Sequential horseshoe formation in the birth and death of chaotic attractors

In periodically driven systems, the intersection of stable and unstable manifolds of saddle orbits forms two distinct (topological) horseshoes: The first horeshoe is associated with the destruction of a chaotic attractor, while the second horseshow creates a new chaotic attractor. A laser model is used to illustrate how sequential horseshoe formation controls the birth and death of chaotic attractors.