with the creation of mathematical models describing change (“dynamics”) in parts of the world (“systems”) as time progresses. For our purposes, a few of the methods of dynamical systems theory (DyST) are particularly worth flagging.
First, it’s important to note that DyST takes change as its primary object of interest. This might seem obvious given the name of the field, but it is vital that we appreciate the degree to which this assumption colors the DyST approach to scientific model-building. Rather than focusing on particular instantaneous states of systems—say, the position and momentum of each particle in a box of gas, or particular weather-states (the like of which were the focus of the qualitative approach to weather forecasting discussed in Chapter Four)—DyST focuses on ensembles of states that describe a system over some time period, not just at a single instant. The central mathematical tool of DyST is an equation that describes how different physical quantities of a system (e.g. force, mass, and velocity in Newtonian physics; populations of predator animals and prey animals in ecology; presence and concentration of certain atmospheric chemicals and global temperature in climatology) vary in relation to one another over time. That is, DyST is concerned with modeling how physical quantities differ with respect to one another at different times in a system’s lifetime—in most systems, this is accomplished through the use of differential equations, which describe how variables change in response to one another. The familiar Newtonian equation of motion (F = ma) is a simple differential equation, as it relates the
- Strictly speaking, differential equations are only applicable to systems in which the values in question can be modeled as varying continuously. In discrete-time systems, a separate (but related) mathematical tool called a difference equation must be used. For our purposes here, this distinction is not terribly important, and I will restrict the rest of the discussion to cases where continuous variation of quantities is present, and thus where differential equations are the appropriate tool.