Fixed-point theory, linear and nonlinear stability, bifurcation analysis, and state-space reconstruction lay the groundwork for discrete maps and continuous flows that exhibit sensitive dependence on initial conditions. Lyapunov exponents, entropy rates, strange attractors, and control algorithms are computed, while data-driven approaches—Koopman mode decomposition and reservoir computing—demonstrate modern forecasting of chaotic time series in finance, climate, and biomechanics.