By Michael Pohst
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Additional info for Algorithmic Methods in Algebra and Number Theory
We have a local bifurcation when the qualitative changes in the orbit structure can be analysed in a neighbourhood of a fixed or a periodic point of a map, or a fixed point of a flow. ) Global bifurcations are characterised instead by changes in the orbit structure of the system which are not accompanied by changes in the properties of its flxed or periodic points. Local/global bifurcations occur when a local bifurcation has also global repercussions that qualitatively change the orbit structure far from the fixed point.
1 -> 1 ^ 1 ^0 specifying what state the cell should take in the next period for each of the eight possible states of its neighborhood (including itself). Wolfram proposes to assign to each rule the number which corresponds to the binary digits of the assignment, so the rule described would be 30. This system can be generalized to any number of states and any range. We can visualize the evolution of a 1-dimensional cellular automaton by graphing its states (using colors) across a number of pixels, and then moving down as time moves forward.
They have played an important role in the development of economics and social science in recent years as a device for simplifying and disambiguating claims about the properties of interactive systems. Cellular automata are a particularly simple class of automata that live on a graph or network. ) At least in the simplest cases, all the nodes of the lattice are identical "cells" which can take on a finite range of "states". To begin with I will consider cellular automata that live on a one-dimensional lattice (the integer numbers, or the integer numbers modulo some number n, which makes the integer number line effectively into a circle by making cell n the immediate lefthand neighbor of cell 1).
Algorithmic Methods in Algebra and Number Theory by Michael Pohst