Coexisting Attractors

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

Theory

Coexistence means that the equations and parameters stay fixed, but changing only the initial condition can lead to a different long-term behavior. In the Lorenz case used by the toolbox, the parameter set is not the classical chaotic one with rho=28. It 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 is not described only by its equations; it is also shaped by the region of phase space where the experiment begins. For students, the important lesson is that one simulation is not enough to describe a multistable system.

Steps

1. Open the coexistence workflow in the toolbox.
2. Select the registered Lorenz coexistence case.
3. Load the case parameters so sigma = 10, rho = 24.4, and beta = 8/3 are applied.
4. Compare the registered initial conditions (5,5,20) and (-5,-5,20).
5. Simulate both trajectories in the same figure.
6. Read the final destinations: one trajectory approaches the positive stable fixed point and the other approaches the negative stable fixed point.

How Parameters Change This Figure

• 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.

Common Confusion

• Coexistence is not a basin: a coexistence plot compares selected trajectories; a basin map classifies many initial conditions on a grid.
• Coexistence does not always mean two chaotic attractors: the Lorenz example here shows two stable fixed points for the same parameter set.
• Same parameters matter: if you change parameters between runs, you are no longer testing coexistence.