Discrete mathematics

A pseudo-phase-space is a variation on a phase-space. A pseudo-phase-space allows a 1 dimensional data-series to be converted into a 2D (or higher) representation. It is used in non-linear systems to look for periodicity or stability within a data-series.

For a 1D dimemsional data-series, such as:

s(0), s(1), s(2), s(3), ..., s(n), ...,

A 2D representation of the data can be produced by plotting s(i) against s(i+1).

For example:

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 173 | 184 | 127 | 82 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 0 | 0 | 0 | 20 | 135 | 129 | 92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 103 | 94 | 81 | 0 | 0 | 0 | 0 | 0 | 0 |

3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 34 | 112 | 95 | 0 | 0 | 0 | 0 |

4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 125 | 131 | 0 | 0 |

5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 46 | 199 | 0 |

6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 7 | 209 |

7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 203 |

8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 200 |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 232 |

A | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 | 182 | 0 |

B | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 108 | 84 | 0 | 0 |

C | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 25 | 110 | 98 | 0 | 0 | 0 | 0 |

D | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 100 | 105 | 95 | 0 | 0 | 0 | 0 | 0 | 0 |

E | 0 | 0 | 0 | 32 | 121 | 116 | 122 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

F | 393 | 192 | 152 | 107 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**A pseudo-phase-space of a logistic map { x(n+1) = 3.995 * x(n) *
(1 - x(n)) }**

Here, the first 5000 values of x(n), which varies between 0 and 1, have been assigned
to sixteen intervals. The rows represent values of s(n), the column values
of s(n+1). Each entry in the table shows the frequency of events which turn a
given s(n) to a given s(n+1). This convention is adopted throughout
these webpages.

- Astrophysics
- A pseudo-phase-space of pi
- Graphs of pseudo-phase spaces
- Blind mathematics
- Logistic map

Links at other sites...

Created 27/9/99

Last modified 1/8/00