Catastrophe theory
From GenerativeArt
[Click here to run the Hysteresis App]
[Click here to run the Catastrophe App]
Contents |
Hysteresis
Hysteresis is a mapping from input to output which is different depending on whether the input is increasing or decreasing. Hysteresis can be use to transform a continuous input into a binary output "debouncing" the output signal at the transition points. In electronics and electronic music a simple gate with hysteresis is called a Schmidt Trigger.
Note that hysteresis cannot be modeled by a mathematically well defined function because it doesn't maps an input into one and only one output. The implication is that computer code implimenting hysteresis will either have to have some form of state memory which saves the previous input, or will have to have two arguments, one for the input, and one for the direction of change.
Elementary Cusp Catastrophes
An elementary cusp catastrophe is one of 7 elementary forms from the classification theorem developed by Rene Thom based on complicated proofs from the mathematical field of topology. The cusp catastrophe is perhaps the most useful and understandable form, as it exists in a low dimensionally, and it models a number of day-to-day phenomena in an intuitive way. The cusp catastrophe can be applied to situations where 2 input parameters result in a single output, and the response seems to "morph" from a smooth output to a discontinuous output which exhibits hysteresis.
Example - Fear and Aggression in Dog Behavior
The following example is from the (longer) draft for E.C. Zeeman's article which popularized Catastrophe Theory in the April 1976 issue of Scientific American. Generative artists seeking to model complex animal behaviors where there are conflicting drives and motivations should find it directly of interest, but keep in mind that Catastrophe Theory can be used to model any situation where multiple inputs generate one or more discontinuous outputs.
Catastrophe Characteristics
The elementary cusp catastrophe shows the following qualitative features. Catastrophes at higher dimensions show similar behavior:
Bimodality
As the inputs are increased the output behavior will tend to split into 2 (or more) modes rather than being clustered around a central value such as in the case of a normal distribution.
Inaccessibility
Intermediate behavior between the bimodal behavior is so instable as to be inaccessible.
Sudden Jumps
At higher input levels behavior will tend to jump from one mode to another.
Hysteresis
This jumping behavior will exhibit hysteresis such that the threshold for jumping from one mode to another will vary depending on past history. The same instantaneous input parameters can at times result in either mode of output behavior being exhibited.
Divergence
Given two nearly identical situations, as the splitting factor increases the two instances may exhibit rapidly increasing differences, and thus a dramatic divergence in their state.
Example - Boom and Bust Economics
Note that in the dog behavior example the two input factors (fear and rage) are congruent with the vertex formed by the catastrophe threshholds. More typically (as shown in the above generalized model) a single input factor represents a splitting factor while a second factor (the normal factor) drives the hysteresis behavior. The two orientations are mathematically equivalent, but for the purposes of generative art quite different in effect. The following example from economics illustrates the more typical splitting/normal factor orientation.
Pragmatics
The divergence aspect of the elementary cusp catastrophe shows a sensitivity to initial conditions which is simpler than, but in some ways similar to, chaos. For use in generative art it is possible to precisely model the cusp catastrophe. However, it may be easier to create a very functional approximation by breaking the bifurcation set into 2 distinct linear functions, and tracking the history (currently active sheet) with a flag which is used to select and switch between the 2 functions.
In situations where group behavior is the issue it might be tempting to centralize the group behavior using a single catastrophe model. But it should also be possible to give each agent their own model, perhaps with slightly differing parameters and environmentally driven inputs, to create more realistic individual behavior contributing to emergent group behavior.
[Click here to run a hysteresis demonstration applet.]
[Click here to run a fight / flight catastrophe model applet.]
