CONTINUOUS CHANGE AND SUDDEN JUMPS

Chapter 4 dealt with systems that followed one of two courses. They either moved smoothly toward a point of equilibrium or careened off toward infinity. They also had the nice property of being predictable. Most of the time we can develop an analytic solution in order to see where a system is going. When analytic solutions are not possible we can, thanks to modern computers, work out enough specific numerical examples so that we have a good idea of how the system behaves.

Another characteristic is less obvious, but in practical situations may be just as important. Every system has a starting state and some parameters that control the process of transition from one state to another. When we are dealing with an actual phenomenon, like the spread of rumors or exchanges of insults, we never know for sure what the starting state and the parameters are, and so we use statistical methods to approximate them. Statistical methods work because it is possible to approximate the behavior of the real-world linear system by a tractable mathematical system that has the same functions and almost, but not quite, the same starting state and parameters. In order to apply the Gottman-Murray equations to a particular pair of romantic partners you do not need to know the exact starting state of their relationship, nor do you need to know the exact values of the influence parameters.