-from weather forecasting to cryptography-that have a potential application.