Introduction to the Modeling and Analysis of Complex Systems

38 CHAPTER 4. DISCRETE-TIME MODELS I: MODELING(which is called the Fibonacci sequence) can be converted into a first-order form by introducinga “memory” variable y as follows:y t = x t−1 (4.6)Using this, x t−2 can be rewritten as y t−1 .follows:Therefore the equation can be rewritten asx t = x t−1 + y t−1 (4.7)y t = x t−1 (4.8)This is now first-order. This conversion technique works for third-order or any higherorderequations as well, as long as the historical dependency is finite. Similarly, a nonautonomousequationx t = x t−1 + t (4.9)can be converted into an autonomous form by introducing a “clock” variable z as follows:z t = z t−1 + 1, z 0 = 1 (4.10)This definition guarantees z t−1 = t. Using this, the equation can be rewritten asx t = x t−1 + z t−1 , (4.11)which is now autonomous. These mathematical tricks might look like some kind of cheating,but they really aren’t. The take-home message on this is that autonomous first-orderequations can cover all the dynamics of any non-autonomous, higher-order equations.This gives us confidence that we can safely focus on autonomous first-order equationswithout missing anything fundamental. This is probably why autonomous first-order differenceequations are called by a particular name: iterative maps.Exercise 4.4 Convert the following difference equations into an autonomous,first-order form.1. x t = x t−1 (1 − x t−1 ) sin t2. x t = x t−1 + x t−2 − x t−3Another important thing about dynamical equations is the following distinction betweenlinear and nonlinear systems: