Orbit diagram
The orbit diagram iterates a map at many values of one parameter and draws, above each value, the iterates left after a transient (Strogatz §10.2, Figure 10.2.7). It shows the period-doubling route to chaos at a glance. Choose Orbit in the View panel. It is offered for a map with a parameter.
Example project: orbit-diagram.sepx, the logistic map for r from 2.8 to 4, with 1,000 transient iterates and 200 drawn in each column.
Read the plot
Each column of pixels is one value of the parameter. Every column starts at the first start point, runs the transient, and then plots the next iterates. A column is more opaque where the orbit returns more often. A stable fixed point shows as one curve, a 2-cycle as two, and chaos as a filled band. The vertical line marks the parameter's current value.
Click a column to set the parameter to that column's value. The other views and the parameter's slider follow.
Read the period doublings
For a one-dimensional map, the Orbit diagram panel lists the period doublings it finds in the columns shown. Each is where the cycle of period 2ⁿ⁻¹ loses stability and the cycle of twice the period is born, as its multiplier passes −1. The panel refines each to 10⁻¹². For the logistic map:
| Separatrix | Exact or published | |
|---|---|---|
| r₁ | 3.000000 | 3 |
| r₂ | 3.449490 | 1 + √6 |
| r₃ | 3.544090 | 3.544090360 |
| r₄ | 3.564407 | 3.564407266 |
The chaos gate checks these to 10⁻⁶. The ratio (r₃ − r₂)/(r₄ − r₃) = 4.656 already approaches Feigenbaum's δ = 4.669.
Set the diagram
| Field | Meaning |
|---|---|
| From, To | The parameter's range across the plot. |
| Transient N₀ | Iterates run before plotting, up to 100,000. |
| Drawn N | Iterates plotted in each column, from 1 to 976. |
Zoom into a window, such as r from 3.82 to 3.86, to see the period-3 window open with a saddle-node bifurcation (Strogatz §10.4).
Export
The LaTeX figure draws the diagram with pgfplots, as up to 5,000 dots. Above that, Save… writes a density image and Copy LaTeX draws fewer dots. CSV writes one row per kept iterate. See Export figures and data.