Chaos theory and catastrophe theory are both important in understanding the dynamics of complex systems, like tornadoes, stock markets or brains. When a system is "chaotic", it can go in any of a number of different directions, depending on tiny differences in the starting conditions. So a butteryfly flapping its wings in Brazil today can make the difference in whether or not there is a storm in Mongolia one month later. The gravitational force of a raindrop ten miles away can affect the outcome of a game of billiards. The different end results that depend so much on the initial conditions are called the "attractors" of the system.
Catastrophe theory studies what happens when a system goes from one state to a radically different state without passing through any intermediate states. This sudden, discontinuous jump is called a "catastrophe".
NTT Systems assisted Dr. Martin Taylor in investigating the use of chaotic neural networks to efficiently store memories (perhaps in a manner much like the human brain). The work was done in the C programming language, on top of the Rochester Connectionist Simulator. Our simple model neural networks were designed to be chaotic, but in a controlled manner. The system was placed on a boundary close to many quite different attractors, so that it could be given a "push" in the right direction (perceptual input from the outside world), and fall readily into the right "memory" state.
Many interesting features of the space of these attractors were explored, including the discovery of catastrophes. In a preliminary attempt to get the system to learn, we tried "mating" different neurons in a "genetic algorithm"--a computer modelling of the evolutionary process via sexual reproduction. The goal was to develop genetic algorithms that would grow chaotic memory networks from the ground up. This was part of a larger speech recognition project.
A report is available:
Allan F. Randall (NTT Systems) and M. Martin Taylor (DCIEM). Studies of Self-Organised Dynamic Behaviour in Neural Networks, 1992.
The following diagram from the above paper shows the "phase diagrams" for a space of different memories in a single neural net: