Introduction to the Modeling and Analysis of Complex Systems

102 CHAPTER 6. CONTINUOUS-TIME MODELS I: MODELINGConvert the following differential equation into an autonomous, first-Exercise 6.2order form.d 2 x− a cos(bt) = 0 (6.15)dt2 For your information, the following facts are also applicable to differential equations,as well as to difference equations:Linear dynamical systems can show only exponential growth/decay, periodic oscillation,stationary states (no change), or their hybrids (e.g., exponentially growing oscillation)a .a Sometimes they can also show behaviors that are represented by polynomials (or products of polynomialsand exponentials) of time. This occurs when their coefficient matrices are non-diagonalizable.Linear equations are always analytically solvable, while nonlinear equations don’thave analytical solutions in general.6.3 Connecting Continuous-Time Models with Discrete-Time ModelsContinuous-time models and discrete-time models are different mathematical models withdifferent mathematical properties. But it is still possible to develop a “similar” continuoustimemodel from a discrete-time model, and vice versa. Here we discuss how you canjump across the border back and forth between the two time treatments.Assume you already have an autonomous first-order discrete-time modelx t = F (x t−1 ), (6.16)and you want to develop a continuous-time model analogous to it. You set up the following“container” differential equationdxdt= G(x), (6.17)and try to find out how F and G are related to each other.