Chaotic Evolution via Generalized Probabilistic Automata (Probabilistic Arrays) Azaria Paz 1
|
Abstract:
The finite-state probabilistic array was proposed some time ago [Paz68] as a theoretical model of polynomially nonlinear statistical interactions among the members of populations---multiway, multi-machine, or multi-parent interactions---producing state-distribution variabilities from generation to generation in the population, analogous to the variabilities studied in genetics or population biology. These machine-like stochastic systems are revisited here, with attention directed toward new details about asymptotic behavior. It is now possible to obtain more definitive results about the potential of such systems to exhibit profound disequilibrium, asymptotically over time, as the state-distributions evolve. The concepts and techniques used are suggested by chaos theory, aided by exploratory computational visualization of examples. The 2-state case is explored in detail, with brief remarks on systems with larger state sets. The principal results for 2-state systems are as follows: (i) there exist infinite-periodicity (i.e., chaotic) two-state systems of polynomial degree d (modeling d-way interactions), in particular for d as small as 5; (ii) two-state systems with quadratic nonlinearity
Citations
| 4387 | Probabilistic Reasoning in Intelligent Systems – Pearl - 1988 |
| 64 | Simple mathematical models with very complicated dynamics – May - 1976 |
| 48 | Queueing Systems, Vol. 1: Theory – Kleinrock - 1975 |
| 42 | An Introduction to Chaotic Dynamical Systems, 2nd ed – Devaney - 1989 |
| 21 | Universal behavior in nonlinear systems – Feigenbaum - 1983 |
| 19 | On the predictability of coupled automata: An allegory about chaos – Buss, Papadimitriou, et al. - 1990 |
| 2 | Probabilistic arrays – Paz - 1968 |
| 1 | et al , "On Devaney's definition of chaos – Banks - 1992 |
| 1 | Stochastic arrays and chaos – Farhangi - 1992 |
| 1 | A computational view of population genetics (prel. vers – Rabani, Rabinovich, et al. - 1995 |
| 1 | Iteration of maps, strange attractors, and number theory--- an Ulamian potpourri – Stein - 1989 |
| 1 | Yet another definition of chaos – Touhey - 1997 |
