A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART I: THRESHOLD OF COMPLEXITY
- 1 December 2002
- journal article
- Published by World Scientific Pub Co Pte Ltd in International Journal of Bifurcation and Chaos
- Vol. 12 (12), 2655-2766
- https://doi.org/10.1142/s0218127402006333
Abstract
This tutorial provides a nonlinear dynamics perspective to Wolfram's monumental work on A New Kind of Science. By mapping a Boolean local Rule, or truth table, onto the point attractors of a specially tailored nonlinear dynamical system, we show how some of Wolfram's empirical observations can be justified on firm ground. The advantage of this new approach for studying Cellular Automata phenomena is that it is based on concepts from nonlinear dynamics and attractors where many fuzzy concepts introduced by Wolfram via brute force observations can be defined and justified via mathematical analysis. The main result of Part I is the introduction of a fundamental concept called linear separability and a complexity index κ for each local Rule which characterizes the intrinsic geometrical structure of an induced "Boolean cube" in three-dimensional Euclidean space. In particular, Wolfram's seductive idea of a "threshold of complexity" is identified with the class of local Rules having a complexity index equal to 2.Keywords
This publication has 8 references indexed in Scilit:
- Cellular Neural Networks and Visual ComputingPublished by Cambridge University Press (CUP) ,2002
- Methods of Qualitative Theory in Nonlinear DynamicsWorld Scientific Series on Nonlinear Science Series A, 2001
- UNIVERSAL CNN CELLSInternational Journal of Bifurcation and Chaos, 1999
- CNN Genes for One-Dimensional Cellular Automata: A Multi-Nested Piecewise-Linear ApproachInternational Journal of Bifurcation and Chaos, 1998
- CNN: A Paradigm for ComplexityWorld Scientific Series on Nonlinear Science Series A, 1998
- Methods of Qualitative Theory in Nonlinear Dynamics - Part IWorld Scientific Series on Nonlinear Science Series A, 1998
- CNN: A Paradigm for ComplexityWorld Scientific Series on Nonlinear Science Series A, 1998
- Autonomous cellular neural networks: a unified paradigm for pattern formation and active wave propagationIEEE Transactions on Circuits and Systems I: Regular Papers, 1995