Main topic: Cellular automata

In 2002 Stephen Wolfram, in the book A New Kind of Science, proposed an idea that nature is simplistic and apparent complexity is just impression of the observer. Within the book Wolfram rediscovered so-called simple programs, which were previously known as cellular automata. Those programs, as the nature itself, as proposed by Wolfram, are by definition very simple, but their evolution might be very complex and in some cases chaotic. In this sense cellular automata appear to be very similar to stochastic processes.

But there is one significant difference – cellular automata are most usually deterministic! Those simple programs could have no random noise within them, but they would still be able to evolve chaotically. In this sense cellular automata seem to behave very similarly to dynamical systems exhibiting dynamical chaos. Though let us remind you that cellular automata are by definition simple programs in contrast to complex mathematics behind the dynamical description of systems.

Thus in this section of Physics of Risk website we will demonstrate that chaos-like behavior might be obtained even by using very simple programs – cellular automata.

More models »

# Belousov-Zhabotinsky reaction

Belousov-Zhabotinsky reaction [1] is a chemical reaction, or more precisely a reaction family, known for exhibiting temporal and spatial oscillations.

This reaction is one of the classical examples of the natural non-linear oscillations. Another prominent example is the previously analyzed prey-predator interactions in the ecosystem. Interestingly enough despite being of a very different nature both of these example can be modeled using Lotka-Volterra equations.

In this text we will also consider certai cellular automaton, which replicates the spatial oscillations seen in some of the Belousov-Zhabotinsky reactions. Continue reading “Belousov-Zhabotinsky reaction” »

# Wolfram’s elementary automatons

In mathematics and computation theory there are a class of cellular automatons which are known as elementary automatons. This class of cellular automatons is restricted to the one dimensional grid (in the figures below the second dimension, ordinate (vertical) axis, is time) with cells either on or off. Another important simplification is that the actual state of the cell at given time, $$x_{i,t}$$, depends only on the previous state of the same cell and the previous states of its immediate neighbors, i.e. on $$\{x_{i-1,t-1},x_{i,t-1},x_{i+1,t-1}\}$$. Due to these restrictions and simplifications, generally speaking cellular automatons might evolve in the infinite dimensions, have infinite neighborhoods and have limitless number of possible cell states, these cellular automatons appear to be very simple, though as we show bellow they can replicate very complex and even chaotic behavior. Continue reading “Wolfram’s elementary automatons” »