Iterating a formula, it is possible that the results walk to a fixed value. Raising a parameter, the behavior changes to two periodically repeating values. With a more raised parameter you can see another bifurcation into four different values, that change to the next at every new iteration. A little bit later, there is one more bifurcation, etc. Beginning with a critical parameter value no more order can be seen; the output values are chaotical and the next iteration result depends on minimal variations behind the decimal point in the input.

→ B = ?, ↑ V = ?

Formula: $x_(n+1) = B*x_n*(1 - x_n)$

Move the zoom rectangle, resize it with the red
point and click **Zoom in**. Use the blue circle to examine the system
parameter **B** (x axis) and the iteration result **V** (y axis) along the graph.
If the graph looks washed out zooming in, raise the number of iterations and hidden dots.
After that refresh the graph (round arrow button right).

**System Parameter B**

Iterator | 1. Bif. |
2. Bif. | 3. Bif. |
Ratio |
---|---|---|---|---|

Verhulst | 3.000 | 3.445 | 3.540 | 4.684 |

Sinus | 2.019 | 2.130 | 2.156 | 4.269 |

Square | 0.749 | 1.251 | 1.370 | 4.218 |

May | 23.554 | 24.226 | 24.380 | 4.364 |

Logarithmus | 1.996 | 2.396 | 2.471 | 4.938 |

Cosinus | 4.174 | 4.272 | 4.294 | 4.445 |

Henon | 0.365 | 0.911 | 1.025 | 4.789 |

Exploring the bifurcation scenery

Every iterator formula in the machine above depends on the system parameter B. In the region of order, iterating the formula leads to a fixed value. For every iterator there are specific values of B, where a bifurcation accurs.
These values change with the equation.

But the values show a quantitative universality for a class of nonlinear transformations,
when you do the following calculation:

Use the blue point in the machine to read the values $b_1$, $b_2$, $b_3$ for three successive bifurcation points and calculate the quotient $(b_2 - b_1)/(b_3 - b_2)$. The result doesn't really depend on the iterator, look at **Ratio** in the table.
The deeper you zoom into the bifurcation graph, the closer you get to the specific limit value

**δ = 4.669201...**, known as **Feigenbaum Constant**.

Chaos & order and... fractals!

Feigenbaum diagrams show regions of chaos and regions of order. In the examples above, the diagrams start with order, where the results are a single number, and with raised values of the parameter B the iteration of the formula bifurcates to two numbers, then to four, eight etc. That means the results change to fixed values every single iteration step. In a chaos region the result is no longer predictable. In the diagram you see a cloud of points as results of the iteration steps. There are small regions of order embedded in the chaos regions. Zooming in there are structures that look like the whole diagram. This is a fractal property. Changing the orientation of the diagram most of the graphs look like a tree.

© 2021 Ulrich Schwebinghaus