Basins of Attraction

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

Question Where does each start go?

Use This When You need a grid classification

Question and result

Calculation performed

Build a two-dimensional map of initial conditions, compare the computed destination classes, and verify the reading with representative trajectories.

GUI tab

Cuenca de atracción

Configuration and calculation

Control sequence

Select a supported 3D flow and define its parameters, integration method, step, and total time.
Set the x0 and y0 limits, the fixed z0 coordinate, and a moderate Nx by Ny grid for the first calculation.
Keep Superponer equilibrios enabled when equilibrium markers will help orient the map; the Equilibrios panel reports the coordinates used by the view.
Calculate the basin, inspect escape or unresolved regions, and refine the window, time, or resolution only after the first map is interpretable.
Simulate representative starts from different colors before assigning a dynamical meaning to each class.

Record

Computed data

A color-coded grid in which each sampled initial condition is assigned a computed destination class.
When requested, equilibrium markers over the map and selectable equilibrium coordinates in the adjacent information panel.

Scientific reading

Interpretation criteria

The colors are numerical class labels for the selected plane, grid, solver settings, and finite integration time. State-variable values remain available through the trajectory data.

Equilibrium overlays provide geometric reference points. Combine them with representative trajectory checks and convergence tests near a boundary.

Files

Experiment record

Use Guardar gráfica... to save the basin figure after its legend, limits, and equilibrium markers are readable.
Record the system, ordered parameters, method, dt, total time, x0/y0 window, fixed z0, Nx, Ny, and whether equilibrium overlays were shown.

Applications

Questions addressed by the calculation

Studying sensitivity to initial conditions and multistable destination regions.
Selecting representative starts for later trajectory comparisons and teaching basin concepts.
Comparing how a sampled basin changes under controlled parameter or resolution variations.

Scope

Evidence conditions

A finite grid can miss small or intricate regions, and a finite horizon can misclassify slow transients.
The displayed slice documents the selected plane of state space. Establish neighborhood attraction through trajectory tests and refinement around the marker.
Hidden-attractor localization and certification use the separate Hidden Attractors FO mathematical-engine workflow and its dedicated evidence record.

Current Toolbox Chaos Cuenca de atracción tab showing a Lorenz classification — Current interface: each cell is an initial condition on the declared plane, colored by its finite-time destination class.

Bistable basin map with two destination classes and representative initial conditions — Map reading: each color encodes an observed destination for the declared numerical contract. Select points on both sides of the boundary, integrate again, and test whether the classification persists after refining the grid, step, and horizon.

Basin-map geometry and destination classes

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.

Basin-map procedure

Open the attraction-basin workflow in the toolbox.
Select a supported continuous 3D flow such as Lorenz, Chua, or Rossler.
Choose the plane limits with x0 min, x0 max, y0 min, and y0 max.
Set the fixed coordinate, for example z0 = 1 for a Lorenz x-y initial-condition plane.
Use Superponer equilibrios when you want equilibrium markers on the map, and read their coordinates in the Equilibrios panel. Disable the option when the markers would obscure a dense boundary.
Start with a modest grid such as Nx = 120, Ny = 120. Increase resolution only after the region is meaningful.
Run the classification and inspect stable classes, recording divergent or unresolved points as part of the result.
Pick representative initial conditions from different color regions and simulate their trajectories to confirm the basin reading.

Grid, horizon, and classifier controls

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.

Basin maps, coexistence, and trajectories

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.