When modeling dynamical systems, which mathematical tools are commonly employed?

When modeling dynamical systems, differential equations and state-space models are essential tools because they provide a framework for describing how a system evolves over time. Differential equations allow for the modeling of the relationships between a system's variables and their rates of change. For example, in mechanical systems, Newton's second law can be expressed as a second-order differential equation that relates forces acting on a system to the acceleration of the system.

State-space models represent dynamical systems in a way that is particularly useful for control systems engineering. They express the system using a set of first-order differential equations, which are compact and can be manipulated in various ways that facilitate analysis and design. State-space representation is powerful because it encapsulates the entire system behavior in a way that can easily incorporate multiple inputs and outputs, making it a versatile tool for both analysis and controller design.

In contrast, algebraic expressions alone are not sufficient for capturing the dynamics, as they do not account for time variance and how system responses evolve. Geometric shapes may sometimes be used for visualization or to represent static properties of systems, but they do not directly model dynamic behavior. Statistical models can be useful for data-driven approaches or predictions based on observed behavior, but they generally do not provide the foundational understanding of the