Introduction to the Modeling and Analysis of Complex Systems

36 CHAPTER 4. DISCRETE-TIME MODELS I: MODELINGthis notation makes it easier to see how many previous steps are needed to calculate thenext step (e.g., if the right hand side contains x t−1 and x t−2 , that means you will need toknow the system’s state in previous two steps to determine its next state).From a difference equation, you can produce a series of values of the state variable xover time, starting with initial condition x 0 :{x 0 , x 1 , x 2 , x 3 , . . .} (4.4)This is called time series. In this case, it is a prediction made using the difference equationmodel, but in other contexts, time series also means sequential values obtained byempirical observation of real-world systems as well.Here is a very simple example of a discrete-time, discrete-state dynamical system.The system is made of two interacting components: A and B. Each component takes oneof two possible states: Blue or red. Their behaviors are determined by the following rules:• A tries to stay the same color as B.• B tries to be the opposite color of A.These rules are applied to their states simultaneously in discrete time steps.Exercise 4.1 Write the state transition functions F A (s A , s B ) and F B (s A , s B ) for thissystem, where s A and s B are the states of A and B, respectively.Exercise 4.2 Produce a time series of (s A , s B ) starting with an initial conditionwith both components in blue, using the model you created. What kind of behaviorwill arise?4.2 Classifications of Model EquationsThere are some technical terminologies I need to introduce before moving on to furtherdiscussions:Linear system A dynamical equation whose rules involve just a linear combinationof state variables (a constant times a variable, a constant, or their sum).