Introduction to the Modeling and Analysis of Complex Systems

6.3. CONNECTING CONTINUOUS-TIME MODELS WITH DISCRETE-TIME... 103Here, let me introduce a very simple yet useful analogy between continuous- anddiscrete-time models:dxdt ≈ ∆x∆t(6.18)This may look almost tautological. But the left hand side is a ratio between two infinitesimallysmall quantities, while the right hand side is a ratio between two quantities that aresmall yet have definite non-zero sizes. ∆x is the difference between x(t + ∆t) and x(t),and ∆t is the finite time interval between two consecutive discrete time points. Using thisanalogy, you can rewrite Eq. (6.17) as∆x x(t + ∆t) − x(t)= ≈ G(x(t)),∆t ∆t(6.19)x(t + ∆t) ≈ x(t) + G(x(t))∆t. (6.20)By comparing this with Eq. (6.16), we notice the following analogous relationship betweenF and G:F (x) ⇔ x + G(x)∆t (6.21)Or, equivalently:G(x) ⇔ F (x) − x∆t(6.22)For linear systems in particular, F (x) and G(x) are just the product of a coefficientmatrix and a state vector. If F (x) = Ax and G(x) = Bx, then the analogous relationshipsbecomeAx ⇔ x + Bx∆t, i.e., (6.23)A ⇔ I + B∆t, or (6.24)B ⇔ A − I .∆t(6.25)I should emphasize that these analogous relationships between discrete-time andcontinuous-time models do not mean they are mathematically equivalent. They simplymean that the models are constructed according to similar assumptions and thus theymay have similar properties. In fact, analogous models often share many identical mathematicalproperties, yet there are certain fundamental differences between them. Forexample, one- or two-dimensional discrete-time iterative maps can show chaotic behaviors,but their continuous-time counterparts never show chaos. We will discuss this issuein more detail later.