Structural Chaos

A dynamical system is called “chaotic” if small changes to its initial conditions can create large changes in its behavior. By analogy, call a dynamical system “structurally chaotic” if small changes to the equations describing the evolution of the system produce large changes in its behavior. Although there are many definitions of “chaos”, there are few mathematically precise candidate definitions of “structural chaos.” I propose a definition, and I explain two new theorems that show that a set of models is structurally chaotic if contains a chaotic function. I conclude by discussing the relationship between structural chaos and structural stability. Suppose a scientist wishes to predict the behavior of a dynamical system, such as the evolution of an ecosystem, the motion of a pendulum, or the spread of an epidemic. To do so, the scientist might estimate the current state of the system (e.g., the number of predators in an ecosystem), develop a mathematical model of how the system evolves (e.g., equations describing how the number of predators changes over time), and use her model to predict the future given the estimated current state. Thus, there are at least two potential sources of predictive inaccuracy. First, predictions may be inaccurate because the scientist mismeasures or misestimates the system’s initial conditions. Call this initial conditions error (ice). Alternatively, error may arise from an inaccurate model of how the system changes over time. Call this structural model error (sme).1 Frigg et al. [2014] argue that the distinction between sme and ice is crucial for both scientific practice and policy-making. They claim that, although there are methods that generate accurate predictions in the presence of both ice and chaos, there are no known methods for doing the same with For a discussion of other sources of error in modeling, see Bradley [2012].