Coexisting Attractors

Compare the registered Lorenz coexistence case: same parameters, two initial conditions, and two stable coexisting destinations.

Case Lorenz, rho = 24.4

Use This When Same parameters, different starts

Question and result

Calculation performed

Compare registered trajectories under one unchanged parameter set and use their different initial conditions to study distinct computed destinations.

GUI tab

Coexistencia

Configuration and calculation

Control sequence

Select a registered coexistence case and review its reference and parameter set. Coexistence simulations read that registered parameter_set directly as part of the calculation setup.
Choose one registered attractor or simulate all of them with the same dt and total time.
Confirm that the equations and parameters remain unchanged while only the initial conditions differ.
Use Calcular cuenca para este caso for a direct follow-up: the application switches to Cuenca de atracción, loads the case, and immediately runs the current basin window, grid, step, and total-time configuration.
Use Enviar a Atractor 3D, Enviar a Cuencas, or Enviar a Bifurcación to copy the selected system, parameter values, and selected initial condition into the corresponding workspace.
Review the destination tab's own controls before calculating there; sending a case changes the workspace, and the destination tab retains its numerical setup.

Record

Computed data

One or more trajectories showing the destinations reached from registered starts under the same model parameters.
A destination workspace preloaded with the selected coexistence case for 3D inspection, basin sampling, or a parameter sweep.
When the direct basin button is used, a newly calculated basin for the loaded case using the Cuenca tab's current sampling controls.

Scientific reading

Interpretation criteria

Different computed destinations under identical equations and parameters are finite numerical evidence of coexistence for the tested initial conditions and settings.

The transfer buttons support a consistent follow-up workflow. Interpret each basin map or bifurcation diagram through its own numerical contract alongside the original trajectory comparison.

The direct basin result must be interpreted from its declared plane, grid, and finite integration time, even though it was launched from the coexistence case.

Files

Experiment record

Use Guardar gráfica... to save the coexistence view after all trajectories and initial-condition labels are distinguishable.
Record the case, system, ordered parameters, every tested initial condition, method, dt, total time, and any destination-tab settings used after a transfer.

Applications

Questions addressed by the calculation

Teaching multistability and the role of initial conditions under a fixed parameter set.
Selecting cases for 3D views, basin sampling, or bifurcation follow-up with the shared model inputs transferred directly.
Designing controlled comparisons of candidate destinations and their numerical robustness.

Scope

Evidence conditions

A comparison of a few registered starts documents the declared initial-condition sample. Map the basin and its destinations through a grid-based study with its stated coverage.
Coexistence documents distinct destinations for the tested starts. Identify the class of each destination and verify settling behavior with a sufficiently long transient analysis.
The transfer workflow supports follow-up visualization and sampling; Hidden Attractors FO provides the separate mathematical workflow for hidden-attractor localization and certification.

Current Toolbox Chaos Coexistencia tab showing the registered Lorenz case — Current interface: the registered Lorenz case keeps one parameter set and compares the trajectories from both documented initial conditions.

Animated classification of initial conditions between two stable destinations — Pedagogical coexistence analogy: one dynamical law and one parameter set produce two destination classes when only the initial condition changes. The figure explains basin logic, while the registered Lorenz case provides the reproducible interface experiment.

Initial conditions and coexisting destinations

Coexistence means that the equations and parameters stay fixed, while changing the initial condition can lead to a different long-term behavior. The Lorenz case used by the toolbox uses sigma=10, rho=24.4 and beta=8/3, where the registered starts approach two different stable equilibria.

This is a central idea for multistability. The system combines its equations with the region of phase space where the experiment begins. A multistable study compares a declared collection of initial conditions.

Coexistence comparison procedure

Open the coexistence workflow in the toolbox.
Select the registered Lorenz coexistence case.
Review the registered values sigma = 10, rho = 24.4, and beta = 8/3. The simulation actions read this case parameter set automatically.
Compare the registered initial conditions (5,5,20) and (-5,-5,20).
Simulate both trajectories in the same figure.
Read the final destinations: one trajectory approaches the positive stable fixed point and the other approaches the negative stable fixed point.
For an immediate grid study, first review the window, fixed coordinate, resolution, dt, and total time in Cuenca de atracción. Return to Coexistencia and press Calcular cuenca para este caso; the application loads the case in that tab and starts the basin calculation directly.
Select one registered attractor and use Enviar a Atractor 3D, Enviar a Cuencas, or Enviar a Bifurcación to copy its system, parameters, and initial condition. Review the receiving tab's controls, then run the requested analysis there.

Initial state, horizon, and solver controls

Initial condition: this is the main control in coexistence. Changing only the start can change the destination.
rho: rho=24.4 shows two stable Lorenz equilibria; rho=28 shows the classical chaotic Lorenz attractor and is a different lesson.
Total time: too short can show only transients. The destination is read after the trajectory has settled.
Step size: too large a step can move a trajectory into the wrong destination numerically.
View angle: rotation changes how clearly the symmetric destinations separate in the 3D figure.

Coexistence results and basin sampling

Coexistence and basin maps serve distinct tasks: a coexistence plot compares selected trajectories; a basin map classifies many initial conditions on a grid.
Identify the destination class: the Lorenz example here shows two stable fixed points for the same parameter set.
Keep parameters fixed across runs: this condition preserves the coexistence experiment and isolates the role of initial conditions.