Graph theory

Pseudo-phase spaces

A pseudo-phase space of a discrete number stream is in similar to the adjacency matrix of a graph. This similarity is based on two factors. First, taking the digits of the number stream as the vertices of the graph. Secondly, taking the digit changes (d(n),d(n+1)) as edges connecting the vertices. The main difference is that the pseudo-phase space makes the number of connections between each vertice pair explicit, whereas an adjacency matrix usually records only the presence or absence of an edge.

**Example**

This is a graph of the same equation that I used for the pseudo-phase space example (the logistic map for lambda = 3.995). The main difference is that this example groups the x values into 50 intervals rather than the 16 used before. The vertices are the 50 points grouped in a circle. The edges are the lines connecting the points.

**Conjectures**
This similarity may be useful in cases were the adjacency matrix is
sparse.

- It may reduce the memory needed to analyse a chaotic system.
- It may allow some higher order pseudo-phase spaces (i.e. spaces such as d(n),d(n+1) against d(n+1),d(n+2) or d(n),d(n+1),d(n+2) against d(n+1),d(n+2),d(n+3)) to be generated with reduced memory requirements. (Perhaps it will also allow other representation methods for these higher order spaces.)
- It may allow mixed order pseudo-phase spaces to be generated. (i.e. enable us to extract repeated streams of unequal length from a number stream)

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Created 13/10/99

Last modified 13/12/99