Introduction to the Modeling and Analysis of Complex Systems

Chapter 6Continuous-Time Models I: Modeling6.1 Continuous-Time Models with Differential EquationsContinuous-time models are written in differential equations. They are probably moremainstream in science and engineering, and studied more extensively, than discrete-timemodels, because various natural phenomena (e.g., motion of objects, flow of electric current)take place smoothly over continuous time.A general mathematical formulation of a first-order continuous-time model is given bythis:dxdt= F (x, t) (6.1)Just like in discrete-time models, x is the state of a system (which may be a scalar orvector variable). The left hand side is the time derivative of x, which is formally defined asdxdt = limδt→0x(t + δt) − x(t). (6.2)δtIntegrating a continuous-time model over t gives a trajectory of the system’s state overtime. While integration could be done algebraically in some cases, computational simulation(= numerical integration) is always possible in general and often used as the primarymeans of studying these models.One fundamental assumption made in continuous-time models is that the trajectoriesof the system’s state are smooth everywhere in the phase space, i.e., the limit in thedefinition above always converges to a well-defined value. Therefore, continuous-timemodels don’t show instantaneous abrupt changes, which could happen in discrete-timemodels.99