Introduction to the Modeling and Analysis of Complex Systems

Velocity v3.2. PHASE SPACE 313.2 Phase SpaceBehaviors of a dynamical system can be studied by using the concept of a phase space,which is informally defined as follows:A phase space of a dynamical system is a theoretical space where every state of thesystem is mapped to a unique spatial location.The number of state variables needed to uniquely specify the system’s state is called thedegrees of freedom in the system. You can build a phase space of a system by havingan axis for each degree of freedom, i.e., by taking each state variable as one of theorthogonal axes. Therefore, the degrees of freedom of a system equal the dimensionsof its phase space. For example, describing the behavior of a ball thrown upward in africtionless vertical tube can be specified with two scalar variables, the ball’s position andvelocity, at least until it hits the bottom again. You can thus create its phase space in twodimensions, as shown in Fig. 3.1.Velocity vPosition xPosition xFigure 3.1: A ball thrown upward in a vertical tube (left) and a schematic illustration ofits phase space (right). The dynamic behavior of the ball can be visualized as a statictrajectory in the phase space (red arrows).One of the benefits of drawing a phase space is that it allows you to visually representthe dynamically changing behavior of a system as a static trajectory in it. This providesa lot of intuitive, geometrical insight into the system’s dynamics, which would be hard toinfer if you were just looking at algebraic equations.