Catastrophe Theory

Catastrophe theory refers to a type of behavior among some nonlinear dynamic mathematical models that experience nonlinear dynamics such that sudden or rapid large-magnitude changes in the value of one [Page 95]variable are a consequence of a small change that occurs in the value of a parameter (called a control parameter). In this sense, catastrophe theory can model phenomena that loosely follow a “straw that broke the camel's back” scenario, although catastrophe theory can be very general in its application. The modern understanding of catastrophe theory has its genesis in work by Thom (1975).

Nearly all early work with catastrophe theory employed polynomial functions in the specification of differential equation mathematical models. In part, this was an important consequence of the generality of Thom's (1975) findings. Because all sufficiently smooth functions can be expanded using a Taylor series approximation (which leads us to a polynomial representation of the original model), it is possible to analyze the polynomial representation directly (see Saunders, 1980, p. 20). However, scientists can avoid using one of Thom's canonical polynomial forms by working with their own original theory-rich specifications as long as it is clear that the original specification has catastrophe potential (Brown, 1995a).

Fundamental to catastrophe theory is the idea of a bifurcation. A bifurcation is an event that occurs in the evolution of a dynamic system in which the characteristic behavior of the system is transformed. This occurs when an attractor in the system changes in response to change in the value of a parameter (called a control parameter because its value controls the manifestation of the catastrophe). A catastrophe is one type of bifurcation (as compared with, say, subtle bifurcations or explosive bifurcations). The characteristic behavior of a dynamic system is determined by the behavior of trajectories, which are the values of the variables in a system as they change over time. When trajectories intersect with a bifurcation, they typically assume a radically different type of behavior as compared with what occurred prior to the impact with the bifurcation. Thus, if a trajectory is “hugging” close to an attractor or equilibrium point in a system and then intersects with a bifurcation point, the trajectory may suddenly abandon the previous attractor and “fly” into the neighborhood of a different attractor. The fundamental characteristic of a catastrophe is the sudden disappearance of one attractor and its basin, combined with the dominant emergence of another attractor. Because multidimensional surfaces can also attract (together with attracting points on these surfaces), these gravity centers within dynamical systems are referenced more generally as attracting hyper surfaces, limit sets, or simply attractors.

The following model, which is due to Zeeman (1972), illustrates a simple cusp catastrophe and is used to model the change in muscle fiber length (variable x) in a beating heart. The control parameter A (which in this instance refers to the electrochemical activity that ultimately instructs the heart when to beat) can change in its value continuously, and it is used to move trajectories across an equilibrium hyper surface that has catastrophe potential. The parameter q identifies the overall tension in the system, and f is a scaling parameter. The two differential equations in this system are dx/dt =−f(x3−qx + A) and dA/dt = x−x1. Here, x1 represents the muscle fiber length at systole (the contracted heart equilibrium). Setting the derivative dx/dt = 0, we will find between one and three values for x, depending on the other values of the system. When there are three equilibria for x for a given value of the control parameter A, one of the equilibria is unstable and does not attract any trajectory. The other two equilibria compete for the attention of the surrounding trajectories, and when a trajectory passes a bifurcation point in the system, the trajectory abandons one of these equilibria and quickly repositions itself into the neighborhood (i.e., the basin) of the other equilibrium. This rapid repositioning of the trajectory is the catastrophe.

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