A dynamical system is a mechanical, electrical, chemical, or biological system that evolves in time.Dynamical systems theory provides one of the most powerful and pervasive applications of matrix methods in science and engineering.These qualitative and quantitative tools and methods allow for the determination and characterization of the number and types of solutions, including their stability, of complex, often nonlinear, systems.These methods include phase-plane analysis, bifurcation diagrams, stability theory, Poincare diagrams illustrated using linear and nonlinear physical examples, including the Duffing equation and the Saltzman-Lorenz model.