Introduction to the Modeling and Analysis of Complex Systems

Chapter 5Discrete-Time Models II: Analysis5.1 Finding Equilibrium PointsWhen you analyze an autonomous, first-order discrete-time dynamical system (a.k.a. iterativemap)x t = F (x t−1 ), (5.1)one of the first things you should do is to find its equilibrium points (also called fixedpoints or steady states), i.e., states where the system can stay unchanged over time.Equilibrium points are important for both theoretical and practical reasons. Theoretically,they are key points in the system’s phase space, which serve as meaningful referenceswhen we understand the structure of the phase space. And practically, there are manysituations where we want to sustain the system at a certain state that is desirable for us.In such cases, it is quite important to know whether the desired state is an equilibriumpoint, and if it is, whether it is stable or unstable.To find equilibrium points of a system, you can substitute all the x’s in the equationwith a constant x eq (either scalar or vector) to obtainx eq = F (x eq ), (5.2)and then solve this equation with regard to x eq . If you have more than one state variable,you should do the same for all of them. For example, here is how you can find theequilibrium points of the logistic growth model:(x t = x t−1 + rx t−1 1 − x )t−1K61(5.3)