A collection or network of objects or entities is called a system if they are interconnected or can interact to produce outputs or results according to given inputs. Dynamical system is a natural or man-made system that evolves with time. In other words, the instantaneous description of the system, known as the state of the system, changes with respect to time. Control systems are those dynamical systems that can be controlled using suitable inputs, called control variables. So, a control system can be considered a particular type of dynamical system.

The motion of a planet can be described as a dynamical system, while the motion of a vehicle driven by a driver can be described as a control system. Dynamical systems, whether natural or man-made, are present in every facet of our day-to-day life. The solar system, wind motion, epidemics, population dynamics, economy, human metabolism, missiles, and automobiles are some examples of dynamical or control systems.

Mathematical modeling is a procedure for translating a real-world problem into a mathematical form and obtaining information about the problem using the solution of the mathematical model. If a proper procedure is followed in formulating a mathematical model, one can expect reliable information about the real-life problem from the model.

In Section 1.1, general mathematical forms of continuous-time and discrete-time dynamical systems are introduced. Simple mathematical models of population dynamics and prey–predator population models are presented in Section 1.2. Mathematical models of some mechanical systems, namely simple harmonic motion and the simple pendulum, are discussed in Section 1.3. The concentration dynamics of a fluid while mixing a substance into it is discussed in Section 1.4. A brief discussion on mathematical models of decay of mass of a substance and temperature changes in a substance are given in Sections 1.5 and 1.6, respectively. In Section 1.7, the inverted pendulum model is discussed in detail.

1.1 Dynamical Systems

In this section, continuous-time and discrete-time dynamical systems are introduced.