Counting and Classifying Attractors in High Dimensional Dynamical Systems

✍ Scribed by R.J. Bagley; Leon Glass

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 381 KB
• Volume: 183
• Category: Article
• ISSN: 0022-5193
• DOI: 10.1006/jtbi.1996.0220

✦ Synopsis

Randomly connected Boolean networks have been used as mathematical models of neural, genetic, and immune systems. A key quantity of such networks is the number of basins of attraction in the state space. The number of basins of attraction changes as a function of the size of the network, its connectivity and its transition rules. In discrete networks, a simple count of the number of attractors does not reveal the combinatorial structure of the attractors. These points are illustrated in a reexamination of dynamics in a class of random Boolean networks considered previously by Kauffman. We also consider comparisons between dynamics in discrete networks and continuous analogues. A continuous analogue of a discrete network may have a different number of attractors for many different reasons. Some attractors in discrete networks may be associated with unstable dynamics, and several different attractors in a discrete network may be associated with a single attractor in the continuous case. Special problems in determining attractors in continuous systems arise when there is aperiodic dynamics associated with quasiperiodicity of deterministic chaos.