Introduction to the Modeling and Analysis of Complex Systems

4.5. BUILDING YOUR OWN MODEL EQUATION 51Exercise 4.7 Simulate the above two-variable system using several different coefficientsin the equations and see what kind of behaviors can arise.Exercise 4.8 Simulate the behavior of the following Fibonacci sequence. Youfirst need to convert it into a two-variable first-order difference equation, and thenimplement a simulation code for it.x t = x t−1 + x t−2 , x 0 = 1, x 1 = 1 (4.18)If you play with this simulation model for various coefficient values, you will soon noticethat there are only certain kinds of behaviors possible in this system. Sometimesthe curves show exponential growth or decay, or sometimes they show more smooth oscillatorybehaviors. These two behaviors are often combined to show an exponentiallygrowing oscillation, etc. But that’s about it. You don’t see any more complex behaviorscoming out of this model. This is because the system is linear, i.e., the model equationis composed of a simple linear sum of first-order terms of state variables. So here is animportant fact you should keep in mind: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.In other words, these behaviors are signatures of linear systems. If you observe suchbehavior in nature, you may be able to assume that the underlying rules that producedthe behavior could be linear.4.5 Building Your Own Model EquationNow that you know how to simulate the dynamics of difference equations, you may wantto try building your own model equation and test its behaviors. Then a question arises:How do you build your own model equation?