Introduction to the Modeling and Analysis of Complex Systems

4.3. SIMULATING DISCRETE-TIME MODELS WITH ONE VARIABLE 39Linear equations are always analytically solvable, while nonlinear equations don’thave analytical solutions in general.Here, an analytical solution means a solution written in the form of x t = f(t) withoutusing state variables on the right hand side. This kind of solution is also called a closedformsolution because the right hand side is “closed,” i.e., it only needs t and doesn’t needx. Obtaining a closed-form solution is helpful because it gives you a way to calculate (i.e.,predict) the system’s state directly from t at any point in time in the future, without actuallysimulating the whole history of its behavior. Unfortunately this is not possible for nonlinearsystems in most cases.4.3 Simulating Discrete-Time Models with One VariableNow is the time to do our very first exercise of computer simulation of discrete-time modelsin Python. Let’s begin with this very simple linear difference equation model of a scalarvariable x:x t = ax t−1 (4.12)Here, a is a model parameter that specifies the ratio between the current state and thenext state. Our objective is to find out what kind of behavior this model will show throughcomputer simulation.When you want to conduct a computer simulation of any sort, there are at least threeessential things you will need to program, as follows:Three essential components of computer simulationInitialize. You will need to set up the initial values for all the state variables of thesystem.Observe. You will need to define how you are going to monitor the state of the system.This could be done by just printing out some variables, collecting measurementsin a list structure, or visualizing the state of the system.Update. You will need to define how you are going to update the values of thosestate variables in every time step. This part will be defined as a function, and itwill be executed repeatedly.