Dynamical systems theory is a branch of mathematics that studies how systems evolve over time. It focuses on understanding the behavior of complex systems, which can be described by equations that represent their dynamics. These systems can be anything from physical phenomena, like weather patterns, to abstract concepts, such as economic models. In this theory, systems are often classified as either linear or nonlinear, depending on how their components interact. Tools from calculus, geometry, and algebra are used to analyze stability, periodicity, and chaos within these systems, helping researchers predict future states and understand underlying patterns.