Fig.8: Smooth basin boundary with three attractors where a=1.5.
Since the phase space is 2-dimensional in this case, observation of basin structure is useful for understanding the crisis leading to the intermittency. We demonstrate qualitative changes in basin structure with focusing on metamorphoses [3] or fractalization [15] of basin boundaries. Under a coexistence state of multiple attractors, the phase plane is divided into multiple basins of attraction. Basin boundary is the separatrix of the basins of attraction, which can be smooth or fractal.
(1) Smooth basin boundary
There coexist three attractors including the two off-diagonal attractors and the 2-periodic chaotic attractor over the diagonal as shown in Fig. 8. The basin boundary separating the basins of attraction is given by stable invariant manifolds of saddle cycles [16]. The two regions of the 2-periodic chaotic attractor approaches each other with increase of a, and finally contact at the basin boundary. The boundary crisis of the 2-periodic chaotic attractor leads to a fractalization of the smooth basin boundary.
(2) Fractal basin boundary
After the boundary crisis, the region which was the basin of the destabilized attractor is separated into the basins of the remaining attractors by a fractal basin boundary. The largest component of the basin including an attractor is called an immediate basin. The basin boundary is obtained by cutting out all the preimages of the immediate basins from the phase space. Since a set has multiple preimages in non-invertible maps, the procedure is like how to make Sierpinski's gasket and there remains a set with fractal nature as illustrated in Fig. 9.
The simply connected immediate basin turns into a multiply connected one with further increase of a as shown in Fig. 10. The crisis inducing intermittency occurs due to a contact between the attractors and the fractal basin boundary. A fractal basin boundary typically embeds a non-attracting chaotic set (e.g. chaotic saddle or chaotic repeller). Therefore, the contact can also be viewed as a contact between attracting and non-attracting chaotic sets.