Basins of Attraction

Compute and read grid-based basin maps that classify initial conditions by their final destination.

What a Basin Means

A basin of attraction partitions initial conditions by their final behavior. In a two-dimensional basin plot, the horizontal and vertical axes are two initial coordinates. The third coordinate is held fixed. The solver launches one trajectory from each grid point, integrates it, and classifies the destination after the retained integration window.

The important reading habit is to separate the map from the trajectories. The map tells you which destination belongs to many starts. The representative trajectories show why the colors are meaningful: starts chosen from different regions fall into different final zones.

Steps

1. Open the attraction-basin workflow in the toolbox.
2. Select a supported continuous 3D flow such as Lorenz, Chua, or Rossler.
3. Choose the plane limits with x0 min, x0 max, y0 min, and y0 max.
4. Set the fixed coordinate, for example z0 = 1 for a Lorenz x-y initial-condition plane.
5. Start with a modest grid such as Nx = 120, Ny = 120. Increase resolution only after the region is meaningful.
6. Run the classification and inspect whether the classes are stable rather than dominated by divergence or unresolved points.
7. Pick representative initial conditions from different color regions and simulate their trajectories to confirm the basin reading.

How Parameters Change This Figure

• Grid resolution: higher Nx and Ny reveal finer basin boundaries but increase computation time.
• Integration time: too short can misclassify slow convergence; too long can make high-resolution grids expensive.
• Fixed coordinate: changing the fixed coordinate changes the plane being sampled, so the basin shape can change.
• Plane window: a wide region gives context; a narrow region is better after you know where the boundary lives.
• Classifier thresholds: hit radii, escape radii, and unresolved labels control how final destinations are assigned.

Common Confusion

• Basin map: many initial conditions are classified over a grid.
• Coexistence plot: a small set of selected trajectories is compared under fixed parameters.
• Attractor plot: the long-term geometry of one trajectory is shown in phase space.